49

Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

  • Upload
    others

  • View
    1

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

����������� �������� ������������������� ��!#"$��%&�' �

()�*,+.-&/103254'-�+768-&9;:&<>=?+A@2B+DC=?E>=F9GIHKJMLBN'OQPSRBJMTUHVP;WYXJMZ�[]\?OVX^ T_TUH;X\?`aWbJcTdWbZbJ�XJcefXg WbJVWUW�` ^ RYJMOQO hUJ

i�j>kFlSm]npocqr k,sMtvuxwcsMy{z| o}o~t�zsM��npjboAn~�f�| y�o~t�zsM�~�KsMt�ndscn | n~n}u�� �'�?�lS��j>m&tbl�z| o#wMj>k]n | �Anpjb�~j�n� �����D��� ��t>jbm�wMj>t>jbo~ocqr k��5j_s�npj>�A��zjbo~tbjbnpjbo�oAkcscy�zsM�AlSo�oAt | �At{scn | n{�&wcsM�M�?l�o�s���� �7 ]¡�¢{£{£>£>¢p  � � ¢¤ � �7¥�¡U¢>£{£{£>¢~¥ � ��wDj>k]n | � | k¦o~kVsMy�zsc�~lSo�o~t | �~t{sVnds§ySjb�M�Djbm©¨.� ¢ ¤«ª �$¬ �­¯® ¡   ­ ¥ ­ �I°²±.jbmFn~lfo~kVsMy{zsM�~l�ooAt | �At{scnAndsMy�jbySy�zsVn~wcs�u ��jb�M��j r kFySl��&j>oAt>l�wDjbk]n | ��npj>�Ajbn�k�zjb³BjbtM�����M�,�´wMj>k]n | ��µ | oAo~t�zsMm � ��zsMo�o~t�z| wcsMym | �~��zsU¶&zsMm'��s¸·d��·}�$¹ ¨.� ¢ � ª o~t�zsM� | n«zj>�An.¶cqr k'�°�tvu � wDjbk]n | ��n�zj>��npjb�~��zj>oAt>jbn~j>o���zsMt>l�ops�ySjb�M�Djbm = ¡U¢>£{£{£>¢ = � £ ��o~tbj>�~l�m]nºn~jbn~o~t�| ySj>�Mj>o�u � �?�5j>y�l��� �7 ]¡�¢{£{£>£>¢p  � ��wMj>k]n | �~�pscy�±.jbm&m�zsMy�yY���»¬ �­¯® ¡   ­ = ­ �

¼½o~��j>�Ajbn~j>o>�1µ | �M�¾sMt�u � �Fn�npjbk?l�m]npµ&j�n7¶cqr k¿zr �M�ÀlSo>����lSm]n_sMt�u � wDj>k]n | �An�zj>�I±.jbySj�n~npl�sVÁKm�npjb�~j�n{���t�zsMy�npsMy sMtKu � j r kFySl��&j>oAt>l�n�zj>�Aµ&j>t�¶ r n r mk5�fsM�Kjby��?m&jbkÂjbySjb�Kjbl � wcsM�M�Â�©zscoIoAt�z| wcsMy ³ | mFn.¶~sMlÃ����zsM�jb�M�DjbmÄzj>��n�zjbkf�r jbk'� °�tÄu���j r kFySl��&j>oAt>lFn�zj>�A�BjbmÅs / ¢�Æ�Ç uÄ�I³ | mFn | kvn'zscw | ySo&zsM��zscn;s©È � / ¢dÆ ���É· /�Ê Æ ·oAt�zsM�Ësc�V¶~s��Kjb�&�

°�t�u � j r kFySlS�j>o~tblfn�zjb�vlSt | �Kj�np�~l{zsU¶&zsMmÌj>�M� | y��]scmÂÍ�Î'u ��Ï u � �&lжAjbk]n�zÑ w¦ySjbk�zjb³Bjbt5zjboAnÂzj>�AnMqr m&k'�sc�Kjby��ºn'zscw | ySo&zsM�Mndsc�An'z| ��°8k | �&zsM�&�&lDnpsMm r yS��zsMm]� | kF��z| yD�©zsc�YlSoA�Kjb�An>��µ | �M��npjbn~o~t��| ySjb�Dj>o1ÍÒlSt | �Kj�np�~l{zsMµ | tjb�M��zj>��npj>y��Ò�r jbmÀndsM��n | tblSk«j>�c� | yÓ�]sMmÂÔ§Î'u �¦Ï u � | �An | � | m�zsMy�lSo�ySl�m&j�zsc�~lSoºySjbk�zj>³5j>t5zjbo�zjbo�j>�c� | yÓ�]sMmu � �?�BjbySlY�ÕwDjbkFn | �>�'��j>yÓ�Dj>kFkcj>y�±.j>m&m}zsMySyÍ � / ���ÖÔ � / ��×Ò� ��Ø � Ø �

�}zsM�~��j>yÓ� / Ç u � j>o~j�n�zjbm��°ÙÔ | �An | � | m�zscySlSo�ySlSmj�zsM�AlSo�ySjbk�zjb³Bjbt5zjboAn�sctKu � jb�M�ÂÚ ¡d¢{£{£{£b¢ Ú � | ��n | m | �~��zsMy�n��}zsMt>l�o�zsM�~s�mÄzjbtbwMjzÑ �.¶~s�ySj_sMt * �'�?jb�&�~jbm&� �r k]wcsM�&�~scnpl�k r o���zscn~�~lÓÛB�ÄÜÌl�mFn�sMt_kBq| tbn r � | nAn{�5sMt * ��zscn~�~lÓÛ *KÝ n~�psMm&oAt�Þ

³ | m�zscy�n7¶&zsVwMscy#±.j>m&m}zsMySy�sMt *KÝB* �àßÂq| o~oAt>j�±Uqr �D�1zj>ob�}sMµ | y}ß«sMt«�©áÒ�'�?jboajb�M�?o?zj>�M�©zscn~�~l¯Û | n�¶Aj>y½q| ySl3�°��Kjbm&m]�?lS�5j>m ÈFâ>ã * � Ø n~j>yжAjbocqr yä��sMkFk | �ÂsåÍ lSt | ��jbnp�Al�zscnÌlS�&zsMm]�BzÑ n'zscoAndsc�An'z| m�sMkå� | m&�V¶ r k'�ÈFâ>ã * � Ê Ø ±.j>m&m}zsMySy{zsMops�jbo~jbn�zj>m¦sKÍælSt | �Kj�np�~l{zscn�l���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| m&sMkKmjbwDjbt>t�qr k5�GIHK\?O�WYXJMZbT ^ Z ^ Hç\]W_N ^�è [JcZ>é?J è5ê P�X\�h�X\ è \?N²XJMZ�WbJMOQ`ÌJcH1XJMTUJ

° / Ç uÄ�¦³ | mFn�ë � ëKìxí]��o r �]sc�Yzr �'q| ���&kBq| �~m]�Mj>t>j�n�zjbm8sKî�ï � / �}�Éð ÆIÇ u �´ñ?È � / ¢vÆ ��òåë#ó³ | m]npµ&sMyS�KsMtbn§zjb�An.¶cqr k5�ô�j>�M�Mj>m�sMtÕõöj>�M��u � �F�BjbySl_sMyQsck?t>scn�zj>oÀsctÕë�j>�c�»³ | tbl�n{zÑ w©wcsMy�z| oÌoAt�zsM���Ë°�t ÷�ï � õI�Ò�øfùMú>û î�ï � / �åm]�BzÑ y�n�µ�scyS�KsMtbn�sct�õ²³ | m]n~µ�sMyS�KsMt�ë�o r �DsM�Bzr �'q| ���kYq| �~m]�Djbt>jbn�zj>m&jbkÌmjbwDjbt>t�qr k5�ÜÌl�mFn�l�o~��j>�Ajbnpjbo>�&wcsMySsM�Kjby��Kõ ¢�ü � õþý�»ÿ ¢ºü ý�»ÿF��³ | mFn~µ�sMy��asMt | kÅn'zscw | ySo&zsM��zscn�sÈ � õ ¢pü ���þlSm�±Yð;È � / ¢pÆ �,ñ / Ç õ ¢�ÆÌÇ�ü óÕq| o~oAt>j�±Uqr �D�1zj>o§zÑ �.¶~s«ySjc����t�qr kFo?zj>�Yqr m&kÂwcsMm8j>�M�Â��zsMo~l�kn�zscw | ySo�zsc��zjb�AnpjbyS��j>t5zj>o?zjb�~j�lSob���������� J� è Xg ê PVX^ � ô�jb�M�Djbm&j>k�sM�wcsÂsMtÀõ ¢Kü � õ ý� ÿ ¢Kü ý� ÿF�Åk | �~y{zscn | oÒzjboat�zsM�AnasMyQsck?t>scn | ku � �?�Bjbm�� ��t>j>k��s r oA� | � ���F±{zjbySj�n'zsVw | y�o�zsM��zscmÌs�È � õ ¢pü ����l�m±Yð�ë�ì�í«ñVõ���÷�ï �7ü � ¢ºü �å÷_ï � õI�Bó m&j>��m&j>�Dscn�zÑ w§o~t�zsM� | n,zj>��n7¶cqr k'�� JMéçhUJMé��'H1XJcT � �fw?lS�&jbm&o>�5µ | �M� �È � õ ¢pü �Ä� í�npjbyжAj>ocqr y�sMkFk | ��zj>o��>opsck«sMkFk | �b�Bµ�sű.j>m&m}zsMySy�õæ� ü ���q| mmF�Mj>m�lS�DsMt | y�µ�scn�z| ��µ | �M�ÌsMt_u � �?�5j>y�l�k | ��³�sMk]nvsMySsMkFt{scn | k��¸µ�sMy��asMt�zsMmÌs �È�Î��¸á�� Ï u±dqr �M�Mw�zj>m]�,j>�c�,�Kj�np�AlSkÄzscnºsM���

Ø

Page 2: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

°��s r oA� | � ���F±{zjbySjIn�zscw | ySo�zsc�a±.j>y�µ�sMo~tbm�zsMy{zsMo�zscwcsMyvzj>��npjbyS��j>t>ml1y�j>µ&j�n�j>�c� | yÓ�]sMm¦o | � | t{scn�k | m]wMj>��Þ�Mj>m �bl�zsU¶&zscn�l�o>��sc�Kjby��?m&j>k§sMtvj>y�j>��j>l�m&jb�Âqr �Aj>o�k | �K³�sckFnº³ | m]npµ�scyS�KsMt | k'������ � � J � è Xg ê PVX^ � ô�j>�c�Dj>m´sM� | nAn�sct�u � �?�BjbySl�k | �~y{zscn | o¾zj>o_t�zsc�AnIsMyQsck?t>scn | k?�}z| yKzsMySy{z| õ�� ��� Ç�� �o | � | t{scn>� °�tbnº� | m&�V¶ r k'��µ | �M�«sMt_õ��©o | � | t{scnºk | m]wDj>�A��zsMy�s ü k | �~y{zscn | o§zj>oºt�zsM��nºsMyQsMkFt>scnpµ | tM��µ�ssct�� � �È � õ�� ¢dü �}q| oAo~t>j�±Uqr �D�1zjbo~o~jbyY��j>�Dµ&scn'zsM� | t | n~n��� ���åÇ � �}o~t�zsc�Ko | � | t>scnÄj>�M�Åm r y�ySo | � | t>scn{�

ô�j>�M�Mj>mÖsM� | nAn�sMt¾u � n�zj>���]�?�&l��Kjbm&t>l{z| o � �$ò ���,ySl�m&j�zsc�~lSoasMy�n~j>�Aj>lSmj>k©jb�M����� ��� Ç�� �o | � | t{scnps��i�j>kFlSm]n~oMqr k�sMt«u � jb�M�?o?zjb�MwDjbkFn | �pscl¾zsMyÓndsMy}scySk | n | nAn�� ��� ¡ �öð�� Ç u � ñ�·d��·Ì� Ø óoAtb±{zjb��zscn>�'n | w�zsc�&��zs�sMt�sMy�n�zj>�Ao | � | t{scn�jbySjb�Kjbl�zsMy�npsMy�sct�� ��� ¡ ��q| ���±.jby~qr ySj�np�f�| yBkFlS��jbn~o~tbjbn~n� � Ê Ø � �?�lS��j>m&tbl�z| o������ ¡� ���;��� ¡ � ��� jb�M�?o?zj>�M�'q| ����q| kMjbn>����"! ��� J� è Xg ê PVX^ � °�t��#� ���ÉÇ$� �Åy�lSm&j�zsM�~l�o�sMyÓn�zjb�~o | � | t>scnp�&z| y�sMt�nI� | m&�ç¶ r k'��µ | �M��sMtÅk | m]wDj>�A��zsMys��]�?�&l��Kjbm&t>l{z| o ���ySlSmj�zsM�AlSo}sMyÓn�zjb�~µ&jbtM�'µ�sasct8zscy�ndscyQs���j>�Dµ�sVn'zsM� | t | nAn � ��� ¡� ���ÖÇ%� ��o~tb±{zjb�psMo | � | t{scnk | m]wDjb�~��zsMy1s � � Ê Ø �3�?�&lS��j>mt>l�z| o�� �&� ¡ � ��åo~t�±�zj>�&zsMµ | tc�

°�t�u � n�zjb�~�5j>m©wMj>�M��qr m&k j>�M�'�]�F�&lS��j>m&tbl�z| o � ��òà����scÁ�m sMy�n~j>�Aj>kF�f�| y¦zsMySy{z|)( � ��� Ç*� �o | � | t{scn | n{��°�t ( �©scÁ�m¦sMyÓn�zjb�~m&jbk§�Kjb�M±.j>y�j>yA�| ySl�m&j�zsM�AlSo}scy�npjb�~j�n;¶Aj>y½q| yжAj�#������,+ � � J � è Xg ê PVX^ � °�t ( � ����Ç-� �_scy�n�zjb�~o | � | t>scnp�&z| y�sctbnº� | m&�V¶ r k'�&µ | �M�«sMt�k | mFwMj>�A��zsMy�s�D�?�&lS��j>mt>l�z| o �( scÁ�m�sMyÓn�zj>�Aµ&j>tc��µ�s�npj>y ¶Aj>ocqr y�sMt_sMy{zsM�&�&l�kfzj�nº±.jby�n�zjbnpjby3Î� Ø �Ò°�t�#� ���åÇ � ��ySl�m&j�zsc�~lSo�scy�n�zjb�~o | � | t>scnÄk | m]wDjb�~��zscy�sMt �( �?m�sMkÅ��j>�M±.jbySj>yA�| ��©ySl�m&j�zsM�AlSo�sMyÓn�zjb�~µ&jbtM�� �D��. sMm | yÓ�]sMm / � ���xÇ/� ��³ | mFn~o | � | t{sVn{�BsM�Kjby��?�~j�±.j>mm�zsMySy / � Ç ( ���}zj>o�sM��j>y��§k | m]wDjb�~��zscyfsMt�( sMyÓn�zj>�}jb�M� �/ ³ | m]n7¶&zsMµ | tM�G10�Zb\�`32'`xX\]W�Z{P54¾RBJcW�JMZ>`¦P è X\ è TYX\ è \FN�é?J ^ `ÌJcW�Z>PS\?PhUJDOSJ è WYXJcTUJ°�t�u �F�?�5j>m,ySjb�M�Djbm&j>kasM�wcs�s�y�lSm&j�zsM�~l�opsMműUqr �D�Dj�npySjbma� ¡U¢>£{£{£>¢ �6� � Ø879�:7 ��� wMj>k]n | � | k'��°�tzscy�ndscy r k��Dj>mj>��zscy�n�ySl�m&j�zsc�~lSo�sMy�n~j>�~j�n;¶Aj>y q| yжAj � � � ¡�¢>£{£{£>¢ �6�#���°<;$�ÉðK¬ �­¯® ¡>= ­ � ­ ñFí 7 = ­ 7²Ø,��? � ØD¢{£{£{£�¢@� �vóų | m]n~µ�sMyS�KsMt�n�sÅ� ¡�¢{£{£>£>¢ �6�åwMj>k]n | � | kzscy�ndscy�kFl�±.j>oAtDzÑ npjbnAn � �?�&l��Kjbm&t>l{z| o�³�sM�~sMySy�j>y�j>³&l�³Bjb� | m&m�sckKm&j�wDjbt>t�qr k'�°�tA� � � ¡U¢{£{£>£>¢ �6�#��ySl�m&j�zsM�AlSoÅsMyÓn�zjb�~�5j>m�wDj>�c��qr mk jb�M� Ú ¡�¢>£{£{£�¢ Ú�� | ��n | m | �~��zsMy�n���zsMt>l�oAn>��B�j�Þ

¶Ajbt>t�qr kÂkFlÄs��DC ��E � ØD¢{£{£{£b¢@� ��wDj>k]n | � | kVscn�j>tbj>mÒ��zsMtblSo�wDj>k]n | � | kF��z| y3Î �DCÅ� ¬ �­¯® ¡GF ­ CBÚ ­ �ö°y�lSm&j�zsM�~l�o;k | ���&l�m�zs �>l{z| o;jb�M��qr n~n~µ�scn'z| kI��j>�Dµ&scn'zsM� | t>m&sMkIjb�M� � �Fj>�&�Aj>m&� �r k]wcsM�&�psVnplSk r o * ��zscnp�AlÓÛ | n>�° ñ È?â>ã * ñ´zj>��n�zjbkam&jb� ±Uqr �D��s�Ú ¡U¢{£{£>£>¢ ÚH� | ��n | m | �~��zsMy�nÄ��zsMtblSo���j>�cw�zscyQsMoAtbn'zsco�zscn�z| y3�?�KlÓwDjby�sMt � ��lS��j>m&tbl�z| oI;$³�sc�psMy�ySj>y�j>³&l�³Bjb� | man�zj>��± | �]scnps´zjb³&³5j>m ñ È?â>ã * ñ �

i�j>kFlSm]npocqr k,sMt�n�sct � �?j>�&�Aj>m&� �r k]wcsM�&�psVnplSk r o%J ��zscnp�AlÓÛ | n{���Kjby��?m&jbk§j>y�j>��j>l K ­ C}� ¨7� ­ ¢ �DC ª��?~¢LE � ØD¢{£{£{£b¢@� ����°MJ��Fnºs�� ¡�¢{£{£{£b¢ �6�©wMj>k]n | �~�~jbm&�&oAt>j>�%N��~sM���?�©zsVnp�~l¯Û�zsMm�sMkÅm&j�wDj>tbt�qr k'��fw?lS�j>m&ob� µ | �M��±.j>m&m}zsMySy sOJ � *KÝB* q| oAo~tbjb±Uqr �D�1zjbo>��scµ | yÄsMt *KÝ sMt * �©zsVnp�~l¯Û�np�~sMm&o~tb³ | m�zsMy�n�Þ¶&zsVn#¶Aj>y½q| ySl3����m&m&j>k¾kBq| wMjbn~kMj>t�n�zjb�Bjbm ÈFâ>ãPJöì¿í n~j>y ¶Aj>ocqr yä��zj>o_sRQ ÈFâ>ãDJ o~t�zsc�à�Kjb�Dj>�M�Mj>tblSk¾s3;³&sM�psMy�ySjbySj>³lS³5j>� | mKn�zj>��± | �]scn�zscwcsMy3�GIHK\?O�WYXJMZIPSZ�X\ è �1Xg W�X\FTU\

ô�j>�M�Mj>m�sMt��xj>�M� � �?�&l��Kjbm&t>l{z| oºySlSmj�zsM�AlSoºsMy�n~j>�Aj�sMt�u � wDj>k]n | �An�zj>�~mj>k ��� � Ø �U��i�j>kFlSm]npocqr ksct%�©sMyÓn�zjb� - ¡�¢>£{£{£>¢ - � zjbo�Ú ¡U¢{£{£{£b¢ ÚH����zsct>lSo~sMl�n>��B�j ¶Ajbt>t�qr kKkFlBs��©zsco | �&lSk���zsct>lSo wDjbkFn | �pscl�n}sct�j>y�oM�|�}zsMt>l�o�wDj>k]n | �psMl�m�sMk´y�lSm&j�zsM�~l�o�k | ���&l�m�zs �>l{z| ¶~sMkfzj>m]n>ΧÚSC«�ଠ�­ ® ¡>T ­ C - ­ �5E � ØD¢>£{£{£�¢@� ���VUMjby½q| y ¶AjW s T ­ C_j>�M��qr n~n~µ�scn�z| kF��z| y;kfzjb³&t>j�n~n � á � �?j>ov��zscnp�AlÓÛ | n>�Y��j>y��Mjbn�sKk�zj�n���zsMtblSo�kBq| tDq| n~n~lÅzscnp��j>mjbn��zscn~�~lÓÛ�zsMm&sMk � lSySy�jbnAwDjvs���zsMtblSo�np�pscm&o~t�± | �~��zs �>l{z| �©zsVnp�~l¯Û�zsMm�sMk��ÄoAt | kÄzsMoºm&jbwMj>tbm&l3�°���j>mmF�?l��BjbmÀ±.j>m&m�zscySy�ÈFâ>ã W ìÖí��sck?k | �vsMt�n�� | m&�V¶ r k5�Bµ | �M�¦sMt - ¡�¢{£{£{£b¢ - � zj>o�Ú ¡�¢{£{£>£>¢ ÚH��}zsMt>l�o | k,sctX��sMyÓn�zjb�~m&jbk§j>�M�DsMt | mÌl���zsMm]�BzÑ n�zsMo�zscn�k�zjb³]w�l�o~jbySlSk'�

Page 3: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

��q| m&m]�Djbm§�5j>y�zsVnpµ�scn�z| �]µ | �c��µ�sIjbt>jbmaj>k]w?l�wcsMy�j>m �blQsM�Aj>y{zs �>l{z| sMyQsM³F¶&zsMmasct-� sMyÓn�zj>����zsct>lSo~sMl�n | o~tbn�zsVÞyÓ� | kF��sÅo | � | y ¶ r k'��sMkFk | ��kfzjbn���zsMtblSo | o~t�n'zsMyÓ�?n�m]�Mj>�cqr mk5������ � � J� è Xg ê PVX^ � °�t � y�lSm&j�zsM�~l�o�sMyÓn�zjb��lS��zscmF�BzÑ n'zsMo&zsMm�sMtbnÒzjb�An7¶Vqr k'� µ | �M�8kFl�nMqr m]n~jbn7¶Vqr k8sMt«sMt | m | ol���zsMm]�BzÑ n�zsMoAn�kfzjb³Fw?l�o~j>yA�| ���?�BjbySl��}zsMt>l�o | kKj>�M�?lSk | o~t�n'zsMyÓ��zsVn{�

°�tÄu � j r kFy�lS�&jbo~t>l?n�zjb�~�5j>m�wDjb�M��qr m&kIjb�M� � �?�&lS��j>mt>l�z| o ( scÁ�mÅsMyÓnpj>�Ajbn>�1°�t ( �?m�sMk_��j>�M±.jbySj>yA�|� �F�&lS��j>m&tbl�z| o�y�lSm&j�zsM�~l�o�sMy�n~j>�~j�n�¶Ajby½q| y ¶Aj/����°�t ( sVÁKmÀsMy�n~j>�~j�nÂzr �M�,lS�&zsMm]�BzÑ n7¶ r k5�'µ | �M�«�Kjb�]sM� r m&kjb�M�«lS�&zsMm]�BzÑ n'zscoAn�sMt8�åySl�m&j�zsc�~lSo#sMy�n�zj>�Aj>m��� JMéçhUJMé��'H1XJcT � °�t�u � wDjbkFn | �An�zj>�Am&j>k�sMt = ¡�¢{£>£{£>¢ = � npj>�A��zjbo~tbjbnpjbo1�}zsMt>l�o�zsMy�npsMyM�Aj>³&�Aj>tbj>m]n'zsMyÓn�lS�&zsMm]�BzÑ n'zsMo&zscn� | m&�ç¶ r k,sMt�u � npjb�~��zj>o~tbjbn~j>oºlS��zscmF�BzÑ n'zsMo&zsMm�sMk � wcsM�M�a�©zsMo}oAt�z| wcsMy�n~j>�~�Òzj>oAt>j�npj>o | �AlSj>m]n�zs �>l{z| ¶&zsMm�sMk��U�

G [JcN&W ^ Z{P�X\?OQPST�TUH ^ Z>H;X\FTi�j>kFlSm]npocqr m&kÌsMt�u��?�F�Bjbm�saySl�m&j�zsM�AlSo~sMm�±dqr �M�Djbn~ySj>m�� ¡�¢{£{£>£>¢ � ��� ¡ wDjbk]n | � | kVscn{��ô�j>�c�Dj>m ¤ sMt�s

wMj>k]n | �>�5sM��j>y��a�~jbm&�&j>y�kMjbt>lSk«sct�sMy�zsM��&lYn r yQsç¶A� | m&o�zsM� | kFkcscy3Î� Ø � ° ¤ µ | o~o~t>s¾�Kjb�Dj>�M�Mj>tblSkÕsÌ� ¡U¢{£>£{£>¢ � �&� ¡ wMj>k]n | � | kÉzscy�ndscy kFlÓ±.j>o~tMzÑ n~jbn~n � � Ê Ø �3�?�&l��Kjbm&t>l{z| o³&sM�psMy�ySjbySj>³lS³5j>� | mKn�zj>��± | �]scn�zscwcsMy3�� �D��° ¤ wMj>k]n | �º��j>�M�| ySj>�Mj>o}sMt8� � � ¡U¢{£>£{£>¢ � �&� ¡ ��ySl�m&j�zsc�~lSo#sMy�n�zj>�A�~jM���� ��° � ¡d¢�£>£{£�¢ � �&� ¡d¢ ¤ ��zsMt>l�oºsMt�u���n~j>�A��zjbo~t>j�npjbo�l���zsMm]�BzÑ n�zsMo�zscn��~j>³�~j>tbj>m]n'zscyж~s�� JMéçhUJMé��'H1XJcT � ÜÌl�mFn#lSoA�Kjb�~jbn~j>ob�sMt��Ì� � jbo~jbn~�Bjbm«sI±.j>m]n~l5n r yQsç¶A� | m&o�zsc� | kFkVsMy'�Kjb�Dµ�scn�zsM� | t | n~n ¤wMj>k]n | �An�s�� ¡ zj>oºs�����wDjbkFn | �~l{zsMySl�o}o~t | �~t>scn'zsMm&sMk§� | m&�V¶ r k'�

B�j ¶Ajbt>t�qr k�kFl�s�� ­ ��? � ØD¢{£>£{£b¢ � Ê Ø ��wDj>k]n | � | kVscnºsct�u � npjb�~��zj>o~tbjbn~j>o}��zsMtblSo&zsMm�sMkÅwDjbkFn | �pscl�wcsMys � ­ � ¬ �C ® ¡GF ­ C�� = C ± | �A�©zsc��sMm��°�y�kcscyS�KsMt>t r kÕ� | o�nÅsMt � � � Ê Ø ¶Aj>y½q| y�zj>o�n{� ��q| m&m]�DjbmjbySy�j>mf�| �AlStbµ&jbnM�| �&µ | �c�,sMt F ­ C�ySl�m&j�zsM�AlSo�k | ���&l�m�zs �bl�z| o�jb�M��qr n~n~µ�scn'z| k?kVsMy�±.jbm&m�zsMy�y�s

¤ ����������

F ¡½¡ F ¡ � £{£{£ F ¡ ���� ��� � � � ���F � ¡ F �� £{£{£ F � �= ¡ = � £{£{£ = �

���������

��Ø � �D�

jb�M�Djbm&y~�| o?zj>��5��j>yÓ�?m&j>k_¶ | �&� | yS��scy�zsMm,jb�M�,��áÀ�'�Fj>o��©zsVnp�~l¯Û§�jbnpjb�~��lSm�zscm&ops�o~tbj>�Aj>³5j>y3�

Page 4: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

GIHKJcé ��[�X\?O�W ^ HfX^ TIR�P���JMZbJ è5ê P�X\?O��B\]W�X^ OSJcNºXJMe�JMH�XJcT�JcN°xn | w#zsM�&�lQsMkF��sMmÌsMt��ajb�M�§uf�?�5j>y�l � m]�BzÑ y�n#wMsc�M�,t�zsc�AnU��l�mFn~j>��wMscySy r � | n ± | ��¶Aj>y q| ySml3�f��tbj>m¶Ajb�M�?t>j�np�5j>måj>�M���ÉÎ�� Ï uËwcsMy�z| o�±dqr �D�cw�zjbm]� �l �Bj>�~jbm �>l{zsMySµ&scn'z| o&zsM��zsMm sMt�n zj>��n7¶Vqr k'��µ | �M� sMt�n~jbn~o~t�| ySj>�Mj>o~jbm�o | kFoAt | �f�j>�~lÓw#zsMySµ�sVn'z| �csMt{sct�sMt�űUqr �D�Mw�zjbmF� T�� � | oAtbn�zsMy��ºzr ��°�o~t | kÄzsMo | k?m&sMkI��j>�c±.j�Þy�j>yA�| j>mÕsMt �´±Uqr �D�Mw�zjbmF�Â�&jb�~lÓw�zscy�n7¶&zsVn����1± | ��¶~s�¶Aj>y q| ySm&lä�§���M�ÕsM� | n~n��¾lSm]npjb�AwcsMy�y r � | m zj>��npj>y��Kjbt>j�n~n

�l �Bj>�~jbm �>l{zsMySµ&scn'z| ±dqr �D�cw�zjbm]�Dj>k,µ&sMyS�KsMt�zsc�ps�sMt�� � ����¶Aj>y q| y�zjboAn�scySkVsMyS�KsMtbt r k'�i�j>kFlSm]npocqr m&kÖjb�M��� Î�� Ï u � ySjbk�zjb³Bjbt5zjboAn{� °�t�� ­ Î�� Ï u ��? � ØD¢>£{£{£>¢ ���«wcsMy{z| o±dqr �M�Mw�zj>m]�DjbkMjbn>����j>yÓ�Dj>kFmÄzjby_±.jbm&m�zsMy�y�� ­ � ãA�8� ¨�� � ãA� ¢ = ­ ª ��zsc�~��j>y�� ã Ç �M�?�AjM��s�� k |?| �~�lSm�zscnpsç�±dqr �M�Mw�zj>m]�DjblSm&jbk�m&jbwMj>tbtqr k'����t>jbk?kcj>y?m]��l�y�w#zsMm�n~j>yжAjbocqr yMs�� � ãA��� ¬ �­¯® ¡ � ­ � ãA� = ­ � ã Ç ����q| o~oAt>j�±Uqr �D�1zj>ob�°���ySjbk�zjb³Bjbt5zjboAn1�&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| m�sMk�� | m&�V¶ r k'�çµ�s}s}k |?| �~�lSm�zscnpsç�F±dqr �D�cw�zjbm]�Dj>l T�� � | o~t�n'zsMyÓ�ºzr sck5�°��Kjbm&m]�?lS�5j>m s�É�&l �Yjb�~j>m �>l�zscySµ�scn�z| �ÄsMkFk | �Ks � � � ãA�,� ¬ �­¯® ¡ � �­ � ãA� = ­ wDjbk]n | ��n,s�Éj>y�oM�| �&j�Þ�Al�w#zsMyÓn7¶&zsMm�sckK� | m&�V¶ r k,s�ã Ç �§³ | m]np��scm����º�?lSyÓw�zscmFwcsMy{z| �µ | �M�§±.jbm&m�zsMy�y� � � ãA��� y�lS��! �"

� � ã�×�#'� Ê � � ãA�# £

¼½yÓ�«�©z| � | m�zjb�An~j>yS��j>tbm&l�n r �ç¶ r k�s$� � Î%� Ï u �Ì�&jb�~l�w#zsMyÓnºySj>kfzjb³Bjbt5zj>o�n{�º°�m�scy�z| ���©z| � | m��&j!&�ml�zsMy�m&ly�j>µ&j�n�s'���asc�]sMopsc�&�«�~j>m�Ä�r �j>�~lÓw#zsMy�n}y�j>kfzj>³5j>t5zj>oAj>l�n�lSob�ô�j>�M�Mj>måsM� | n~naj>�M���àÎ(� Ï u � �&l �Bj>�Aj>m �bl�zsMy�µ�scn�z| y�j>k�zj>³5j>t5zj>o>��.�jb�M��qr m&k�j>�M� u � �?�5j>y�l�Íl�t | ��jbn~�~l{zscn{����j>y��MjbnKsMt ��Ø � Ø ��kFl�±.j½¶Aj>t5zj>oÌzÑ ��ySjc���fw?lS�j>m&ob��µ | �M�Òs8Í*)�� Î�� Ï u � ySj>kfzjb³Bjbt5zj>o

oAt>l�mFn�zj>mÌ�&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| zjbo}±.j>m&m�zscySy � Í�)�Ä� � � ãA���ÖÔ � � � � ãA�~�U�°�t+� lSm]npjb�AwcsMySy r � | m n~j>kFlSm]npocqr m&k©jb�M� ��zsMo~l�k-, Î�� Ï u��å�&l �Bj>�Aj>m �bl�zsMy�µ�scn�z| ySj>kfzjb³Bjbt5zj>o�n{�.�j/&�m&l{zsMyж r k©sMt0� Î�� Ï u ±dqr �M�Mw�zj>m]�FnÀsMt0� � ãA�8� ¨1� � ãA� ¢ , � ãA� ª � ã Ç ���§k?lÓ±.j ¶Ajbt5zj>oAo~jby3�þ°

k |�| �~�&l�m�zscnds��F±Uqr �D�Mw�zj>m]�Mj>kMj�n�scySkVsMyS�KsMt�wMs_kBq| m&m]���r �5j>y�zsVnpm&lä�µ | �c�§±.j>m&m}zsMySy�sct� � � ãA� �¿¨2� � � ãA� ¢ , � ãA� ª ×Ö¨1� � ãA� ¢ , � � ãA� ª ��Ø � � �

q| o~o~tbjb±dqr �D�1zj>o>�G W13^5464 [�X\�OÓW ^ HfX^ TIR�P���JcZ>J è5ê PçX\?O��B\]W�X^ O�JMNºXJMe�JcH1XJcTUJMN

°�tÕu � j r kFy�lS�&jbo~t>l_n�zjb�Àj>�M�Öm]�BzÑ y�n»zj>o�q| oAo~tbjb±Uqr �D���| �?zj>oAt>µ�scyS�KsMt�zscnÌu � �?�5j>y�l_npsM�An | �©zscmF�?m�sckmjbwDjbt>t�qr k5�É°��Kjbm&m]�?lS�5j>m�s´ndsM��n | ��zsMm]�Fn§kFlS�f�| wBzÑ n7¶Vqr k µ�scn�zsM�~³ | m]n7¶~sMl�m�sMk s´µ�sMy��asct�zscwcsMyä��sMkFk | �jb�M�,u � �F�BjbySl�t�zsM�An}npsM�An | �©zscmF�Fn�kVsM³ r m&k'�i�j>kFlSm]npocqr m&kÒj>�c� sMt�u � �?�BjbySl ��� �à�D�87 m]�BzÑ yÓnÅndsM��n | ��zsMm]� | mÉzj>��npjbyS��j>t>j�n~n9��Î:7 Ï uwcsMy{z| o�zjb�An�zj>kf�r ±Uqr �D�Mw�zjbmF�Fn>����t>jbmK¶Aj>�M�?t>j�np�5j>m�sÅnçq| �&�]w#zsMyÓn | t�z| o��±dqr �M�Mw�zj>m]�FnvsMkFk | ��� | m&�V¶ r k«�&lÓ±QÞ±.jb�~jbm �>l{zsMySµ�sVn'z| m�sck5�Bµ�s§sMt;���?m&jbk�n~jbn~o~t�| ySj>�Mj>o��Aj>m&�&�5j>m�y�zjbnpjbt>m&jbk¦sK³�sM� �bl�zsMy�lSo��&jb�~l�w#zsMyÓn7¶~sMlä��wMsc�M�?lSoµ&sÌsMt�� T � � | o~tbn�zsMy���zr ��°�t9� ? �?jb�&lSk̳�sM� �bl�zsMy�lSo��j>�~lÓw#zsMy�n�±dqr �M�Mw�zj>m]��zj>�~jKs=< ­ � ��? � ØD¢{£{£{£b¢@� �¶Ajby½q| y�zj>o�nºsMySkVsMy��asMtbt r k'����oAt>j>�AlSm]n�s�< ­ ��Î>7 Ï ux±Uqr �D�Mw�zjbmF�?�Aj�±.jbm&m�zsMy�y�s

< ­ � �@?�¡p¢{£{£{£b¢A? �#���¿ySlS��/ ("Ø#0B � ��?�¡U¢{£>£{£>¢A? ­ ×C# ¢{£{£{£b¢A? ��� Ê � ��?�¡U¢{£{£>£b¢A? ­ ¢{£{£{£b¢A? �Ä�ED

q| o~o~tbjb±dqr �D�1zj>o�npj�npo~t��| ySjb�Dj>o �@?�¡d¢{£{£{£b¢A? ��� Ç 7 j>oAjbn�zj>m��f°�7xnpsM�An | �©zscmF� | m��&j!&&m&l�zsMyÓn��&l �Yjb�~jbm �>l{zsMySµ�sVn'z|±dqr �M�Mw�zj>m]�Djbk,µ�sMyS�KsMt�zscn�s�k�zj>oM�| �&�&lSjbk,o | �&zsMm=� � 7,�ı | �{¶~sº¶Aj>y½q| ySm&lä�ô�j>�M�Mj>m�sM� | nAn�sF7 � u � npsM�An | �©zscmF� | m�j>�M��G1Î:7 Ï u � wDjbkFn | �?zjb�An�zj>kf�r ySjbk�zj>³5j>t5zjbo>� °�tG8k |?| �~�&l�m�zscnpsç�F±Uqr �D�Mw�zjbmF�Mj>l�m»sct | kVscn�sMt�LC5ÎH7 Ï u �5E � ØD¢{£{£{£�¢ ���§±dqr �M�Mw�zj>m]�DjbkMjbn»zj>��n7¶cqr k'�

I

Page 5: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

��j>yÓ�Djbk?�Aja±.j>mm�zsMySy �LC ��?�¡U¢{£>£{£b¢ ? �#�I�ਠG �@?�¡p¢{£{£{£b¢A? �#� ¢ = C ª ��zsM�A�Kjby�� ��?�¡U¢>£{£{£>¢ ? �Ä� Ç 7 �KjbySySj�n~n>�°�k |�| �A�&lSm}zscndsç�]±Uqr �D�Mw�zj>m]�DjbkFkMj>y�n~j>µ�zscn�n~j>yжAjbocqr y�sct

G �@?�¡U¢{£{£{£b¢A? �Ä��� ��C ® ¡ �LC �@?�¡U¢{£{£{£b¢A? �#� = C

jb�M�Djbm&ySj�n{�°�t G ySjbk�zj>³5j>t5zjboAn1�&l �Bj>�Aj>m �bl�zsMy�µ�scn�z| m�sMk�� | m&�V¶ r k'�çµ�s}s}k |?| �A�&lSm�zsVndsç�F±dqr �M�Mw�zj>m]�Djbl T � � | o~t�n'zsMyÓ�ºzr sMk'���t,jbo~jbn~�BjbmÒnpjbn~o~t��| ySjb�Dj>o - � � ¡U¢{£{£>£>¢ ���� Ç 7 ³ | m]np��sMm´wMj>µ&j�n7¶Vqr kÕsÀ³�sM� �>l�zscySlSo��&jb�~lÓw�zscy�ndsck?�}z| ym]�Mj>�An��Âá � �?jbo

�6G � - ��������< ¡ � ¡>� - � < �A� ¡b� - � £>£{£ <���� ¡>� - �< ¡ � � � - � < �A� � � - � £>£{£ <���� � � - ���� ��� � � � ���< ¡ � � � - � < �A� � � - � £>£{£ <���� � � - �

�����

��zscn~�~lÓÛ | n>���j>yÓ�Djbn�sMt�GIwDj>k]n | �A±dqr �D�cw�zjbm]� - �?�5j>ySl UDs � | �l¯�?��zscnp�AlÓÛ�zsMm�sck�mjbwDjbt�qr m&k'�°�t�GI�&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| y�j>kfzj>³5j>t5zj>o ? �?jb�&lSka³�sM� �bl�zsMy�lSoÄ�&j>�Al�w#zsMy�n.¶&zsMm«s

< ­ G ��?�¡U¢>£{£{£>¢ ? �#��� ��C ® ¡ <

­ �LC ��?�¡U¢>£{£{£>¢ ? �#� � = C

± | �~� r y{zscwcsMy���j>�Dµ�sVn'zsM� | t | nAn < ­ GKÎ 7 Ï u � wDjbk]n | ��±Uqr �D�Mw�zj>m]�Fn¾zj>��n7¶cqr k ��? � ØM¢{£{£{£b¢@� ���º�fw?lS�&jbm&o>�µ | �M�§±.jbm&m�zsMy�y

< ­ G ��?�¡d¢{£>£{£>¢A? �#��� y�lS��/ �"Ø# B G ��?�¡U¢{£>£{£>¢A? ­ ×C# ¢{£{£{£b¢A? ��� Ê G �@?�¡d¢{£{£{£�¢A? ­ ¢{£>£{£>¢A? �Ä� D £

i�j>kFlSm]npocqr m&k§jb�M�,��zsMo~l�k Æ Î�7 Ï u � �&l �Bj>�Aj>m �bl�zsMy�µ�scn�z| ySjbk�zjb³Bjbt5zjboAn{� .�j>�c��qr k§sMtbn�sMt��Î>7 Ï u´±Uqr �D�Mw�zj>m]�Fn{�csM��j>yÓ�Djbn�sMt � ��?�¡U¢{£{£>£>¢A? �Ä���²¨2G ��?�¡d¢{£>£{£b¢ ? �#� ¢dÆ���?�¡U¢>£{£{£>¢ ? �Ä� ª j>�M�Mj>m&y�jbnzÑ �ºy�j_n~jbn~o~t�| ySj>�Mj>o �@?�¡d¢{£{£{£�¢A? �#� Ç 7 jbo~jbn�zj>m���ôIzsVnpµ�scn�z| �'µ | �M�ÌsMt;��±dqr �D�cw�zjbm]�Ì�l �Bj>�~jbm �>l{zsMySµ&scn'z| zj>osų�sM� �>l{zsMySl�o#�&jb�~l�w#zsMyÓndsMkF�~s�npj>y ¶Aj>ocqr y�s< ­ �Ì� ¨ < ­ G ¢dÆ ª ×Ö¨2G ¢ < ­ Æ ª ��Ø � I �

q| o~o~tbjb±dqr �D�1zj>o ��? � ØD¢>£{£{£>¢ � ������� � � J � è Xg ê PVX^ � ô�j>�M�Mj>mÀsc� | n~n�jb�M� | yÓ�]sMm=G�Î>7 �©u � Ï u � �l �Bj>�~jbm �>l{zsMySµ&scn'z| y�j>kfzj>³5j>t5zj>ob�sc�Kjby��?�~jÂlS�]sct � 7 �1� °�t G>�FnÌ�~j>� r y�zsM�AlSoAm�sMk©� | m&�V¶ r k5��µ�s�UDs � | �l¯�?��zscnp�AlÓÛ�zsMm�sck��pscm&��¶&zsc�ps�}zsM�~��j>yÓ� � Ç 7˳ | m]n~��sMm«±.j>m&m�zscySy������ G � �;��� � £

Page 6: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

� ���!Ä���Ò����Ë��½����������������� �Ä��!Ä" ��%�'�����?��B)�a=?4 � +]C- G� E�E 68-¾41@2 GVÚ C=F9 - �¦= � 9�+ �� =?E�� �� C= GçÚ =��G é63^ Z 4 JMRB\FZ>\ 4 Xg [�� ^ TUTUHç\� ����� � J� è Xg ê PVX^ � u��?�?�5j>y�l�± | y��Fn | m | oº�'q| �~�Äzj>m¦jb�M�=�åÎ � Ï u��̱ | y��Fn | m | o�ySj>kfzjb³Bjbt5zj>o�n¦zj>��nMqr m&k'�}°� � ���� ð � � ãA�1ñ½ã Ç ��óº³ | m]n~µ�sMyS�KsMt�n�s �Â�'q| �A�Bj�³}zsMy��#zsU¶&zsMm&sMk � wcsM�M�_np�~sç¶Aj>k]n'z| �~l�zsd¶&zsMm�sMk���mjbwDjbt>t�qr k5�

� ��� ��� J� è Xg ê PVX^ � ��l��as¾�'q| �~�Äzjbm©sMt«u � � �É�¸�D��j r k?y�lS�&jbo~tbl�n�zj>�A�Bjbm�j>�c� � Î� Ï u � �l �Bj�Þ�Aj>m �bl�zsMy�µ�scn�z| ySj>kfzjb³Bjbt5zj>o�nÌzjb�AnMqr mk5�° �©oAlS�Ks��'q| �A�#zjbnº�~jb� r y{zsM�~l�o~m�sMk�� | m&�V¶ r k'��µ�s�±.jbm&m�zsMy�y6� � � ãA�vý���Ìnpj�npo~t��| ySjb�Dj>o}ã Ç �§j>oAjbn�zjbm��

ô�j>�M�Mj>m�sM��wMs�j>�M� ��Î>� Ï u � ± | yÓ�Fn | m | o��'q| �~�5jM��.�j>�M��qr m&k«j>�c��� ¢���� �?zj>oAt>lSm]n~j>�AwcsMy�y r � | n�sMt�M�?�5j>mæzj>o��;�?m�sMk�sMt�� ¢���� �?�Aj�n�q| �An�zj>mf�| ySjbo~t'�r k5zÑ n�zj>o?zjbn>�§°��Ë�Ëð�¾� ã " ò ã ¡ ò � � �fò ã � � � ów�zjb�Dj>o�o~t�zsM��o | � | t{scn | n�sct � ¢���� jb�M�_±.j>y | o~tbn�zsMo�zscm�sMk�� | m&�ç¶ r k'�;°�� � ã " � ¢ � � ã ¡ � ¢{£{£>£>¢ � � ã ����³ | m]n | kq| o~o~tbj>kBq| n�zj>o?zj�wDjby#m]�Djb�An"!¿nçq| �{q| n~nAw | m&sMyQscn�s �vñ#� ¢���� �'q| �~�5j>�&sM�psM���s¾zÑ ��nvm | �A�©zsMy�lSo�nçq| �{q| n~nAw | m&sMySm�sckmjbwDjbt>t�qr k5�#��t>j>m�nçq| �{q| n~nAw | m&sMy�µ | oAo~t>s%$ � !����x¬ �­¯® ¡ · � � ã ­ � Ê � � ã ­ � ¡ �ç·ç�� �"! � � J � è Xg ê PVX^ � ° �vñ#� ¢���� ��q| �~�Äzj�n��~jbkFn~l�&�kÄzsMy�µ�scn�z| m�sMkK� | m�V¶ r k'��µ�sÅsMtð&$ � !��ºñ'!©s;�vñ(� ¢���� �'q| �A�Äzj>�5j�zÑ ��n}m | �A�©zsMy�lSo nçq| �{q| n~nAw | m�scyBó�wcsMy�z| o1o~t�zsM�Kµ&sMyS�KsMtı.jbyAqr y��M�| y]k | �~y{zscn | ob�°���j>mmF�?l��Bjbm¦s;�vñ#� ¢���� �'q| �A�Bjbo~t>jb�D��j>m&o��~jbk]npl�&&k#zscySµ�scn�z| ��sMkFk | �ºs�m&jb�Kmj>�]scn{zÑ w

$ � �vñ(� ¢���� ���»o r ³vð)$ � !��Iñ�! s'�vñ#� ¢���� ��q| �~�Äzjb�Bj�zÑ �An�m | �~��zsMySl�oÄnçq| ��q| nAn~w | m�sMy;óoAt�zsM� | n�s �vñ(� ¢���� �'q| �~�5j>�&sM�psM�ÂzÑ w�µ | o~oAt�zsMm�sck,m&jbwMj>t>t�qr k'�� ������* XJcW�JDO �,+.-0/�1�-�24365�7839-�/:1 � Î � Ï u �<;0=?> 3@/BAC'DFE -HGJILK3 D > -�M(1 � ¢���� � �ON K3 DFP = 2 P - D 7Q3�M?MSR >- ; - P K-�243 �vñ(� ¢���� /BACHDHE -�5�3 D 3 E)D -�T P =VU TWK3�M?X�3 P KC K- ;ZY -�2�2[K3:MVM

$ � �vñ#� ¢���� ���]\_^` · � � � ãA�ç·YÈDã £

G é63^ Z 4 J�X\]W�eY\FZb\�`ÖXJcW�JMZbJMH1XJMTUJ°xn | w#zsM�&�lQsMkF��sMm�o~tblSm]npj�kFlSt�zsM��z| yQsM���Aj>� r y{zsM�~l�o�o~l��as���q| �~�ÄzjbkFkMj>y�± | �MyQsMy�k | t r m&k'�

� �,+ � � J� è Xg ê PVX^ � ô�j>�M�Mj>mÒsc� | n~nIj>�c��²ÎH� Ï u � �~jb� r y�zsc�~lSovoAlS�Ks���q| �~�5jM�,��m&mj>k zj>�AlSm]nD�| ¶Fzjbm´sã Ç ��µ&jby��Djbm sMtbn�sMt§j>�c�Dj>mj>oAn´zjb�An.¶cqr k5�fsM�Kjby��¿zscn~µ�sMyQsc�´s � � ãA�I³ | mFn | m zjbo�sM��j>y��?m&jbk8sctaj>�c��l�kl���zsMm]�FwDjbk]n | �~s$� � � ãA���� ��� � � J � è Xg ê PçX^ � ô�j>�M�Mj>m�sM� | n~n�j>�M�=�»Î � Ï u��À�Aj>� r y�zsM�AlSoºo~lS�KsÅ�'q| �~�5jM��.�j>�c��qr mk�jb�M�ba �xul�m]npj>��wcsMySy r � | nKzj>oºsMt | m�j>�M� | y��]scm�cÒ�&l �Yjb�~j>m �>l�zscySµ�scn�z| ±Uqr �D�Mw�zjbmF�Fn>�'sM�Kjby��?�~j�±.jbm&m�zsMy�yc � a���� �Õzj>o,cH� �ed �vý�»í �ed}Ç a��U��°%f�Ò� ��).c�o~lS�Ksv�'q| �A�#zj>�c�| y5sMtbn � | m&�V¶ r k'�?µ | �M��s�� c zsMy�npsMySlzsVnp³�sM�~sM��zjbnpjb�~jbt5zj>o?zjbwMj>y&¶{q| n~n�y�zjbnp�AjM�°gc��Fnºsų�sM�~sM��zjbnpjb�An~�psMm&oAtb± | �A�©zs �>l�z| ±Uqr �D�Mw�zj>m]��zj>m&jbk«o~t | k#zscoºmjbwDjbt>m&lä�°���j>mmF�?l��Bjbm_cH��ìþí¾n~j>y ¶Aj>ocqr yä��sMkFk | �Ksctåzscnp³&sM�psM�Òzjbn~j>�Aj>t5zjboAn�lS�&zsMm]�BzÑ n'zscoAndsc�An'z| m�sMk�� | m&�V¶ r k'���y�ySj>mkMj>t��| jbo~j�np�5j>m � wcsM�M�?lSo�µ�shc �5ò©íD��sctÂzscnp³�sc�psM��zjbn~j>�~jbt5zjboAn�lS��zscmF�BzÑ n'zsMo�w�zscy�n'z| m�sMk�m&jbwMj>t>t�qr k'�

i

Page 7: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

� JMéçhUJMé��'H1XJcT � ����l�y�w#zsMm]wcsMy�z| �Yµ | �c�¾s � zjbo f�»� ��) c ��q| �~�Äzjbk¦³�zscy��#zsU¶~s§��j>�Djb�M�Djbt>lSk'������j>y�ySjbnAn±.jbm&m�zsMy�y f� � � d � �]c � �ed � � � � � c � d �~� ¢ � ��� Ø �f� � � �ed ��� c � � � d � � � � � c �ed �~��×_c � �ed � � � � � � � c �ed �~� £ � ��� �D�

° � ��� Ø �Äq| o~oAt>jb±dqr �M�;zj>oA�f�| y�sM��z| �&l�k'�Mµ | �c��s f�;�?m�sMk_s d�Ç a§µ&jby��Mj>m�wDj�n~n�zjb�~l�mFnM�| j>�c�Dj>mj>o~j}��j>�Mj>�M�Mj>t>l�ks'� �'q| �~�5j"c �ed �#³ | mFn~�BjbySl�zjb�~lSm]nM�| j>�M�Mj>m&jbo?zj�wDj>yä�� JMéçhUJMé��'H1XJcT � °Ò�~j>� r y�zsM�AlSo���q| �~�5jf�Dj | �Kj�np�AlQsMlp¶Aj>y�ySj>��t�| lSm�zjb�An.¶cqr k�sMt | kVscn�s��'q| �A�Äzj>�;�| yDo~t�zsM�~�KsMt�ndscn | n~n�Kscn~j>�Kscnpl�kcsclM± | �]sMy��asckcsVn{�çsM��j>yÓ�Dj>kv³&sM�psM�Òzjbn~j>��np�pscm&o~t�± | �~��zs �>l{z| wcsMyDoAt>j>���Bjbm8zj>o�u � �?�5j>ySl]l�t | ��jbn�Þ�Al�zscwcsMyYo~t>jb���5j>m�lSm]wcsM�~l{zsMm&opsck5�GIHKJcé �'TUHçJMZ �L Xg [�� ^ é?\?OQ`Ì\� �� � � J � è Xg ê PVX^ � ô�jb�M�Djbm sMt=�´m]�BzÑ yÓn�wMsc�M�Òt�zsc�AnKlSm]npjb�AwcsMySy r �KsÀsMt,uf�?m&j>k5�Ö° � Î � Ï u ��Aj>� r y�zsM�AlSo o~lS�Ks��'q| �~�ÄzjbnÄj>�M�?o~tbj>�F�r m&j>kK� | m�V¶ r k'��µ&s�s���y�j>k�zj>³5j>t5zj>o�jb�M��µ | ��j | � | �E&&t>� r o�n�sc�§sMt�alSm]npjb�AwcsMySy r � zj>oºs;�©³�zsMyÓ�#zsU¶~s�kBq| t]q| nAn{�� JMéçhUJMé��'H1XJcT � ô�jb�M�DjbmÂsM� | nAnvj>�M� | yÓ�]sMm � Î � Ï u � �~jb� r y�zsc�~lSo��'q| �A�Bjc��sMµ | yfsMt;��lSm]n~j>�AwcsMy�y r �m]�BzÑ yÓn{� �ºj>�þm&jbµÄzj>t�l��]sMt | ySm&lä��µ | �M�§��zsc�~��j>y���ã Ç �aj>o~j�n�zjbmÌwcsMm | y��DsMm aÂm]�BzÑ y�n}�?zj>oAt>lSm]n~j>�AwcsMy�y r � sMt�M�?�5j>mY�'sM�Kjby��?mÄzjbyYã Ç aæzj>oºs;�;�?m�sMk§s a'�?�~j�n�q| �An�zj>mf�| ySjbo~t'�r k5zÑ n�zj>oAjvj>�M�«jb�M�?o~tbj>�F�r �'q| �A�#zjbnºsM�Y�� ����� � J � è Xg ê PVX^ � °�t}u � �?�5j>ySl��8sMySsMkFt{scn | n�j>�c��oAt>jb�?�r zÑ w?m&j>k�� | m&�ç¶ r k'�?µ&svwcsMm | yÓ�]sMm'�©Î � Ï u �jb�M�?o~tbj>�F�r �~j>� r y�zsM�AlSo���q| �~�5jM��sc�Kjby��?m&j>k§sų}zsMy��#zsU¶~s���j>�Djb�M�Djbt>lSk����FwDjby3���tIjbo~j�np�5j>m�s$�;�]n�s��j>�M�?oAt>j>�F�r zÑ w«jb�M�?lSk,³�sM�~sM��zjbnpjb�~jbo�jby~�| zsMy�y zÑ n'zsMo&zsMm�sMk � wcsM�M�«³&sM�psM�Òzjbn~j>�Aj�Þt5zj>o?zj>m&jbk��m&jbwMj>tbtqr k'�� JMéçhUJMé��'H1XJcT � °�tÌjb�M�?o~tbj>�F�r zÑ w´jbySm&j�wDj>t5zj>o§µj>y��MjbnAn§sMy�kVsMyS�KsMt>ml#o~t | k]n'zsMk�sctÌjb�M�?o~t>jb�F�r ��q| �~�5jDzÑ wjbySm&j�wDjbt5zj>o�n�l�o>�

#jby�zscn~µ�scn'z| ��µ | �M��j>�M���«j>�c��oAt>jb�?�r zÑ w�k�qr y q| mF��q| t�| ³&sM�psM�Òzjbn~j>�Aj>o�j>yA�| zsMySy3zÑ n�zsMopsMlÓn�zscnp³&sM�psM�Òzjbn~j>�Aj>t5zjbo�ÞoAj>y�ySjbµ&jbn���j>�Mkcsc³&m&l�j>�c����zsMoA��z| y3�� � � � � J � è Xg ê PVX^ � ° � Î9� ¢���� Ï u � �Aj>� r y�zsM�AlSo,�'q| �A�Äzjbn�t�zsM�An~m�sMk©� | m&�V¶ r k5��µ�s�wcsMm | yÓ�]sMm���ÎDu Ï u��«�&l �Yjb�~jbm �>l{zsMySµ�sVn'z| ySjbk�zj>³5j>t5zjbo>�5sM��j>yÓ�§�~jbm&�&jbySkMjbt>l�k«sMt�sMy�zsc�&�&lYn r yQsç¶A� | m&o�zsc� | kFkVsMy3Î� Ø � B�jbm&m�zsMy�y�s ��vñ#� ¢���� � � q| oAo~tbjb±Uqr �D�1zjbo>�� �D��i�jbn~o~t��| ySjb�Dj>o�ã Ç u�jbo~jbn�zj>m�npj>y ¶Aj>ocqr y �� � ãA��� �� � ã�× � � Ê ?�A���� � � � � J � è Xg ê PçX^ � °C��Î�� ¢���� Ï u � t�zsM��nf�'q| �~�Äzj�nfj>�c��oAt>jb�?�r m&jbk�m&j�wDj>tbt�qr k'�]µ�s�s �vñ � ¢�� ��ySj>kfzjb³Bjbt5zj>ol�mV¶Aj>k]n{zÑ w�zjboº�}zsM�~��j>yÓ�Kã Ç�� ¢�� ����j>y�ySj�n~n�±.jbm&m�zsMy�y6� � ãA��ý��� � ?���� JMéçhUJMé��'H1XJcT � ô�jb�M�Djbm&j>kÀÚ ¡ zj>ovÚ ��j>�M�?��zsMo~�~s«��j>�c�| ySjb�Dj>o�j>�M�?o?zj>�MwMj>k]n | � | k¾u � �?�5j>mY� . sMySsM�Kjby��� Ç u � ³ | m]n | n zj>o�Öì íÂo~t�zsc� | n�w�zjbwMj¦npjbk?l�m]npocqr k sMt�n§sF�KÎ � í ¢ ��� � Ï u����&l �Yjb�~jbm �>l{zsMySµ�sVn'z|y�j>kfzj>³5j>t5zj>o�n{�vsM��j>yÓ�Dj�n«s�� � ãA�À� � × � | o'ã�Ú ¡ × §o~l�m�ãÚ � � ã Ç � í ¢ ��� � �Àq| o~oAt>j�±Uqr �D�1zj>oÂzÑ �aySjc��fw?lS�j>m&ob�5µ | �M�«s;�©jb�M� | yÓ�]sMm«jb�M�?o~t>jb�F�r t�zsM�An���q| �~�5jM��sc�Kjby��?m&j>ka³�zsMy��#zsU¶~s�jb�M�AKo r �]sM�Bzr kBq| �b�G é63^ Z 4 3LYOSJVW8XJcZbW�JDOS`ÌJMH�XJcT�J

ô�j>�M�Mj>m sM� | nAn�j>�M� � Î9� Ï u � oAlS�Ks©�'q| �A�Bjc� i1jbkFlSm]npocqr m&kxj>�c� sMt �»lSm]npjb�AwcsMySy r � | mzjb�AnpjbyS��j>tbjbn~n�� Î � Ï u � �l �Bj>�~jbm �>l{zsMySµ&scn'z| ySj>kfzjb³Bjbt5zj>o�n{�´°��Kjbm&m]�?lS�5j>m���zsM�A�Kjby���ã Ç �Âj>o~j�n�zjbmsct� � ãA��wDjbk]n | ��n zr �M�Ìnpjbk?l�m]n7¶cqr k'���Kl�mFn�j>�M� � � ãA��kMj>tb�f�| ³ | m]nYzr l���zsMm]�BzÑ n | n~n�o~t{sckcsco~tbn�sMt�u � j r kDÞy�lS�&jbo~tbl�n�zj>�A�Bjbm��'sMkFk | ��sct����Fn�j>�c�,o~l��as�wDjbk]n | �A�Kjbt�| m&jbk,� | m&�V¶ r k,s;���'q| �~�5jv��j>m]n�zjbm���

Page 8: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

°©n | w#zsM�&�lQsMkF��sMm�s��±Uqr �D�Mw�zj>m]���j>�~lÓw#zsMy�n.¶~sMl�nazr �M�In~j>kFlSm]n.¶cqr k5�]��lSm]n �;�F�KjbmFn~lwDjbk]n | �A�Kjbt�| kcjbn{�° � o~jb�Bjbo~o?zj>�M±dqr �D�cw�zjbm]��zjbmÀsMt�n�s   Î1� Ï uÖySjbk�zjb³Bjbt5zjboAn«zj>��n7¶cqr k'�scµ | y  B� ãA� �É· � � � ãA�ç·�npjbyжAj>ocqr yn~jbn~o~t�| ySj>�Mj>o�ã Ç �M�?�~jc�

� ����� ��� J � è Xg ê PçX^ � ô�jb�M�DjbmÀs;�©oAlS�Ks��'q| �~�5j��~jb� r y�zsc�~lSob�fi�j>kFlSm]npocqr k,s;�©��j>m]n�zjbm¦sctbn�s� Î � Ï u �«wMj>k]n | �A�Kjbt�| n>�Ysc�Kjby��?�~jv±.j>mm�zsMySy � � Ø

  � � �}��t>jbm � wDj>k]n | �~��j>t��| nvs'�Ùzj>�AlSm]nD�| ySj>�Mj>ojb�M�?o?zjb�MwDjbkFn | �~��j>t��| ¶Fzj>mj>k�m&jbwMj>tbtqr k'� � ° � y�j>kfzj>³5j>t5zj>o�n�npsMm&�Djbm �>l{zsMySl�oIjb�M�?o?zjb�MwDjbkFn | �~��j>t��| m&jbkÒlSooAt | kÄzsMoº� | m&��sMml3� �� ��� � ��� J � è Xg ê PçX^ � ° ��Î2� Ï u ���Aj>� r y{zsM�~l�o�o~l��as���q| �~�Äzjbm&j>k«s�ã Ç �§³ | mFn~��sMm«wDj�n~nº�'q| �A�fqr ySj�n�zj>m�s� � ãA� � Ø

 Y� ãA� · � � � ãA��·�m&j>��m&j>�Dscn�zÑ w§o~t�zsM� | n«zj>�An.¶cqr k'�° ±.jbmFn~l�kFl�±.j½¶Aj>t5zj>o~oAj>y���j>�Dµ&scn'zsM� | t | n~n � Î�� Ï u ± | y��Fn | m | oÀy�j>kfzj>³5j>t5zj>o�nÂs�� ��q| �~�;qr y�jbn~l±dqr �M�Mw�zj>m]��zj>m&jbk�� | m&�ç¶ r k'�

� JMéçhUJMé��'H1XJcT � ��jb�Ùm&jbµ#zj>t_�Bjby�zscn~m&l3�?µ | �c�,s;� �~j>� r y�zsM�AlSo���q| �~�5j�ã Ç �a³ | m]n~��sMm,wDj�n~nº�'q| �A�fqr ySj�n�zj>�Aj±.jbm&m�zsMy�y�sMt_sMy{zsM�&�&l�q| o~o~tbjb±dqr �D�1zj>o

� � ãA��� ¹ · � � � ãA��· � �&· � � � � ãA��· � Ê ¨2� � � ãA� ¢ � � � � ãA� ª � Y� ãA��� £ � ��� � �

� JMéçhUJMé��'H1XJcT � ��q| m&m]�Mj>m�lS�]sct | y�µ�scn'z| �1µ | �M�´s ���~j>� r y�zsM�AlSo_�'q| �A�BjK³�zsMy��#zsU¶~s�j>�M�Õj>�c�Dj>mj>o~�Aj«j>o~l�ksck?k | �«zj>o �>o~sMk«sMkFk | �b��µ�sÅs�� �'q| �A�fqr ySj�npl5±Uqr �D�Mw�zj>m]�Dj_j>yÓn?�r m&l�k'�� ��� � ��� J � è Xg ê PçX^ � ô�jb�M�DjbmÀsM�wcs�j>�c� ��Î � ¢���� Ï u � �~jb� r y�zsc�~lSo#�'q| �~�5jM�° � � �Ä����� ^` � � ãA�  B� ãA�&ÈDãåo~t�zsM� | nºs'� npjbyжAj>o}��q| �~�;qr y�jbn�zj>m&jbkam&j�wDj>tbt�qr k'�

ô�j>�M�Mj>m�»Î � Ï u � j>�M� | y��DsMmÀ�Aj>� r y{zsM�~l�o���q| �~�5jM�YsM�Kjby��?m&jbkÌs�ã Ç ��µ&j>yÓ�Dj>mÀwMjbnAn��'q| �~�;qr ySjbn~jmj>�þn?�r mlSk�j>y � wcsM�c��l�o � � ãA�vý�Öí]�U��° � ��� � �vq| o~oAt>j�±Uqr �D�1zj>oA�f�| y�sM�}z| �&lSk'�&µ | �M�«sMt� � ãA��� � � � � ãA� Ê ¨1� � � � ãA� ¢ � � ãA� ª � � ãA� wMj>k]n | �ºmj>� ySjbµ&jbn�m r y�y�wMj>k]n | �b�� ���&! � � J� è Xg ê PVX^ � °�t � ãA��� ¡

� ��������� � � ãA� wDjbkFn | �Anvs����'q| �~�5jIã�³ | m]np�5j>y�l�±U�| m | �~��zsMy�lSo�j>�M�?o?zj>�cÞwMj>k]n | ��zsMm�sck,m&jbwMj>t>t�qr k'�°x±U�| m | �~��zsMySl�o#j>�c��o?zj>�MwMj>k]n | �~�pscy�n~j>yжAjbocqr y�m&j>kas � � � ãA���  B� ãA� � � ãA�� � ãA� zjbo

� � � � ãA���   � � ãA� � � ãA��×  B� ãA� � � � ãA�� � ãA� � ��� I �q| o~o~tbjb±dqr �D�1zj>o~jbk'�#¼½y��a�©z| � | m�s;� ã3�?�BjbySl���q| �~�;qr y�jbn�zj>�~j�±.j>m&m�zscySy�sMt_sMy{zsM�&�&lYj>�M�Mj>m&yA�| o?zjb�

� � ãA��� ¨1� � � � ãA� ¢ � ãA� ª Y� ãA� � £ � ��� � �G é63^ Z 4 3LYOSJVW�é?J ^ `ÌJVWbZ>PS\?P�hUJDOSOSJMé?J

ô�j>�M�Mj>m»sM� | nAn«j>�M�C� Î�� Ï u � �~jb� r y{zsM�~l�oa�'q| �A�Bjc��n | w#zsM���zsÒj>�M�©u � �?�BjbySl�Í lSt | ��jbnp�AlQs�sc�Kjby��Dj�nIsMt ��Ø � Ø �Åq| o~oAt>j�±Uqr �D�1zj>oKzÑ �vy�jM�Åi�j>kFlSm]npocqr k�s=� Í oAt>jb�~lSm]n~l�k�zjb³Äzjbn>�1wcsM�M�?l�oIs ��x�ÙÍ�) ��Aj>� r y�zsM�AlSo��'q| �~�Äzj�n{���fw?lS�&jbm&o>�'µ | �c�a±.j>mm�zsMySy�m�sMkKs �� � �»Ô0)�� � ¢ �  �  '¢ �� �ÖÔF) � ¢ �� ��xÔ+) � �

Page 9: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

q| o~o~tbjb±dqr �D�1zj>o~jbk'����m&m&jbk,kYq| wDjbn~kMjbtbn�zjb�Bjbm¾lS�]sctvsMt_sMy�zsc�&�&l�kFlжAjbySjbmFn�zj>ob�� ����� XG�OQO Xg W�X\?T ��� � K- ;�����ÖÍ0) � D -�/ R M�K3 D =V; /BAC'DFE K-�T � K- ; �� /BACHDHE AR Me- P = Y AR�/ /�7 K-�2 1�- = > -�/ -0/�1�- N 2 -�T G

°æ��� � � .�j!&&m�zÑ �bl�z| m�zscy�ySjMzÑ �AnpsMkFm�sMk��Kjb�M±.j>y�j>yA�| j>m§wMj>�M��qr k«s;�©�Aj>� r y{zsM�~l�o��'q| �A�Bjvjb�M�f�Ò��� )&c�Î a Ï u � zscn~³�sM�~sM��zj�npjb�~j>t5zj>o?zjbn{��° � �?� Ø �#jb�M�Djbm&y~�| o?zj>�M�f�| y�sM��z| �lSk'�&µ | �c�anpjbyжAjboMqr yf B� d ��� ñ cH� �ed �çñ �  Y� c �ed �~� zj>o f� �ed �_�����?� � c � � � � � c �ed �~� n~jbnpoAt�| y�j>�Djbo d�Ç a5�?�AjM�§¼½yÓ�¾��z| � | mkBq| mmF�Mj>mÀl��]sMt | ySµ�sVn'z| sÅkBq| wDj�npkMjbt�| zscySy3zÑ n'zsMob�� ��� � XG�OSO½Xg W�X\FT ��� � K3 P�� 3 D 3 > K- P - D - N K- ; K-'7H-�M62 1�- D�P f��� ��) c D -�/�R M K3 D =?; /BACHDHE - /BAC'DHE AR�M�- P =�Y AR�/ /�7 K-�2 1WK- D -Y -�2�2[K3�M?M f� � � )&c G� JMéçhUJMé��'H1XJcT � i�j>�c��qr k�±.jby3�Vµ | �M�vs �À�Aj>� r y�zsM�AlSo��'q| �A�Bj �'q| �A�fqr ySj�npj�oAj>µ | yFoAj>� n?�r m&l�k�jby3�1��t j>oAjbn~�BjbmwMj>µ&j�n7¶Vqr k«s;�;�?µ | t_�Aj>m&�j>y�n ,Î � Ï u���±U�| m | �~��zsMy�lSo j>�M�?o?zjb�MwDjbk]n | �A�Kjbt�| n>�° � ��� �M��q| o~oAt>j�±Uqr �D�1zj>oA�f�| y�sc��z| �&l�k5�Yµ | �M�Àsa±U�| m | �A�©zsMy�lSo�j>�c��o?zj>�MwMj>k]n | �Il���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| zscn~³�sM�~sVÞ�Òzjbn~j>�Aj>t5zjbo~mÄzjby1oAj>�þw#zsMyÓn}j>yA�| ¶Aj>y�jbn{�&sMt{sMt f©� 0) c npjbyжAj>ocqr y3�GIH¦Xg [�� ^ T�TUHKT�HçJcZ{P è WbPfeY\FZb\�`ÖXJcW�JMZbJMH1XJMT� ����+ � � J � è Xg ê PVX^ � ���M� �ËÎ � Ï u � �~jb� r y{zsM�~l�o��'q| �A�#zj>�c�| y}sMkFk | �K� | m&�V¶ r k5� µ | �c��sMtÂzÑ w?µ | oAo~toAt>jb�~lSm]n~l�³&sM�psM�Òzjbn~j>�Aj>t5zjbo~�5j>m�wMscm���j>�DsM�wcs�&µ�s¾· � � � ãA�ç·�� Ø npjbyжAj>ocqr yYnpj�npoAt�| y�j>�Djboºã Ç �D�F�~jM�� JMéçhUJMé��'H1XJcT � ° ±.j>m]npl}jbySm&j�wDj>t5zj>o�n,sMt«lSm� | k | yж~s��µ | �c�åzÑ w?µ | oAo~t«o~tbj>�AlSm]npl}³�sc�psM��zjbn~j>�~jbt5zjbo~mÄzjby�s��ñ(� ¢����©� ¢��5Ç � ¢ aò � ���'q| �~�5j>��sM�~sM�ÕzÑ w?µ | oAo~t�zsc�ps�±.j>m&m}zsMySy $ � �vñ#� ¢���� ��� � Ê ��� JMéçhUJMé��'H1XJcT � °ÖoAt{sMkFl�� | ��sMy | ����sMmKsMt�zÑ w?µ | o~oAt�oAt>j>�AlSm]nplB³�sM�~sM��zj�npjb�~j>t5zj>oA�~j8zr �M�ÅoAt | k]n�zsMk�±.jbySµ�zÑ w?m&ls&��c�Dj>y��Kj�n{�çµ | �c��s}�'q| �A�Äzj>mÄzj>y���?oAj>yb¶Aj>y q| ySl�k�s�³�sM�~sM��zj�npjb�An � wcsM�c��l�o�s#w#zsMy�n | t�z| nU������m&m&jbkv�Kjb�M±.j>y�j>yA�| j>m��µ&sÀj>t>jbm¦¶Aj>�M�?t>j�np�5j>m s �æ�'q| �A�Äzj>mÄzj>y�s̳�sM�psc��zj�npj>��n���?oAj>y�¶Ajby½q| y ¶cqr k5�1sMkFk | ��sMt,sMt�nº¶AjbySj>m]n~l3�fµ | �M�· � �3·}� Ø npjbyжAj>ocqr y3�� JMéçhUJMé��'H1XJcT � �fw?lS�&jbm&o>��µ | �M�8zÑ w?µ | o~oAt�o~tbj>�AlSm]npl1³�sM�psc��zj�npj>�Aj>t5zj>ovj>oAjbn�zj>mÂs§�&j>�Al�w#zsMyÓndsMkF�psÅnpjbyжAj>ocqr ys ¨1� � � �ç� ¢ � � � � �ç� ª �æí � � Ç ���Ùq| o~o~tbjb±dqr �D�1zj>o>��° � �*� ��jb�M�Djbm&y~�| o?zj>�M�f�| y;³5j>�lS�,sMtIkBq| wDj�npkMjbt>l�k5�µ | �M�«s;�©��q| �~�5jv�'q| �A�fqr ySj�nplB±dqr �M�Mw�zj>m]��zj>�~j�±.j>mm�zsMySy � �$· � � �3·ç�

i�j>kFlSm]npocqr m&k8jb�M�©zÑ w?µ | o~oAt«o~t>jb�~l�mFnų&sM�psM�Òzjbn~j>�Aj>t>j�n~n'�ÙÎ � Ï u � ��q| �~�Äzj�n{����q| �DtDzÑ npocqr mkÕjb�M� Ç �¦³�sM�~sM��zjbnpjb�?zjb�An�zjbkMj�n{�a°�t� �À� ð�� Ê _ñ�� Ç ��ó§lSm]n~j>�AwcsMy�y r � | m¦wMj>�M��qr kÂsMt�n�sct��»Î� � Ï u±dqr �M�Mw�zj>m]�Fn{��scµ | yvsMt�� ��? � �@? Ç ���K±Uqr �D�Mw�zj>m]��zj>��n�zjbk©�Kjb�Dj>�c�Dj>tblSk sctxzj>�~l�m]nD�| k lS��zscmF�#zscn§�Kjb�]sM��z|� � ?� ¢ � � �× ? �#jb�M�?o?zjb�MwDjbkFn | � | k�µ&sç¶Ay�zsMoAo~tDq| �1zjbwMj>yä�Ä��oAt>j>�AlSm]nºnpj>y ¶Aj>ocqr y� | o�� �@? �;�²¨ � � ?� ¢ � � �× ? � ª zj>o í 7 � �@? � 7 ���� �"! � XG�OQO Xg W�X\?T ��� N�� Y AR�/ /�7 K-�2�2 1�-�M Y -�2�2[K3:MVM 3 y�lS�

� �"� ��? �ñ ? ñ � � � ?� AC ;0; N - Y AR�/ /&K- ; G

GIH¦Xg [�� ^ T�TUH è \FNÕ`ÌJcé �7JDOSJMO �^ X\]WbeB\FZ>\?`ÖXJVWbJcZ>JcH1XJcTô�jb�M�Djbm sc�wcsÌj>�c�F� Î � Ï u � �~jb� r y{zsM�~l�o_��q| �~�5jM����q| �DtDzÑ npocqr mk¾j>�M� Ç �xzj>��n�zjbkMjbn>� .�j>�c��qr kÂs� � ãA��� � �` · � � ��? �ç·�È ? kFl�±.j½¶Aj>t5zjbo~oAj>y5�Kjb�Dµ�scn�zsM� | t | n~n � Î � Ï u©±Uqr �D�Mw�zj>m]�Fn{��°å� | m | n | mÅm�q| wMj>k]w��|� ±Uqr �D�Mw�zjbmF� zj>��n�zj>kFkfzj>oAt>y�jbn�zj�nakfzjb³Bjbt�| lSm]npjb�AwcsMy�y r �¸ySjb�M�Djbm af�Âi1jbk?l�m]npocqr k � �?m�sMk8s c$ÎJa Ï ul�m]wDj>�At_±dqr �M�Mw�zj>m]��zjbn{�°�f��� �;) c¾�'q| �A�Bj�s��;�?m�sck�j>�c��o~³5j �bl�zsMy�lSoÅzscn~³�sM�~sM��zj�npjb�~j>t5zj>oAjM�?�KlÓwDj>y&sMt | mm�sMy�sM�}z| �&lSk'�]µ | �M��}zsM�~��j>yÓ� dºÇ aÕj>o~j�n�zjbm¦±.j>m&m}zsMySy#· f� � �ed ��·º� Ø �Ä��o~t>jb�~l�mFn�s f��zÑ w?µ | oAo~t_o~tbj>�~l�m]npl�³�sM�~sM��zj�npjb�~j>t5zj>oA�BjbmwcsMm���j>�]sM��wMs��

Page 10: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

G TbPS`,LYOVX^ TcXg N XJMT�\«T�PQ`,LYOVX^ NH3^ Zô�j>�M�Mj>m�sM� | n~n�j>�M�8zÑ w?µ | oAo~t�oAt>jb�~lSm]nv³&sM�psM�Òzjbn~j>�Aj>t>j�n~n �xÎ � Ï u � ��q| �~�5jM� � ° �;�?�~saj>oAt>jb�~lSm]n

l��]sMt´· � �ä·Ì� Ø � � ��q| �DtDzÑ npocqr mkÕjb�M� Ç �8³&sM�psM�Òzjbn~j>�?zj>�An�zj>kcjbn{�Kzj>o�npjb�M��qr k8±.j>yä�ĵ | �M� � � ?�Õý�àín~j>y ¶Aj>ocqr yä�f° ����q| �~�5jXM�F�BjbySl�zjb�~lSm]nM�| j>�M�Mj>m&jbo?zj�n�¶Ajby½q| y ¶Aj � �

.�j>�c��qr k�sMt�nvs #�Î � Ï uæ±Uqr �D�Mw�zj>m]�Fn{�YsM��j>y��Mjbn�s # � �ç�}�Ù¨�� � �ç� Ê � � ?� ¢ � ?� ª q| o~oAt>jb±dqr �M�;zj>ozÑ ��ySjc��ôIzsVnpµ�scn�z| �&µ | �M�«s$#?�?�ps�±.j>mm�zsMySy:# � � ?���Öí zjbo # � � � ���� � � ?�}ì©í�°Ò±.j>m]npl�j>kvoAt>j>�AlSm]n�n r � r m&k�wDjbm&m&l?j>�M� | yÓ�]sMm � � ���åÇ-� ��o~t�zsM�Ko | � | t{scn | n{�csM��j>y��?�Ajfnpj>y ¶Aj>ocqr ySm&jbksctIscy�zsM�&�lY±.j>yÓn�zjbn~j>y�j>k'Î� Ø ��i�jbn~o~t��| ySjb�Dj>o �åÇ � �?�~j�±.jbm&m�zsMy�y � � Ç � zj>o � ��ý�9��� �D��° � � � �#��³ | m]n�m&lSm �bo��psç¶�nps�sct � zj>�~l�m]nD�| jb�M�Djbm&j>oAj>m����� ��°�t�� � ���åÇ � ��o~t�zsM�Ko | � | t{scnºk | mFwMj>�A��zsMy�c�?µ | tc�'sMt{sMt ySl��� � � � � ��

UMj>y q| yжAj�� ��sMtbn�s����?�&l��Kjbm&t>l{z| o}o{zÑ k | n>�'sM�Kjby��KndsM��ndsMy��asMtbt{s�sMt � zj>�AlSm]nD�| j>�M�Mj>m&jboAn�zj>o�s�� � � ���³ | m]n | n{�<.�jb�M��qr k n | w#zsM���zs�sMt�� � o>zÑ kF��sMm�sMt�n«s�� �þkBq| ��n{��sM��j>y��þzscn~µ�sMyQsc� s � � ?� ¢ � � � ���³ | m]n | k | m�zj>oºsM��j>y��?m&jbk§sMt � zjb�~l�mFnM�| ¶AjM�� �,+ � XG�OQO Xg W�X\FT � � N�� � ; K� T ; C'DQC N 3 P T C 2 7'- D / K3�M 3�X�X C N 3 � � ?� �.C 2 P C 2%K3 P > -�2C ; K� T6X C N�� 3 > -�M#1 - P3 � � � ?� K- ; � � � � ?� 7H-�T P C'D C T Y - ; N K� P -�2 -�T T = G� ��� � � � J� è Xg ê PVX^ � °�tbn�s � � ?�v³ | m]n | m$zscn~�Kjbmf�| o>zÑ k | n{��sM��j>yÓ��³�zsM�Aµ r t{sM� | o�s=� � � ?�¾zj>o�� � � � ?�wMj>k]n | � | kFkVsMy3�'s'� �'q| �~�5jXKµ&jby��Mj>mÌwMjbnAn�oAlS� r y�z| o>zÑ k>¶&zsMm�sMk�m&j�wDj>tbt�qr k'�� ��� � XG�OSO½Xg W�X\FT � � � � TJAC'D ; C'D C N 3 P T C 2 7'- D / K3�MJ3�X X C N 3OT ACHD XZAC N � 3 > -�M#1 2 -�T ; K� T��Q3 3 N �.C 2 P�E -�M =; =V> R�M�KC ; K� T � ; R�/ 3 D 3 Ø

� � ?� K- ;�� -�2 P�D R > 3h3 � � � � ?��× Ø� � ?� � �� � C 2 P G

� ��� � � J� è Xg ê PVX^ � °�tbnas¾kYq| �An{�#sM��j>y��?m&jbk´o>zÑ k>¶~sÕs� M�F�BjbySl�oAlS� r y{z| o>zÑ k>¶~s��o r �]sc�ps Ø� � ?� zj>o

�bj>m]np� r �KsKs � � � � ���× Ø� � ?� � ?�»³ | m]n{�'s;� �'q| �~�5jXK³ | m]np�5j>y�l�o~lS� r y�z| kBq| �?zj>m&jbka� | m&�V¶ r k5�

GIHKJcé �'TUHçJMZ �L Xg [��5JMHÅWb\FZ�W ^ HfX^ é?J ^ `ÌJVW�Z{PS\?P � ^ é�\?OQ`�\?Nô�j>�M�Mj>m,sc� | n~n#sMt�u��aj r kFySlS�j>o~tbl5n�zj>�A�Bjbmaj>�c� �Õj>�M�?o~tbj>�F�r zÑ w5�f�fw?lS�&jbm&o>��µ | �M�Ks �8npjbn~o~t��| ySjb�Dj>o³ | m]n7¶&zsc��sMm zjb�An~j>yS��j>tbµ&jbnM�| sMtåzjb�~l�mFnM�| j>�c�Dj>mj>o zjbo�sÂ�'q| �~�;qr y�jbn{��°��Kjbm&m]��l��Bjbm�s / Ç � ³ | m]np��sMmmj>� n?�r mlSk�j>y�s«�'q| �~�;qr ySjbn>��sMkFk | �_�j!&�m&l{zsMySmlfySjbµ&jbn_sMtÅjb�M�?o~t>jb�F�r zÑ w / ³ | m]np�5j>y�lfo~l�� r y{z| o{zÑ k{¶&zsVnÀzj>ooAlS� r y�z| kBq| �?zjbn�lSob���t>jbmÒy | kÄzsMy�lSo��Dj | �Kj�np�~lSsMlB¶Aj>y�ySjb�Kt��| k�sMm�sMy�l�n~lSk r o��Kjb�Dµ�scn�zsM� | t�zsMo~s | y�����z| � | mÕw�zj>�Djbt>µ&j�nD�| j>yä�µ | �M��sMySkVsMy��asMtbt r k�sct¦j>�c��oAt>jb�?�r zÑ w j>�M�?l�k ³&sM�psM�Òzjbn~j>�Aj>o§jby~�| zsMy�y zÑ n'zsMo&zscn � wcsM�M�?l�o§sMtÌj>�M�?lSk | yÓ�]sMm��Î � Ï u � jb�M�?o~tbj>�F�r �~j>� r y�zsM�AlSo���q| �~�Äzj�n{��sc�Kjby��?m&j>ka³�zsMy��#zsU¶~s���j>�Mj>�M�Mj>t>l�k ���FwDjbyÃ�U�i�j>�~�Òzj>oAt>j�npj>oAj>m zjb�An~j>yS��j>tbm&lfy�j>µ&j�n_sMt�j>�M�?oAt>j>�F�r zÑ wÀj>�c� q| o~oAt>j�±Uqr �D���| k | ��³�sMk]n���sc�psM�?¶&zscm�sMk¾sµ | o~oAt�zscn�zj>oºs�npj>y ¶Aj>o}�'q| �~�;qr y�jbn�zj�n{�

Ø í

Page 11: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

�B)�*O��C-�+ � -�+]C-���25E � C� / � E C� 41@2 GçÚ C=F9GIH X\?O�W�\?OçX\ è5^ T_é63^ Z 4 J§N�Xg TBXJMZ �^�� ZbJ è JcW@2 4 X\FHVPSTU\! ����� � J � è Xg ê PVX^ � ô�j>�M�Mj>mÀsc� | n~n�jb�M� ��Î � Ï u � oAlS�Ks��'q| �~�5jM��° ���KjbmFn�zj>m«wDj�n~n� ¡�¢{£{£{£b¢ � � Î � Ï u � o~lS�KsºwMj>k]n | �~��j>t�| k?�c�| y�sMt�n�� | m&�V¶ r k'�Dµ | �M��sMt | k��;�?m�sckIjb�M� | �An | m | �~��zsMyÓnk5zÑ o?zj>�c�| �}zsMt>l�o�zscn�scySk | n7¶&zsMk'�çµ�s}��zsM�A�Kjby��ºã Ç ��jbo~j�n�zj>m�s � ¡b� ãA� ¢{£{£>£>¢ � � � ãA�BwDj>k]n | � | k�j>�M� | �An | m | �~��zsMyÓn�}zsMt>l�o�zscnºk�zj>³5j>tblSk,sMtvu � wDjbk]n | ��n�zj>�Am&j>k'�! ��� � � J� è Xg ê PVX^ � °��©Î � Ï u � oAlS�Ksº�'q| �~�Äzj�n�zsMyÓndsMy{zsMm | o�n{zÑ ³ r oYzr m�sMk�� | m�V¶ r k'�Mµ�s�npj�npo~t��| yS�Mj>o1ã Ç �µj>y��Mj>m¦s'� � � ãA� ¢ � � � � ãA� ¢{£{£{£b¢ � � ��� ¡ � � ãA�#wDj>k]n | � | k,jb�M�,y�lSm&j�zsM�~l�opsMma±dqr �M�Djbn~ySj>m,�Aj>m&�o~t>jb�An�sMySk | npm&sMk'�� JMéçhUJMé��'H1XJcT � ° ±.j>m]npl �&j/&�m�zÑ �bl�z| zj>�An~j>y���zjb�BjbmÒsMtKu � �?�5j>y�l��~j>� r y�zsM�AlSo��'q| �A�Äzj>kÂj>�c���5j>m zscy�ndscy�zsMm | on{zÑ ³ r oYzr sMk'����� � j>oAjbn�zj>mÌsMt | kKsMt�u � �?�BjbySl5�'q| �A�#zj>k'��sM��j>yÓ�DjbkK³�zscy��#zsU¶~sIjb�M�§jb�M�Djbm&j>oAo~t>sMkVsMo~tc�Bm&jb�zscy�ndscy�zsMm | o#n�zÑ ³ r oBzr sck5�° k�zjboM�| ��&lSjbk_o | ��zscm�sMtazsMy�npsMy�zscm | o�n{zÑ ³ r oBzr �'q| �A�Äzjbn1��q| w�l��&j>m �>o~sMkÕzsMy�npsMy�zsMm | o��'q| �A�Äzj>mj>k�� | m&�V¶ r k'�

� JMéçhUJMé��'H1XJcT � ô�jb�M�Djbm �öÎ�� Ï u � j>�c� | y��DsMm»oAlS�Ks´�'q| �A�Bjc�vsMµ | y�wcsMySsM��j>y�� ã Ç �©³ | m]nAÞ�&sMm¾s'� � � ãA� ¢ � � � � ãA� ¢>£{£{£>¢ � � ��� ¡ � � ãA�}wMj>k]n | � | k«ySl�m&j�zsM�AlSo~sMm,±Uqr �D�Djbn~ySjbm&j>k'����tIjbo~j�np�5j>mÀwcsMm | y��]scmba�?zj>oAt>lSm]n~j>�AwcsMy�y r �ÙsMt��M�?�Bjbm��'sM��j>yÓ��mÄzj>y�l��]sMt�ã Ç aæzjbo�s �vñ a ��q| �~�5jÂzsMy�npsMy�zscm | o#n{zÑ ³ r oBzr �! �"! � � J� è Xg ê PVX^ � ô�j>�M�Mj>m�sM� | n~n§jb�M� �àÎ � Ï u � zsMy�npsMy�zscm | o�n�zÑ ³ r oBzr ��q| �~�5jM�»° � �Kjbm]n�zj>mwMjbnAn � ¡d¢{£{£{£b¢ � � Î:� Ï u���wDjbkFn | �~��j>t��| k�k5zÑ o?zjb�M�| B'�Aj>m&j�n��?�}zsMt>l�o�zscnIsMy�k | n.¶&zsMk¾s �x�'q| �A�Äzj>mj>k'�fµ�sn~j>y ¶Aj>ocqr y�m&j>kasMt_sMy�zsc�&�&lY±.jby�n�zj�npjbySj>k'Î� Ø ��° � ¡U¢{£{£{£b¢ � � wDj>k]n | �~��j>t��| k«s;���Fm�sMkajb�M� | ��n | m | �~��zsMy�n#k5zÑ o?zjb�M�| ��zsMtblSo�zsVn�k�zjb³Bjbt>l�k5�� �D�¿° � � � ãA� ¢>£{£{£>¢ � � ­ � � ãA��wDj>k]n | � | kxzsMy�npsMy��Kjb�Dµ�scn�zsM� | t | nAn�u��?�?�5j>y�l1y�lSm&j�zsM�~l�o�scy�n�zjb����j>�Mj>�M�Mj>t>l�kÀs� ¡>� ãA� ¢{£{£{£b¢ � ­ � ãA��wDjbkFn | � | k¿zscy�ndscyf�Dj>mj>��zscy�nIsMy�n�zj>�A�~j>y1npjbn~o~t��| ySjb�Dj>o ØA7 ?87 � Ê Ø l�m&�&j�Û zjbo_ã Ç �³&sM�psM�Òzjbn~j>�}jbo~jbn�zj>m����� �,°�t�� � � � � ãA� ¢{£>£{£>¢ � � ­ � � ãA�A��y�lSm&j�zsM�~l�o1scy�n�zjb�~�5j>m�s � � � ãA� ¢{£>£{£>¢ � � ­ � � ãA�©zj>o � ¡b� ãA� ¢{£{£>£>¢ � ­ � ãA����zsMtblSo | kr �M�]sMm&sMtbn�sMtvlS�&zsMm]�BzÑ n'zscoAn�k�zj>³]w?lSoAj>ySl�k§��zsM�A�Kjby�� ?DÇ ð ØD¢{£{£>£b¢ � Ê Ø ó ¢ ã Ç �§��j>ySy�jbnAn{�� I ��° � ¡b� ãA� ¢{£{£{£b¢ � � � ãA�#��zsct>lSoºsMt�u��«npjb�~��zj>oAt>jbn~j>o | �~l�j>m]n'zs �>l�z| ¶&zscn�sM�V¶~s��

° �&j/&�m�zÑ �bl�z| sMySsM³?¶&zsMm«kBq| m&m]�Djbm¦�5j�ySj>µjbnºy�zscn~m&l3�?µ | �c�,lS�]sct�sMt�sMy�zsc�&�&l�kFlжAjbySjbmFn�zj>ob�! ����� XG�OQO Xg W�X\FT � I�K3 D > -�M#1 �©Î � Ï u � K3�M P 3�M�K3�2 C ; P K� � R ; KR /BACHDHE K-�2 -�T -0/�1WK- D�P -�M > R -�2 M6K- P - N = T 3 T�K� ; K- D C� D -�2 - P�� E K3 N =V; 3 G

° �ÉÎ � Ï u �²zsMyÓndsMy{zsMm | ov�'q| �~�5j�B��~jbm&jbn �?��zsMtblSo&zscn�s N��~sM��� ���dµ&��lS�n �F±{zj>y�j | �An | � | m&sMySl�t�zsMy{zsMo~ljbyж&zsM�&zsMo~o~sMy�kVsM³&µ&scn7¶ r ka�Kjb�&��vlSl�m&� r y{zsMo~kfzj>m]n�wMj>�M��qr k�sMt=�ÕlSm]npjb�AwcsMySy r � | m zj>��npjbyS��j>t>j�n~n � ¡ � � �Åzjbo � ¡ � ¡

��� � � ��wDjbk]Þn | �A±dqr �M�Mw�zj>m]�DjbkMjbn>�#°�t � ¡U¢{£{£{£�¢ � ­ � ¡ zjbo � ¡U¢{£{£>£>¢ � ­ � ¡ y�j>kfzj>³5j>t5zj>oAj>k«�Kjb�Dµ�scn�zsM� | t�zsMo~s r n'zsMmÌsMt

� ­ � ãA����� � ­ � � ãA� Ê­ � ¡�C ® ¡ ¨��

� ­ � � ãA� ¢ � C � ãA� ª � C � ãA� � � � Ø �

� ã Ç ���q| o~oAt>jb±dqr �M�;zj>oAo~j>yIzj>�An~j>y��Kjbt>t�qr k«sct � ­ Î � Ï u � ��? �æ� ¢{£{£>£>¢ ���ºySjbk�zj>³5j>t5zjboAn>��°���j>m&m]�?lS�5j>m±.jbm&m�zsMy�y}� 7 ?�7 � Ê Ø � sMkFk | �Ås B'�Aj>m&j�n��?�}zsMt>l�o ? �?jb�&lSkÂj>y�j>��zjbnÅs � ­ � ¡

� ��� � �­ kFlÓ±.j ¶Aj>t5zj>oAo~j>y

m]�Mj>�.¶Vqr k'�-.Ìzj>�Yqr y1s � � wDjbkFn | �~��j>t��| n���zsM�ºj>�M��zjb�AnpjbyS�Ò�r j>m��Kjb�Dµ�scn�zsM� | t>t�zsckÌs � ¡U¢>£{£{£>¢ � �&� ¡ wDjbk]Þn | �A±dqr �M�Mw�zj>m]�Djbk5� ° � � kFl�o~t�zsc� | y{zsMo�zscµ | t �5zj>ySoAt>jb�?�r sct � Ø � �D�ı | �~� r y�zscn�sMySkVsMy��asct>m&lä�ØDØ

Page 12: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

G���\FZ�Wb\ è 2'`xX\]WbZ>P54�XJcT_\«é 3^ Z 4 3LBOSJcWbP � 3LBé?é?[ºXJ è �JcNô�j>�M�Mj>mÌsM� | nAnºs �©Î � Ï u��»zsMy�npsMy�zscm | o#��q| �~�5jM�&��j>yÓ�?m&j>kKk5zÑ o?zjb�M�| B��~j>mjbn��F��zsMt>l�ops � ¡d¢{£{£{£b¢ � � �°��&jb�~lÓw�zscy�zsMoAopsMy�m]�Mj>�An � �­ Î � Ï u � ±Uqr �D�Mw�zj>m]�,m]�?lSyÓw#zsMm«k?lÓ±.j ¶Ajbt>µ&j�nD�| s B��Aj>m&j�n��?��zsct>lSo�jbySj>��j>lÓwDjby�s

� �­ � ��C ® ¡ T

­� C � � C

± | �~��zsM��sMm��VsMµ | y T ­ � C ��?~¢LE � ØD¢{£{£>£>¢ ���Y�&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| ±Uqr �D�Mw�zj>m]��sMt ��³�sM�psc��zj�npj>��ndsM��n | ��zsMm]� | m��! �,+ � � J � è Xg ê PçX^ � ° ���'q| �A�Bj���sM��ndsMm?�F�©zscn~�~l¯Û}zscmÌs�ã Ç �§µ&j>yÓ�DjbmÀsMt�n�sMt����?j>�&�Aj>m&� �r k]wMsc�&�pscn~lSk r oW � ãA�}��zscn~�~lÓÛ | nazj>�An.¶cqr k'��sM��j>y�� ? �?jb�&lSkKo | �&zsM��sMmÌs E �?j>�&l�k§j>y�j>� zj>³&³5j>m T ­ � C � ãA���� JMéçhUJMé��'H1XJcT � ° ±.j>m]nºySjMzÑ �AnpsMkKoAt>jb�~lSm]n�s�ã Ç �,³ | m]np��scm«wMjbnAn���sM��ndsMmF�?�©zsVnp�~l¯Û§jbySjb�KjblS�~j�±.j>m&m�zscySy

T ­ � C � ãA���²¨ � �­ � ãA� ¢ � C � ãA� ª ��?~¢ E � ØD¢{£{£>£>¢ ��� £

° kBq| wDj�npkcj>t�| n�zj�npj>y'sc�~�ps�� r ndscnf��zs�]µ | �M��s���sM��ndsMm?�F�©zscn~�~l¯Û�í��Fn'z| y�k�qr y½q| mF��q| t��| j>y�j>��j>l�nfs � � � Ø �q| o~o~tbjb±dqr �D�1zj>o~oAj>y��&j/&�m&l{zsMy�n � ­ ��? � ØD¢{£>£{£>¢ ����wMj>k]n | �A�Kjbt�| k_±.j>y�µ�sMo~tbm�zsMy{zsMo�zscwcsMy?ySjbµ&jbn;�Kjb�Dµ�scn�zsM� | t>m&lä�! ����� * XJcW�JDO ��� /:1 � Î� Ï u � K3�M P 3�M K3�2 C ; /�AC'DHE - 3 DFP 3�2 � > K3 P D =�� K3 D 3 P -�M ��- ; AR M?2 -�T 3 N 3�M�K3 E E =T = ��-�M�-�2 P K- ; -�T G ���� ILK3 D > -�M(1 ã Ç � - ; - P K-�243 W � ãA� > K3 P D =�� 3:2 P =V; N =V>h> - P D = T R ; G ���� � > -�2�2 1 = E -�2 Y -�2�2[K3:MVM ? × Ø ò E � 3:T T C'D T ­ � C}�Öí G ������ -�M ��- ; AR�M

T ­ � ­�� ¡ �¨�� � ­�� ¡ � ¢ � ­�� ¡ ª· � ­ · � · � ­�� ¡ ·

· � ­ · ��? � ØD¢{£{£{£�¢ � Ê �D� ¢P C 7 K3 E�E K3

T �&� ¡ � � �¨�� � � � ¢ � � ª· � �&� ¡ ·

��� · � � ·· � �&� ¡ ·£

ÜÌl�mFn�l�o~��j>�Ajbnpjbo>�cs�KÎ>� Ï u����'q| �A�Bj   Ç � � ���YoAj>�5j>o~o?zj>�M±dqr �M�Mw�zj>m]��zjbn�s  B� ãA��� · � � � ãA��· � ã Ç ���± | �~� r y{zscwcsMy�sc�n r ka�Kjb�&�! ��� � � J� è Xg ê PVX^ � °�t �ÕlSm]npjb�AwcsMySy r � | m zj>��npjbyS��j>t>j�n~n�� ­ � Ø

  ¨ � �­ ¢ � ­�� ¡ ª ��? � ØD¢{£>£{£>¢ � Ê Ø �±dqr �M�Mw�zj>m]�Fn�s'�þzsMy�npsMy�zsMm | o��'q| �A�Bj ? �?j>�lSk§�'q| �~�;qr y�jbnpl'±Uqr �D�Mw�zjbmF��zjbm&j>k«� | m&�V¶ r k'�G � ZbJ è JVW@2�N�XJce�OSJVWbJcN

° � � Ø � i§zjbnpjby�sMySsM³?¶&zsMm�s©k5zÑ o?zj>�c�| B��Aj>m&j�n��?��zsct>lSo�n�sMy�k | n�z| wDj>k]n | �~��j>t��| kx�&jb�~l�w#zsMyÓn7¶~sMlÓn¦kFlÓ±.j�Þ¶Ajbt>µ&j�n7¶Vqr k,sÅ�'q| �A�fqr ySj�npl5±Uqr �D�Mw�zj>m]�DjbkFkMj>y1sMt�sMy{zsM�&�&lB± | �~��zsM��sMmYÎ

� � ¡ �   � ¡ � � ¢� �­ � Ê   � ­ � ¡ � ­ � ¡ ×   � ­ � ­�� ¡ ��? �x� ¢{£{£{£�¢ � Ê Ø � ¢� �� � Ê   � �&� ¡ � �&� ¡B£

Ø �

Page 13: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

° ±.jbmFn~l q| o~oAt>j�±Uqr �D�1zj>oAj>kcjbn/B��Aj>m&j�n��?kfzj>³ySjbn~j>kFm&jbk�oAt | kÄzsMoºm&j�wDj>tbm&l3�� JMéçhUJMé��'H1XJcT � �fw?lS�&jbm&o>��µ | �c�Õ� � � j>oAjbn�zj>m � ¡ � � ¢ � ���²npjbyжAj>ocqr y3�_zj>o�s � ¡ Î6� Ï u±dqr �M�Mw�zj>m]�«sMt | m | oºs � ±Uqr �D�Mw�zjbm&m]�Dj>yä��°�t��Ì�Ö��o~³5j �>l{zsMySl�o�j>oAjbn~�Bjbm¦³5j>�&l��ű.jbm&m�zsMy�y ñ � ¡ ñc� � �

°�t � � Ê Ø � �Fj>�&l�k��'q| �A�fqr ySj�nplç±Uqr �D�Mw�zj>m]��j>y�n?�r mÄzj>o?zj>m&jbk��Dj | �Kj�np�AlQsMlp¶Aj>y�j>m]n�zjbo?zjbn�sc�V¶~s}�Kjb��s�kYq| wDjbn~kMjbt�|zscySy3zÑ n'zsMob�! ��� � XG�OQO½Xg W�X\FT ��� ��Î�� Ï u � K3�M P 3�M�K3:2 C ; P K� � R ; KR[/�AC'DHE - E -�2�2 -)7Q3�2 -0/�1 � � Ê Ø � � 5 =V> -�2 N = KC ; 3��h23:M P K- DHE -�2 3�T6T CHD K- ; ��; 3�T<3:T T C'D � X 3 � �&� ¡ �Öí P -�M ��- ; AR M GG é63^ Z 4 3LYOSJVW>P � 3LBé?éF[�XJ è ��JMN�é�J ^ `�JcW�Z{P�\�P�h�JMOQOSJcé�J

°�t�sMy�zsc�&�&lQsck,o | �&zsMmÀ��zsc� r ndsVn r m&k�sM�A�ps�Bµ | �M�ÌsMt´zsMy�npsMy�zscm | o��'q| �A�BjI�'q| �~�;qr ySjbn~l�±Uqr �D�Mw�zj>m]�Mj>lfl�m�ÞwcsM�Al�zsMmopsMkasMtvu � �?�5j>ySlYlSt | �Kj�np�~l{zscwcsMy�zj>oºsMtÕzscnp³&sM�psM�Òzjbn~j>�Aj>t5zjbo~oAj>y�o~tbj>���5j>m��ô�j>�M�Mj>mÌsM� | nAn}j>�c�9� Î � Ï u � zscy�ndscy�zsMm | o#�'q| �A�Bjc��i1jbkFlSm]npocqr m&k§sMt�u � j r kFySl��&j>oAt>lYn�zj>�~mj>k§jb�M�;Î�u�� Ï u��§l���zsMm]�BzÑ n�zsMoAnpsM�An�z| lSt | �Kj�np�~l{zsU¶&zscn>�FsM�Kjby��Mjbn#sMt � Ø � Ø ��q| o~o~tbjb±dqr �D�1zj>o�zÑ ��ySjM�f��w?l��&j>m&ob�µ | �M�s ����»Í�)�©kfzj>³�'q| �~�5j��&j>�Al�w#zsMyÓn7¶~sMl��psIn~jbnpoAt�| y�j>�Djbo}ã Ç �§jbo~jbn�zj>m¦±.jbm&m�zsMy�y �� � ­ � � ãA���ÖÔ � � � ­ � � ãA�~�

��? � ØD¢{£{£{£�¢ � Ê Ø �_�v¼½y��Ì��z| � | m¾s ��»��q| �~�5j�l�o�zsMy�npsMy�zscm | o}n�zÑ ³ r oBzr ���º�?lSyÓw�zscmFwcsMy{z| �5µ | �M�¦sak5zÑ o?zj>�M�|B��Aj>m&j�n��?��zsct>lSo | kFm�zsMy�Ô � � ­ � ãA�~�º� �� ­ � ãA� ��? � ØD¢{£{£{£�¢ ���ºnpjbyжAj>ocqr y3�v��m&mj>kÀkBq| wMjbnpkcj>t�n�zj>�5j>mÕlS�]sct�sMtscy�zsM�&�l�kFlжAj>y�j>m]n�zjbo>�! �"! � XG�OQO Xg W�X\?T � � Í = D K3�2 1 K� P K3 ; P 3 D�P KC = N C > - P�D = K3�783:M 2 1 - DFP ����»ÍC)�� /BAC'DFE -Z/BACHDHE AR Me- P = Y AR�/ /�7JK-�2 1 - => -�/�-�/�1 - N 2 -�T 3 � /BAC'DFE AR M�- P = Y AR:/ /�7 K-�2 1�- = 7H-�MSG� JMéçhUJMé��'H1XJcT � °��Kjbm&m]��l��Bjbm sÕÍ lSt | ��jbnp�AlQs¦l���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| �fsMkFk | ��sÀk5zÑ o?zj>�M�| B��~jbm&jbn �?��zsMtblSo | kkBq| tDq| n~nı.j>m&m}zsMySy�m�sMkÅs�Ô=) � ­ � �� ­ ��? � ØD¢{£{£>£b¢ � Ê Ø � ¢ Ô=) � � � Ê �� � q| o~oAt>j�±Uqr �D�1zj>oAj>k'� ��o~t>jb�~l�mFnsÅ�'q| �A�fqr ySj�npl5±Uqr �D�Mw�zj>m]�DjbkF�~j_lS�]sMt �� C���� C �5E � ØD¢{£{£>£b¢ � Ê �M�azjbo �� ��� ¡ � Ê � ��� ¡ �GIH X\]W�eY\FZb\�`ÖXJcW�JMZbJMH1XJMTUTUJDO è �JcZbW_é 3^ Z 4 Jaé63^ Z 4 3LYOSJVW>P � 3L5é�éF[�XJ è ��JDP

ô�j>�M�Mj>mæsM� | n~n¾jb�M� � Î�� Ï u � zsMy�npsMy�zsMm | oÌ��q| �~�5jM�à° �?� � � .�j/&�m�zÑ �bl�z| �&sMmÖy�jDzÑ ��ndsMkFm�sck��j>�c±.j>ySjby~�| m�j>�c� c�Î�a Ï u¸±dqr �D�cw�zjbm&m]�DjbyI³�sc�psM��zjbn~j>�~jbt>t�qr k zscnÀs�� �'q| �A�Äzjbn>� °�t�zÑ �M� m]�Mj>�Anf�Ò���$).c¾�'q| �A�Bj�jbySoM�| k�zjbnÄ�&j>�Al�w#zsMyÓn7¶&zsM�~sv±.jbm&m�zsMy�ySm�sck�s � ��� Ø ��zj>o � ��� �D��q| o~oAt>j�±Uqr �D�1zj>oAj>k'�&s f  Î a Ï uoAj>�5j>oAo?zj>�c±Uqr �D�Mw�zj>m]�?�~j�³5j>�lS�ÅlS�]sct f B�ed ���Éñ c � �ed ��ñ �  B� c � d �~�U�'sMµ | y d�Ç a;���jb� m&jbµÄzj>t,��j>�M� r ndsVnpm&lä�fµ | �M�Ânpj�npoAt�| y�j>�Djbo d,Ç aþzjbo � ��Ø)7 � 7 � Ê Ø �_j>�1zjbo~t«o~t�zsM�jbo~j�n�zj>m8s f� � � d � ¢>£{£{£{¢ f� � � � � d ���&jb�~l�w#zsMyÓn~wMj>k]n | � | k r �M�]sMm&sMtbn_sMt�u��F�?�BjbySl;sMy�n~j>�~j�nv�Djbm&j>�&zsMyж&zsMk���lSm]ns9� � � c � d �A� ¢{£{£>£>¢ � � � � � c �ed �A�ºwDjbk]n | � | k'������f�| y;��zsM�ºkBq| wDj�npkMjbt>l�k5��µ | �M�Às f�ÖÎ a Ï u �¦�'q| �A�BjIlSozscy�ndscy�zsMm | o#n�zÑ ³ r oBzr �;°���j>m&m]�?lS�5j>m«wDjbo~o~t�qr k�s

f� � � � �ed ���»¬ �­ ® ¡ î �­ � d � � � � ­ � � c �ed �~� ��� � ØD¢{£{£{£�¢ � Ê Ø �y�lSm&j�zsM�~l�ovk | ���&lSm�zs �>l�z| o_kFlÓ±.j ¶Aj>t5zj>oAj>kMj�n{��sck?k | ��sct r n | y�o�z| j>�M��qr n~n~µ�scn�z| kF�ps�î �� � d ��� � c � � d �~� � npj>y¯Þ¶Ajbocqr yä�

i�j>�M��qr k�±.j>y3�}µ | �M��sMt zscn~³�sM�~sM��zj�npjb�~j>t5zj>o§l���zsMm]�BzÑ n�zsMoAnpsM�An�z| ��wcsM�M�?lSo�±.j>m&m}zsMySy c ��ì¸í��x° ±.j>m]ny�jDzÑ �Andsck?�}z| y�sM�}z| �&lSk'��µ | �M�´s f�¸zj>o;� �'q| �A�Äzj>kÕk5zÑ o?zjb�M�| B��~jbm&jbn �?��zsMtblSo~sMl�n�sM��z| wDjbk]n | �A�Kjbt�| kF�~j«l��]sMtf� ­ � � ­ ))c ��? � ØD¢>£{£{£�¢ ���U���º�M�]sMm �>opsck kBq| mmF�f�r �5j>y{zscnpml3��µ | �M�ÒsÂ�'q| �~�;qr y�jbnpl ±Uqr �D�Mw�zj>m]�Mj>kkBq| tDq| n~n�±.jbm&m�zsMy�y�s f� ­ ��� ­ ) c ��? � ØD¢{£{£{£b¢ � Ê Ø ��kVsM³ �>o | yQscn>�

Ø �

Page 14: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

i�j>kFlSm]npocqr kv� | oAn�sMt�n;sct jbo~j�npjbn>�]sM��lSk | ��sMtKzscnp³&sM�psM�Òzjbn~j>�Aj>t5zjbo1l���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| � c � ò©í]�U�;��kFk | �sÅk5zÑ o?zj>�M�| B��~jbm&jbn �?��zsMtblSo | kawDjbk]n | �A�Kjbt�| l�m#zj>y�±.jbm&m�zsMy�ySm�sMkasf� ­ � � Ê Ø � ­ � � ­ ) c ��? � ØD¢{£{£>£>¢ � Ê Ø � ¢ f� � � � Ê Ø ����� ��� � � � ) cq| o~o~tbjb±dqr �D�1zj>o~jbk'��° � � � � .�j!&�m�zÑ �>l{z| nvscySkVsMyS�KsMt�wMsKsMtbn�kVsM³F¶ r k'��µ | �M�¾s f� zj>o�� zsMy�npsMy�zscm | o���q| �~�Äzjbk��q| �~�;qr y�jbn~lB±Uqr �D�Mw�zjbmF�Mj>l��~jf� C}��� C ) c �5E � ØM¢{£{£{£�¢ � Ê �D� ¢ f� �&� ¡ � � Ê Ø ���� � � � � ¡ � ��� �&� ¡ ) cn~j>y ¶Aj>ocqr yä�G é63^ Z 4 3LYOSJVW>P � 3LBé?éF[�XJ è ��JMN�N'PSTUH;X\?`´Xg W�X\FTU\

ô�j>�M�Mj>m�sM� | n~nfjb�M� ��Î2� Ï u � zsMy�npsMy�zscm | o��'q| �~�5jM�1°��Â�&j>�Al�w#zsMyÓn7¶~sMlÓn�±.j½¶Aj>tbtqr k�kFls � ¡U¢{£>£{£>¢ � �k5zÑ o?zj>�c�| B��Aj>m&j�n��?��zsct>lSo}wMj>k]n | �~sMl�wcsMy�s� � ­ � � ãA��� ��

C ® ¡ F­ C � ãA� � � C � ãA� ��? � ØM¢{£{£{£b¢ ���

scyQsMkF��sMmY���zj>y�o~tbj>�F�r ��j>�{¶Aj>�M�Mj>t>m;qr m&k'�Vµ | �M��jbt�j>oAjbnp�5j>m�sMt F ­ C1j>�c��qr nAnpµ�sVn'z| kº�&l �Bj>�Aj>m �bl�zsMy�µ�scn�z| ±Uqr �D�Mw�zj>m]�Mj>ksct��,l�mFn~j>��wMscySy r � | mY�Ä°�t F ­ C � ãA� ��?~¢LE � ØD¢{£{£>£>¢ ����jbySj>��j>kFkcj>y��Kjb�Dµ�scn�zsM� | t | n~n��ÕáÀ�'�Fj>o���zscn~�~lÓÛy�j>�M�Mj>m * � ãA� � ã Ç ���U�B�j>yÓn�zjbwMjM�Vµ | �M�v±.jbm&m�zsMy�y]�¾� � �Msct jbySoc�| µ�zsc� | �x�j>�~lÓw#zsMy�n~�ps�s B��Aj>m&j�n��?kfzj>³ySjbn~j>k�±.jbySµ�sMoAt>m�zscy�zsMo&zscwcsMy

sctIscy�zsM�&�l�kFl�±.j ¶Ajbt5zjbo~j>kcjbn�mF�Mj>�öcqr k5Î� � �   � ¡B¢� � � �   � � ¡ ×   � � ¡ � � ¢� � � � � �7  � � Ê   � � � ¡ � � � � ¡ × �7  �   � ¡ × �7  � � ¡ � � � � �1×   � � ¡ � � � � £

° B��~j>mjbn��Fk�zjb³&ySj�npjbk?�;�| y�kYq| wDjbn~kMjbt>lSk'�}µ | �M��sMt * ��zscn~�~lÓÛ´j>y�j>��j>lS�Aj ? ò E j>oAjbn�zj>m F ­ CÌ� ín~j>y ¶Aj>ocqr yä��°Énpj>y ¶Aj>ovl�m&� r k �>l�z| ��z| �&o~tbj>�?zjbwDjby�lS�]sMt | ySµ&scn'z| �Yµ | �M��sct * ±U�| zscn~y�z| �5j>y�l1jbySjb�KjblS�~j�±.jbm&m�zsMy�ysct

F ��� �¿¨1� � � � ¢ � � ª �   � � ¡B£{£{£ � � � ¡q| o~o~tbjb±dqr �D�1zj>o>�5sM��j>mmF�?l��Bjbm � �»���

UMj>y q| yжAj * �$sMt * �?m&sMk´sMt�n�s¾opsM� | kF�©zscn~�~l¯Û}zsVn{� sM��j>y��MjbnÅsct F ­ C ��?~¢LE � ØD¢{£>£{£b¢ � ��jbySjb�KjbkscySk | npm�sMk'����jb�þm&jbµ#zj>t_lS�]sct | y�m&l3�?µ | �M�KnpjbyжAj>ocqr y

ÈFâ>ã * ���   �� � � � � ¡ � �� � ¡ ­¯® ¡ � �

­ � � � ­ £ � � � �D�

°��'q| �A�fqr ySj�nplc±dqr �M�Mw�zj>m]�DjbkMjbnfk?l]y�j>µ&j�n�o~t�zsc�¦zÑ ndsMm&lDs(�¾�j>�~lÓw#zsMy�n.¶~sMlS��z| yMsMkFk | �1l�o>�Vµ�s�m&j>� l�o~��j>�.¶Vqr ksÅk5zÑ o?zj>�M�| B��~jbm&jbn �?��zsMtblSo}wDjbk]n | �~sMl�n>�Ø I

Page 15: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

° ã Ç ��µ&jby��Djbm wDjbnAn � � � ãA� ¢ � � � � ãA� ¢{£>£{£>¢ � � � � � ãA���&j>�Al�w#zsMy�nAwDjbkFn | � | k�Nv�psM���?��zscnp�AlÓÛs�ySj>�c�Dj>mJ � ãA�U� ��o~t>jb�~l�mFn,s J jbySj>��j>l��~j«±.j>m&m}zsMySy K ­ CÕ� ¨@� � ­ � ¢ � � C � ª ��?~¢LE � ØD¢{£{£>£b¢ �����x° J jbySoM�| �o | ��zsc��sMmÕzjbo;j>y�oM�| � | o~t>y | ³�zsM�&sMm�y�zjbw��| jbySj>��j>kvo~sM� | k?��zscn~�~lÓÛ�zscn�¶Aj>y q| yжAj�J����fÜÌlSm]n1lSoA�Kjb�~jbn~j>ob�Dj>m&mj>k�jbnpjb�~��lSm�zscm&o�zsM�~sÀÈFâ>ãPJ��¿� � ÈFâ>ã * �#� � npj>y ¶Aj>ocqr y3�«° ��� � �M�Kq| oAo~tbjb±Uqr �D�1zjbo�sMySsM³?¶&zsMmÂj>�M�Õ�~jbk r �~tbl�z| ojbyж&zsM�&zsMo~o~sMy���j>�Dµ&scn'zsM� | t>µ&scn7¶ r k§s'���'q| �A�fqr ySj�npl'±Uqr �D�Mw�zj>m]�Djbl�n>Î

� ¡ � Q ÈFâ>ãDJ �  � � ¹ · � � · � �&· � � � · � Ê ¨�� � ¢ � � � ª �  � ¢

� � � ¡ � Q ÈFâ>ãPJ��  �� � � � � ¡ � ��� � � �­ ® ¡ � � ­ � � � ­ ��� � �¢{£{£{£b¢ � Ê Ø � £

Ü�lSm]n�lSoA�Kjb�~jbn~j>ob�cs-� �&� ¡ ��q| �~�;qr y�jbn~lV±dqr �D�cw�zjbm]��m&jb�]scn{zÑ wÌzjb�An�zjbkMjbkMjbn1lSoB±.j>y�wMj>µ&j�n{�;��mm&j>k���j>�Mµ�scn'zsc� | t�zsMopszjb�~�&jbk�zjb�BjbmÌwMj>�M��qr k«s ü ­ C}�²¨1� � ­ � ¢ = C ª ± | �~� r y�zscwcsMyB�Kjb�Dµ�scn�zsM� | t | nAn ü ­ CBÎ � Ï u��?~¢LE � ØD¢>£{£{£>¢ ���I±Uqr �D�Mw�zjbmF�Mj>kcjbn{�Ò° ü ­ C � ãA� � ã Ç ���jbySjb�Kjbk²zsMyÓndsMy#sMySk | n | n~n����?j>�&�Aj>m&� �r k]wcsM��Þ�~scnpl�k r o��©zscn~�~l¯Û | n?¶Aj>y q| yжAj�� � ãA�U����w?l��&j>m&ob�Mµ | �M��s�ñ ÈFâ>ã�� � ãA�çñ�zj>��n�zjbk � ÈFâ>ã�� � ãA��ý��í}j>o~j�n�zjbm'����j>�Mj�Þ�c�Dj>tblSk«s;� � � ãA� ¢{£{£>£>¢ � � � � � ãA�#wMj>k]n | � | k�zsMyÓndsMy�kFlÓ±.j>o~tMzÑ n~jbn~n����?�&l��Kjbm&t>l{z| o}³�sM�~sMySy�j>ySjb³&lS³5j>� | man�zjb�A± | �DsVÞn�zscwcsMy3�;°�t r n | ySo�z| �'q| �~�;qr y�jbnpl5±Uqr �D�Mw�zj>m]�?�~j�³5j>�&l���±.j>m&m�zscySy

� ��� ¡ �ÈFâ>ã��

  �� � � � � ¡ � ��� �&� �­¯® ¡ � � ­ � ��� ­ £

G é63^ Z 4 JMJMOQ`ÖXJMOSJcW�\?OS\Fe5WYXJVW�JDOSJ°�tIsMy{zsM�&�&lYn�zjbn~j>y1sM�~�~s�� r ndscnº��zs���µ | �M��sMt8zsMyÓndsMy{zsMm | o#n{zÑ ³ r oYzr ��q| �~�Äzj�n�l�t | ��jbn~�~l{zscn'z| yBj>yÓnpj>kFl�mFnAwDjjb�M��zj>��npj>y��Ò�r jbm«�Kjb�Dµ�scn�zsM� | tbt�zsMk§sÅ��q| �~�;qr y�jbn~lB±Uqr �D�Mw�zjbmF�Mj>k'�

! ��� � * XJcW�JMO �h+ -�/�1 -�2 -�TL3 5 7Q3 3 �æÎ�� Ï u � K- ; �� Î�� Ï u�� K� 7HX C ;0; N ; N - D = 2 P � 3 D 3 > K- P - D - N - P0PK3:M P 3:M�K3�2 C ; /BAC'DFE K-�T G�� 3 - N -�TO/BACHDHE AR Me- P = Y AR�/ /�7JK-�2 1 - = D - � ­ � �� ­ ��? � ØD¢{£>£{£>¢ � Ê Ø � P -�M ��- ; AR M � 3�T6T C'D7Q3�2 C M(1�3�2 u � � E -�M = Í = D K3�2 1 K� P K3 ; P 3 DFP KC = N C > - P D = 3 � X C /�1 Y -�2�2[K3:MVM ��Ò�ÖÍ�)� G° kBq| wDj�npkcj>t�| k?l ¶Aj>y�j>m]n�zjboAn�s���q| �~�5j>jbyS��zj>ySj�n}sMySsM³n�zjbnpjby�zjbm&j>k§oAt | kÄzsMoºm&j�wDj>tbm&l3�

! �"! � * XJVWbJMO � � N � �©u = 2 P - D 783:MVMSR > C 2 Me-�/�1 -�2 -�T93 5 7Q3 C M#1 3�2 T�� � C ; N P K3�M#1hKR � ¡�¢>£{£{£�¢ � �&� ¡ 7Q3�M KC ;Y AR�/ /�7 K-�2 1�-�T � > -�M#1 -�T D - � C¦ìþí ��E � ØM¢{£{£{£b¢ � Ê �M� P -�M ��- ; AR MSG � N - ; - P0E -�2 3 N = D K3:2 1 K� P K3 ; P 3 D�P KC= N C > - P�D = 39- D - � K- = / �.C 2 P C ; 3�2b-0/�1 C M#1 3�2�K� 7HX C ; ; N ; N - D = 2 P � 3 D 3 > K- P - D - N - P P ��Î>� Ï u � K3:M P 3:M�K3�2 C ;/�AC'DHE - M6K- P - N = T � 3 > -�M(1�2 -�T&/�AC'DHE AR Me- P =�Y AR�/ / 7 K-�2 1�- = 3 N C 2 C ; 3�T 3 > -�/6365 C P P � ¡�¢{£{£{£�¢ � �&� ¡ Y AR�/ / 7 K-�2 1�-�T6T6-�MSGXG�O�W�\�O�X\ èB^ T_WMXg eYLBT1XL8é 3^ Z 4 XJcNÂ\FHÅ\� è \?O�W�JMZ>JcN 4 J èô�j>�M�Mj>m�s��ÖÎ � Ï u � jb�M� | y��]scm¦�~jb� r y{zsM�~l�o��'q| �~�5jM��sc�Kjby��?m&j>k«³�zscy��#zsU¶~sK�5j>mm&j�wcsMm¾j>�c� � ��lS��j>m&tbl�z| o ( sVÁKmasMy�n�zj>�A�Bjbm ��� ���D�U� UMj>y q| yжAj ��sMt ( �Fm�sMk���j>�M±.jbySjby~�| u � �?�5j>y�l�ySl�m&j�zsM�AlSo�sMyÓnpjb�~jbn>��fw?lS�j>m&ob�5µ | �M�§±.jbm&m�zsMy�y6� � � ãA� Ç ����zsM�A�Kjby��Kã Ç �M�?�~jc�

! �� � � J � è Xg ê PVX^ � °�tbn�� | m&�ç¶ r k'�fµ | �M�Õs ��jb�M�²zsMy�npsMy�zsMm | ovn�zÑ ³ r oBzr ��q| �~�5j§sMt � �?�&l��Kjbm&t>l{z| o (scy�n�zjb�~�5j>m��;µ�sÌnpj�npoAt�| y��Dj>o_ã Ç �¾µ&j>yÓ�Djbm�s � � � ãA� ¢ � � � � ãA� ¢{£{£{£b¢ � � � � ¡ � � ãA��wMj>k]n | � | kÂj>�M�ÕySl�m&j�zsM�AlSo~sMm±dqr �M�Djbn~ySj>m,�Aj>m&�o~t>jb�An�sMySk | npm�sck5�i�j>�M��qr k,±.j>yä�Yµ | �c�¦s$�åj>�M�xzsMy�npsMy�zscm | o��'q| �A�Bj�sMt ( scÁ�m¾scy�n�zjb�~�5j>m��ºzj>o���q| �DtMzÑ n~oMqr k«sMt ( sMy�n�zj>�

jb�M�?lSkÂlS�&zsMm]�BzÑ n'zsMo&zscn{�Ì��t§j>oAjbn~�Bjbm s � � � � .�j!&&m�zÑ �bl�z| m�sck¾�Kjb�M±.j>y�j>y~�| j>mæzjb�An~j>yS��j>tbm&l�n r �V¶ r kF�;�?m&sMkØ �

Page 16: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

sct ( scy�n�zjb�~�Aj�w | m�scnpk | t�z| T�K� ; K- D C � D -�2 - P�� E K3 N =V; K3 P �sM��j>y��Mjbn�s � ¡�¢{£{£{£b¢ � �KÎ � Ï �©wDjbkFn | �~��j>t��| kscySk | npm�sMk'�§¼½yÓ�¾��z| � | mÕs � � � �9.�j!&�m�zÑ �>l{z| sMyQsM³F¶&zsMmÂwMj>µ&j�n7¶Vqr k�s � ¡d¢{£{£{£b¢ � � � ¡ Î�� Ï u$��q| �~�;qr y�jbn~l±dqr �M�Mw�zj>m]�DjbkMjbn�lSob�! ����� � J� è Xg ê PVX^ � i1jbk?l�m]npocqr mk¦j>�c� � Î � Ï u �À�~j>� r y�zsM�AlSoº�'q| �A�#zjbn>�Iô�jb�M�Djbm ( sMt�s§ySj>�Mo~t'�r kcj>�&�sVÁKm´sMy�n�zj>�IsMt�u � j r kFy�lS�&jbo~t>l�n�zjb�~�5j>m��fsM�Kjby��¾npsM�AnpsMyS�KsMt>t>s«s ���'q| �A�Bj�³�zsMyÓ�#zsU¶&zscn{�,°�t ( �FnIs=�zscy�ndscy�kFl�±.j>oAtDzÑ npjbnAnºsMy�n�zj>�Am&j>k§mjbwDjbt>t�qr k5�

i�j>�M��qr kÒ±.j>yä�}µ | �M��sF� ³�zsMyÓ�#zsU¶~sÕm&l�m �>oKj>�M� j>�c�Dj>mj>o~jbm��Ìzjbo§s0� j>�c�þzscy�ndscy�zsMm | o���q| �~�Äzj�n«sM�sct©zsMyÓndsMySsakFl�±.jbo~tMzÑ npj�n~n � �?�&l��Kjbm&t>l{z| o ( sMyÓn�zjb�~�5j>m ��� �É�D������q| �DtDzÑ npocqr kÀsMt ( scÁ�mÕsMy�n�zj>�vj>�c��l�kl���zsMm]�BzÑ n�zsMo�zsVn{�;zj>o�wMj>�M��qr k�s � ¡U¢{£{£{£b¢ � � � ¡ �'q| �~�;qr y�jbnpl?±Uqr �D�Mw�zj>m]�Mj>kMj�n{��° � � ��� z°�ySy3zÑ n'zsMo1kBq| wDj�npkMjbtbn�zj>�5j>moAj>���Kl�k�zj>³&³�o~jb�ÙnpjbyжAj>ocqr ySµ&j�nºs � � � ¡ �»í q| o~oAt>jb±dqr �M�;zj>ob�� JMéçhUJMé��'H1XJcT � ���M�I�~jb� r y{zsM�~l�o���q| �~�5j�sMt§zsMy�npsMyQs}kFl�±.jbo~tMzÑ npj�n~nfsMy�n�zj>�~�5j>mIm&j>��±.j>yÓn�zj�npySjbmfqr yFsM��zscy�ndscy�zsMm | on{zÑ ³ r oYzr �'q| �~�Äzjbn{�i�j>kFlSm]npocqr k�sMtbn�s T � � | o~t�n'zsMyÓ�ºzr #�Î&u Ï u�±Uqr �D�Mw�zj>m]�Fn{�sMµ | y # � ãA���Öí�npj>y ¶Aj>ocqr y'ã 7 í�j>oAjbn�zj>m��n | w�zsc�&��zs�ã�ì�í�j>o~j�n�zjbm�±.jbm&m�zsMy�y+# � ãA���»j�Û�³ � Ê ¡� � �U��.�j>�M��qr kKsMtbnºs ��Î&u Ï u � �~jb� r y{zsM�~l�oÄ�'q| �~�Äzj�n{�sc�Kjby��Dj�nºs

� � ãA��� # � ãA� � | o&ã = ¡ ×C# � ãA�}o~lSm�ã = �f×Òã = � � ã Ç u��q| o~o~tbjb±dqr �D�1zj>oazÑ �vySjc�Kô_zscn~µ�scn�z| ��µ | �M�Âs �¸zsMyÓndsMy;k?lÓ±.j>oAtDzÑ n~jbnAn�scy�n�zjb��sMt | m | o�u � �F�asMyä���&j�s=�x�'q| �~�5jmj>� zsMyÓndsMy{zsMm | o n�zÑ ³ r oBzr �GIH�� � 2 4 JMOQP;TcXg N�é 3^ Z 4 XJcN! � � � � J � è Xg ê PVX^ � ° � Î � Ï u � �~j>� r y�zsM�AlSoK�'q| �~�Äzjbn,o>zÑ kF�'q| �A�#zj>m&jbk©� | m&�V¶ r k'��µ�s´wcsMm | yÓ�]sMm�{�?�&lS��j>mt>l�z| o}scÁ�m¦scy�n�zjb�>��sc�Kjby���ndsM��ndsMy��asMtbt{s�s;�©³}zsMy��#zsU¶&zscn>�

ô�j>�M�Mj>m¦sM� | nAn�j>�M� | y��]scm � �'q| �A�Bjc�'sM�Kjby��?m&jbk§³�zsMyÓ��zsd¶~s��Bjbm&m&j�wMscm�jb�M� �åo>zÑ kF��sMm�� �fw?lS�&jbm&o>�µ | �M�Õs � jb�M�¿zsMy�npsMy�zscm | o�n{zÑ ³ r oBzr �'q| �A�BjKsMt �²sMyÓn�zjb�~�5j>m��,°��Kjbm&m]��l��Bjbm8l���zsMm]�BzÑ n.¶ r k�sMt �¿o>zÑ k | n{�sck?k | �#�j!&�m&l{zsMySml�ySj>µjbn#s�� ���?�Aj�w | m�scn~k | t�z| k5zÑ o?zjb�M�| B��~j>mjbn��F��zsMt>l�o�zscn>��ÜÌlÓwDj>y �>o~sMkÅj>�M�Å��q| �~�;qr y�jbn~l±dqr �M�Mw�zj>m]�«sM��z| �&lSk'�&jbtbn%� ¡ µ&jby��Mjbn~n�]�]wMscy�± | ��¶ r k�¶Aj>y q| ySml3�B | m]n | o}��j>�{¶Aj>�c�Dj>tbmfqr m&k'��µ | �c�«sMt�lS�&zsMm]�BzÑ n | nAn}o>zÑ kF��sMm«wDjbnAn(�©�'q| �A�Äzj>mj>k,s

� � ãA��� Ø· � � � ãA�ç· ¨ � � ¡ � ãA� ¢ � � � ãA� ª � ã Ç ��� ± | �~� r y{zscwcsMy¾zj>��npj>y��Kjbt>j�n~nR� Î � Ï u ��q| �~�;qr y�jbn~l

±dqr �M�Mw�zj>m]�Dj_m&j>�Dscn�zÑ w zj>��n�zjbkf�r lSo}y�j>µ&j�n{�� JMéçhUJMé��'H1XJcT � ��w?l��&j>m&ob�Yµ | �M�ÌsMt_u � �?�BjbySl �»o{zÑ k | nvsct | m | o{zÑ npsMm&l�y�j>µ&j�nvsMt�u � j r kFySl��&j>oAt>l�o{zÑ k?kVsMyä�� TJAC 7H- P T6- N C Y - ��- N - P0E -�2 D K- ; N M�- P - ; -�2 Y C / �'R�T P K3 D /�1�3:MV2 = 3 Nvu � ��E -�M = D -�/�R M K3 D =?; /BAC'DFE K-�T6- P G

Ø i

Page 17: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

��)�* E?C� 9 Ú =�+ 41@2 GçÚ C=F9 � �� = G =�� � C-�+74'=F2568= � G: C-{<>-��t>jbm̱.j ¶Ajbt>j�np�5j>mÌsMtvu � �F�BjbySl��Aj>� r y{zsM�~l�o��'q| �A�Äzj>kcjbnºn'zsM�A�M�]scyж r k'�i�j>kFlSm]npocqr m&k�jb�M��� Î � Ï u � �Aj>� r y{zsM�~l�oaoAlS�Ks8�'q| �A�#zjbn>�¿ô�j>�c�Dj>mj>k�� ¢�� Î�� Ï u s �

k |�| �~�&l�m�zscnds��F±Uqr �D�Mw�zj>m]�Mj>l3�I��t>jbk?kcj>y s ��ySj>kfzjb³Bjbt5zj>oIs9� � ãA��� � � ãA� = ¡ × ��� ãA� = � � ã Ç ����sMySsMkF��sMmzÑ �~µ�scn�z| ±.jby3�x°   Î � Ï u oAj>�5j>o~o?zj>�M±dqr �M�Mw�zj>m]�?�~j�m]��l�y�w#zsMm©l��]sMt  Y� ãA�¦� ¹ � � � � ãA�A� � × ��� � � ãA�~� � ��fw?lS�j>m&ob�5µ | �M�«s;�©k5zÑ o?zj>�c�| B��Aj>m&j�n��?��zsct>lSo&zsMµ | t�npsM�An | t�z| � ¡�¢ � �ºwMj>k]n | �~��j>t�| k?�Aj_±.jbm&m�zsMy�y

� ¡b� ãA��� Ø B� ãA�6B � � � ãA� = ¡ × � � � ãA� = � D ¢ � � � ãA��� Ø

 Y� ãA�6B Ê � � � ãA� = ¡ ×�� � � ãA� = � D £° � � ãA�}³ | mFn | m�zscn~�Kjbmf�| � � � ãA�}lS�&zsMm]�ºzr j>�M�Mj>m&jboAn�s�o{zÑ k?��q| �~�5j_ã�³ | m]np�5j>y�l�m | �~��zsMy�lSo}j>�c�Dj>mj>o?zjbm&j>kmjbwDjbt>t�qr k5�+ ����� � J� è Xg ê PVX^ � ° � � ãA�#� Ø

 B� ãA� ¨ � � ¡ � ãA� ¢ � � � ãA� ª kFl�±.j ¶Ajbt5zjbo~o~jbyIzj>��npjbyS��j>t>j�n~n�¦Î � Ï u�±Uqr �D�Mw�zjbmF�Fns'� o{zÑ kF�'q| �~�5j_jby~�| ¶AjbySjbo}�'q| �~�;qr ySjbn�zj>m&jbka� | m&�V¶ r k5�

°�tvu � �?�5j>y�l6�©�'q| �~�5jvj>o~j�n�zjb�BjbmÀs� � ¡ �   � � � ¢ � � � � Ê   � � ¡

jb�M�Djbm&ySj�npjbkÌsc�V¶&zsMk§s$B��~jbm&jbn �?k�zj>³&y�jbn~j>kMj�n{�Ä° �©j>y�oM�| k�zjbn��&j>�Al�w#zsMyÓn7¶&zsM�~sInpjbyжAj>ocqr y� � �   � ¡�¢ � � � �   � � ¡ ×   � � � � £

¼½yÓ�,��z| � | m«s��§j>y~�| ¶Aj>ySjbo}�'q| �A�fqr ySj�np�~j�lS�]sMt�Å� ¨@� � � ¢ � � ª  � � � � � � � Ê � � � � �

  � £ � I � Ø �

� JMéçhUJMé��'H1XJcT � ° ��� � � .�j!&�m�zÑ �>l{z| ��sMm,y�jDzÑ ��ndsMkFm�scka��j>�M±.jbySj>yA�| j>m§wMj>�M��qr k,s'�;�?m�sckaj>�c�f��� � )"c zscnp³&sM�psM�Òzjbn~j>�Aj>t5zjbo?zj�n{�«°���j>m&m]�?lS�5j>m8sMt�zscnp³�sc�psM��zjbn~j>�~jbt5zjboIl���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| � c �ÄòÉí]�U�sck?k | ��sÅ��q| �~�;qr y�jbn~lB±Uqr �D�Mw�zjbmF�Mj>kF�~jIs f��� Ê ��) c q| oAo~tbjb±Uqr �D�1zjboAnºm]�Djb�.¶cqr k'�� JMéçhUJMé��'H1XJcT � ° � � Ø

  · � �3·�kFl�±.j½¶Aj>t5zjbo~oAj>y�zj>�An~j>y��Kjbt>jbnAn��'q| �A�fqr ySj�np�~j�mF�?l�y�w#zsMm,±.j>m&m}zsMySy � � ñ ��ñ �°���j>mmF�?l��Bjbm � � ãA��ý� í�n~j>y ¶Aj>ocqr yä��sMkFk | �¦sÒã3�?�5j>y�l � ãA�§±U�| m | �A�©zsMy�lSoKj>�M�?o?zjb�MwDjbk]n | �A�ps�l��]sMt

� � ãA�� � ãA����� � ãA� � � � ãA� £ °xã�µ&jby��Djbm�wMjbnAn�oAlS� r y{z| kBq| ��kBq| t5zj>³&³ | m]n7¶&zscn�s� � � � ãA��× ¡

� � ��� � ãA�Ä� � � ãA��× ¡

� � ��� � � � ãA�åkFl�±.j ¶Ajbt5zjbo�sM�ç¶~sÅ��j>��G TcXg N�é63^ Z 4 J,JV[ ^ O]XL5W�X\�hU\+ ��� ��� J � è Xg ê PçX^ � ô�j>�M�Mj>m��Î>� Ï u � j>�M� | y��DsMm¾�~jb� r y�zsc�~lSo��'q| �A�Bjc��sM��j>y��?m&jbk¦�'q| �~�;qr y�jbnpjIo~jbµ | yoAj>�$n?�r m&l�k�j>yä�f°�� � ãA��� � � ãA��× Ø

� � ãA� � � � ãA� � ã Ç ����q| oAo~tbjb±Uqr �D�1zjbo~oAj>yY��j>�Mµ�scn'zsc� | t | n~n�xÎ2� Ï u �oAlS�Ks��'q| �~�Äzjbnºs;�©j�w | yczr n'zsU¶&zsMm&sMkKmjbwDjbt>t�qr k5�

�fw?lS�&jbm&o>�'µ | �c�,s�Öj>y�oM�| �&j>�Al�w#zsMyÓn7¶&zsM�~s�±.j>m&m�zscySy� � � ãA��� Ê � � � ãA�� � ãA� � � � � ãA� £ � I � �D�

Ø �

Page 18: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

��oAt>j>�AlSm]nas+� m | �A�©zsMy�lSo�jb�M�Djbm&j>oAj>l�s � jbw | yMzr n'zscm�sMk´sct zjb�~l�mFnM�| l3�Ò° � I � �M��j>�M�Mj>m&yA�| o?zjb�D�f�| y}��zsM�kBq| wMjbn~kMj>tblSk�sMt_sMy{zsM�&�&l�kFl ¶Aj>ySjbm]n�zj>ob�+ ����� XG�OQO Xg W�X\FT � + -�/�1 -�293 NW� ¢���� � � C M#1 3�2 N K3 D�P = 2 P - D 7Q3�M?MSR > � 3 > -�M#1 -�2 3 � -�M�C ��-�M�- ; /BAC'DFE AR M�- P K- D -Y -�2�2[K3�M?M �5� � ãA�vý�»í � ã Ç � ¢���� � G�� N -'7 C M KR P 3 ��ñ#� ¢���� 5�3 D 3 E ��K3�2 3�T K� 7HX C ;0; N K3 D 3 P -�M ��- ; AR M

$ � ��ñ#� ¢���� ��� ���

Ø� � ?� Ê

Ø� ��� � ��� £

G TcXg N�é63^ Z 4 J,eB\?Zb\?OQOSJDO1é63^ Z 4 XJDP�XJcT_JV[ ^ O�[�J è T�JMP+ �"! � � J � è Xg ê PVX^ � ô�j>�M�Mj>m�sM� | nAnÅj>�M� �ËÎ�� Ï u � �~jb� r y�zsc�~lSo���q| �~�5j zj>oÅj>�M�ÒÈ Ç uËo~t�zsM�Ì��°�� � ãA�I� � � ãA��×�È � � � ãA� � ã Ç ����q| o~oAt>jb±dqr �M�;zj>oAo~j>yÄm]�Dj>��n ��$ÎH� Ï u � �'q| �~�Äzj�n�s=��j>�M�?lSk¾³&sM�psMy�ySjby��q| �~�Äzj½¶?zj>m&jbk,� | m&�V¶ r k'�

ô�j>�M�Mj>m sM� | nAnÅj>�M� � Î � Ï u � �Aj>� r y{zsM�~l�o��'q| �A�Bjc� sM�Kjby��?m&jbk´�'q| �A�fqr ySj�npjaoAj>µ | y}o~j>� n?�r m&lSkjby3��zj>o_j>�c� $ Ç u$oAt�zsM��� ��q| �MtDzÑ n~ocqr m&kÌj>�M�O Ç �À³�sM�~sM��zjbnpjb�?zjb�An�zjbkMj�n{��zj>o�wDj>�c��qr k�sctbnIs � Ç � � ���±dqr �M�Mw�zj>m]�Fn{�'sM��j>yÓ�Djbnºs � � ãA��� � �` · � � ��? �ç·YÈ ? kFl�±.j½¶Aj>t5zjboIzÑ ��ySjc�+ �,+ � � J � è Xg ê PVX^ � ° f� � ãA�_� � � ãA�f× � $ Ê � � ãA�A� � ¡>� ãA� � ã Ç ���Åq| o~oAt>jb±dqr �M�;zj>oAo~j>yÄySjMzÑ �An�f�²Î:� Ï u ���q| �~�Äzj�n�s'��j>�M�?l�k§jbw | y�wMj>m&o?zj>m&jbk�� | m&�ç¶ r k'�

° f�©j�w | yÓwDjbm&o��j>�~lÓw#zsMy�n.¶&zsM�ps�sf� � � ãA��� � $ Ê � � ãA�A� � � ãA�  B� ãA� � � � ãA� � ã Ç ��

jb�M�Djbm&ySj�n�sc��z| �&l�k5����oAt>j>�AlSm]n(� � � ãA��� � �>o~sMk,s � � ãA� � $;jbo~j�np�5j>m zscySyB±.jbm&m��+ ��� � XG�OSO½Xg W�X\FT � � > -�2 2 1 = E -�2 = /63 N � � ãA�vý� $ � 3�T6T C'D 3 f� -H7 C M 7'-�2 ; ã X -�M#1 -�2L7'- P P ; =V> R�M�KC TJAC'D K-�2 -�TTJAC N K- � �.C 2 P �Q3 3 � � ãA� � C 2 P G� JMéçhUJMé��'H1XJcT � �vlSoAo?zj�³ | m&�M� | y�zsMm�± | �DsMyS�KsMtbwcs�Ms�±.j>m]npl�zsMy�y zÑ n�zsMo1sMtbn � | m&�V¶~s�kFl3�Fµ | �M��s��8�}zsM�~��j>yÓ�j�w | yÓwDjbm&o?zjbm&j>kÌsMt�jbw | yczr n�zsU¶~s�sMt | m | o��;�FwcsMy3�G H;X\?Z�W�TVXg N�é63^ Z 4 XJMNÂNH3^ Z53LYO�� ^ ZbR�LBOçX\FT�P�TUH;X\?`Ì\+ ��� � � J � è Xg ê PVX^ � ô�j>�M�Mj>mÀsc�wcsÅjb�M� ��Î � ¢���� Ï u � �~jb� r y�zsc�~lSo}��q| �~�5jM�° � � �Ä��� � ^` � � ãA�  Y� ãA�&ÈDã o~t�zsM� | n�s;� npjbyжAj>o�jby~�| ¶AjbySj>o}��q| �~�;qr y�jbn�zj>m&jbkam&j�wDj>tbt�qr k'�°�tvu � �?�5j>m«wDjb�M��qr k,sMt�j>�M�?o?zj>�MwMj>k]n | � | k�� ¡ �æð � Ç u � ñ�· �}·�� Ø ó_kBq| �?zj�n{�° � � ãA��� � | o�ã = ¡ ×KoAlSm}ã = � � ã Ç u#�vq| o~oAt>jb±dqr �M�;zj>oAo~j>y�y�jDzÑ �An � ÎDu Ï � ¡ ySjbk�zj>³5j>t5zjbo�sMt�� ¡ j>�c��o?zj>�DkBq| �r m&l�wMj>�~t�zsMySl�o}ySj�±.j>�Äzjbo?zjbn�sM�V¶~s��i�j>kFlSm]npocqr kÀs � ¡ j>�M�?o?zjb�MwDjbk]n | �A�Kjbt�| n>����lSm]n�j>�M� � ¡ Î�� Ï � ¡ ��u � ± | yÓ�?n | m | o�ySj>kfzjb³Bjbt5zj>o�n{�°æn | ³ | y{z| �Dl�zsc��z| yYl�o~��j>�Ajbnpjbo>�Bµ | �M�«wcsMm | y��]scm ��Î>� Ï u�± | yÓ�Fn | m | o�y�j>k�zj>³5j>t5zj>o>��sM�KjbySyÓ�Dj>y�±.jbm&m�zsMy�y

� )��Â� � ¡ ��¼½�]sMt | ySµ�scn�z| �µ | �M�«sMt �8±dqr �D�cw�zjbm]�,�&l �Bj>�Aj>m �bl�zsMy�µ�scn�z| �;°� ¡�� ãA��� � | o � � ãA� = ¡ × oAlSm�� � ãA� = � � ã Ç ���

kFlÓ±.j ¶Aj>t5zj>o�n��j>�~lÓw#zsMy�wcs�s � � ãA�  B� ãA��� � � � ãA� q| o~o~tbjb±dqr �D�1zj>o~µj>t�¶ r n r mk5� ��o~tbj>�AlSm]n�s � n~j>yжAjbo}j>yA�| ¶Aj>y�j>o��q| �~�;qr y�jbn�zj>�~j�±.jbm&m�zsMy�yH� � �Ä� � � ��� � Ê � � ?���°���j>mmF�?l��Bjbmås �Ù�Aj>� r y{zsM�~l�oK��q| �~�5jÀt�zsc�An{��sMkFk | � � ¡b� ��¦� � ¡b� � �akBq| wDj�npkcj>tbn�zj>�5j>mÖsV� � �#�zjb�An�zjbkMjIs����B�?m&jbk§j>�1zj>o~t_o~t�zsM��o~t | � | o~s�

Ø �

Page 19: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

+ �� � � J� è Xg ê PVX^ � ô�j>�c�Dj>mÀsM�wcs�j>�M� � Î � ¢���� Ï u � �~jb� r y{zsM�~l�o}t�zsM��n���q| �~�5jM� °�t Ø� � � � �Ä�»j>�1zj>o~t

oAt�zsM� | n�s;��t�zsM�Anºo>zÑ kF�'q| �A�BjvkBq| �cqr y�± | �A� r y�zsco~lYoAt�zsM��zsMm�scka� | m&�V¶ r k'�+ ����� * XJVW�JDO � � > -�2�2 1 = E -�243 � Î � ¢���� Ï u � N K3 DFP /BAC'DFE - -�/:1 ; N - D R � 3�T6T C'D 3 � T ACHD AR M Y CHD 5 R M K3 ;0=; N K3 > K3�2�3:T K- D�P K-�T6- Ø 7Q3:/�1 Ê Ø GG N ^�è [J�4ÂH;X\?Z�W�TVXg N�é63^ Z 4 XJMN+ ����� � J � è Xg ê PçX^ � ô�j>�M�Mj>mÂsM�wcs�j>�M� �ÖÎ � ¢���� Ï u � t�zsM��n��'q| �~�5jM�º° �;�Fn�k | m]wDjUÛ�m&j>kÀmjbwDjbt>t�qr k5�µ&s���zsc�~��j>y��Kã " Ç � ¢���� j>o~j�n�zjbmÀs$# � ãA���²¨�� � ãA� Ê � � ã " � ¢ � � � ã " � ª ± | �A� r y�zsVwMscyºzj>��npjbyS��j>t>j�n~n#�Î � ¢���� Ï u ±Uqr �D�Mw�zjbmF�?�Aj #��åí�wcsM�c� # 7 í�npjbyжAj>ocqr y3�+ ��� �)* XJcW�JDO � + -�/�1 -�2b365 C6P0P -�/�1 ��Î � ¢���� Ï u � -0/�1 ; N - D R N K3 DFP /BAC'DFE -HG�� � ; K� T /BAC'DFE - T C 2 7H- �3:T T C'D K- ;���; 3�T<3�T6T C'D � X�3 3 N -�M C ��-�Me- ; /BAC'DFE AR M�- P K- D - �,�åí 7Q3:/:1 � 7 í P -�M ��- ; AR M G+ � � � � J � è Xg ê PVX^ � ô�jb�M�Djbm©sM�wcs�j>�M� � Î � ¢���� Ï u � t�zsM�An��'q| �A�Bjc��° ã Ç � ¢���� zj>��n�zjbkMjbnas��boBzr �bo~³ | m]n7¶&zsMm�sck,� | m�V¶ r k'��sM��j>m&m]�?lS�5j>m«±.j>mm�zsMySy �5� � ãA���»í�

°�t_sMy{zsM�&�&lBkFlжAj>y�j>m]n�zjboAn 3[2"K-�/�1 ��; KR ��; �.C 2 PZP K- P -�Me- mÄzjbwDjbmÀo~t | k]n'zsMkaj>��y zÑ n~j>m&lä�+ �"! �&* XJcW�JMO � �./�1 T C 2 7H- � N K3 D�P ; K� T6/�AC'DHE K-�2 -�T Me-�/ 3:M�K3 E�E 2WK-�/�1 ��; KR ��; �.C 2 P �Q3 783:2 G

Ø �

Page 20: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

�B)�*��ÅC- G 256 � 6Õ=�� � C2BE�= � 9�+ � =?E�� B� C= G 4;@2 GçÚ C= °å�M�]sck | �AyQscn~lBscySkVsMyS�KsMt�zsMo | k_o~tbj>��³ | mFn.¶&zsM��z| y'l��Dj>mű | m]n | o sMt�u � �?�5j>ySl�oAlS�Ksv�'q| �~�Äzj>k�ndscm r yS��zsVÞm]� | t�zsMo~s�1sM�Kjby��?�~j�j>�&�5j>mÂs§±.j ¶Ajbt>jbn~�BjbmÂn�zj>�cqr m&k¦kFl3�_°²n'zsc�~�M�DsMy�zsMo�o | �&zsMm¾±.j>y;± | �{¶ r k�µ�sMoAt>m�zsMy�m&l�s�?��zjbo � �f±.j ¶Ajbt>jbn~j>kF�5j>mÀl�o~��j>�An~jbn~jbn~n}± | �]sMyS�KsMkVscnKzj>oIq| o~oAt>j�±Uqr �D�1zj>oAj>kcjbn{�

GIH�� � 2 4 JDOQP;ZbJMé�LYO�X\?Z>PST_é63^ Z 4 XJMNô�j>�M�Mj>mÅsM��wMs�j>�M���©Î � Ï u � �~jb� r y�zsc�~lSo�o~l��asº�'q| �A�Bjc�1°���k |�| �~�&l�m�zscnds��F±Uqr �D�Mw�zj>m]�Mj>l?ySj>�c�Dj>mj>k� ¢ �5¢�� Î � Ï uº����tbj>kFkMjby;s;�©y�j>kfzj>³5j>t5zj>o�s

� � ãA����� � ãA� = ¡ × �Y� ãA� = ��× ��� ãA� = � � ã Ç ���scyQsMkF��sMm»zÑ �Aµ�scn�z| ±.jby3� ° � zj>�AlSm]nD�| y�j>�Djbo�j>�M�?o?zj>�MwMj>k]n | �A�Kjbt�| ¶Fzj�n«s � � ¡

�� � � = ¡ × � � = ��× � � = � �kFlÓ±.j ¶Aj>t5zj>oAo~j>yYmF�Mj>�öcqr k5�&sMµ | y   � · � � ·ç�f°�t�n�sIo>zÑ k | n{�sM�Kjby�� zscn~�Kjb�M��s�� � ãA� ³ | m]n | m�zjbo#��j>�M�| ySj>�Mj>os'� � � ãA�#wDj>k]n | �~�~s�'s�� �'q| �A�Bjvãĵ&j>yÓ�DjbmÌwDj�n~nºm | �A�©zscySo>zÑ k>¶&zsMm�sMk�m&jbwMj>tbtqr k'�

ÜÌl�mFn_l�o~��j>�~j�npjbo>��s � � ãA�v� Ø B� ãA� · � � � ãA�ç· o~t�zsM� | n�� | m�V¶ r k¾s � ã3�?�BjbySlf��q| �~�;qr y�jbn�zj>m&jbk5�Å°

� ��� � �vq| o~oAt>j�±Uqr �D�1zj>oAm&j>k«��j>�M±.jbySjby~�| jbm«sÅ��q| �~�;qr y�jbn~�~j�±.jbm&m�zsMy�y

� � ãA��� · � � � ãA�}á � � � � ãA�ç· Y� ãA� � £ � � � Ø �

°��Kjbm&m]�?lS�5j>m � � ãA� ý� í�n~j>yжAjbocqr y_wMscyQsM��j>yÓ��ã Ç �M�?�~jc��sMkFk | ��s�� � ãA�̳ | m]n | m zscn~�Kjbmf�| zj>o�s� � � ãA� ¢ � � � � ãA�BwDjbk]n | � | kFkcscy?kFlÓ±.j>o~tMzÑ n~jbn~n�o>zÑ k | n�sº�'q| �~�5jfã3�?�BjbySlDoAlS� r y�z| o>zÑ k>¶&zsMm�sMkºm&j�wDj>tbt�qr k'�f¼½yÓ����z| � | ms � � � ãA��á � � � � ãA�åo~t | �~t{sVn~wDjbk]n | ���Kjb�M�| y�j>�Djbo�s�ã#³ | mFn~�BjbySl�oAlS� r y�z| o>zÑ kF�ps�

� � ãA� ý� í jbo~jbn�zj>mÖs �?� Ø � � .�j!&�m�zÑ �>l{z| m�sMk �Kjb�M±.j>y�j>y~�| j>mÙzj>��npjbyS��j>t>µjbnD�| s´ã§³ | m]np�5j>y�l � ãA�±U�| m | �~��zsMy�lSo;jb�M�?o?zjb�MwDjbkFn | �>��° � � ãA��� � � ãA�Yá � ãA�ÂwMj>k]n | ��n{�?sM��j>yÓ����j>�M�| ySj>�Mj>o�svo~l�� r y{z| o{zÑ k?�~s�]sã3�F�BjbySl&�&l�m | �A�©zsMy�lSo1j>�M�?o?zjb�MwDjbk]n | �Am�sMkÅ� | m&�V¶ r k'�f°�tbnÄs_o>zÑ k | n{�FsM��j>yÓ��npsM�AnpsMyS�KsMt>t>svs�� � ãA�f³ | mFn | nzjboº��j>�c�| ySjb�Dj>o�sct � ãA�#wDjbkFn | �~�~s�5s;�©�'q| �A�Bj�ã#µ&j>yÓ�Djbm�wMjbn~n��~jbk]npl�&&k#zscy�z| o{zÑ k{¶&zscm�sMkamjbwDjbt>t�qr k5�GIH�� � 2 4 JDOQP�X\�OÓWb\?O�X\ è5^ T�WMXg eYLBT1XL8é 3^ Z 4 XJMNi�j>kFlSm]npocqr m&k,j>�c� � Î � Ï u � zsMy�npsMy�zsMm | o#n{zÑ ³ r oBzr �'q| �~�Äzj�n{��° � � �?��.�j/&�m�zÑ �bl�z| oAt>j>�AlSm]n�j>tIsMm&m]�?l�n¶AjbySjbmFn>��µ | �M�Knpj�npo~t��| ySjb�Dj>o}ã Ç �§j>oAjbn�zjbmÀs � � � ãA� ¢ � � � � ãA�#�&jb�~l�w#zsMyÓn~wMj>k]n | � | k§y�lSm&j�zsM�~l�opsMma±dqr �M�Djbn~ySj>mj>k'�°�t � � � Ø �#j>�c�Dj>mySjbn~�f�| y�sM��z| �&lSk'�&µ | �M�«s;�©�'q| �~�;qr y�jbnpj�o~j>µ | y�o~jb�þn��r m&l�kKjby3�° ���Ò±.j ¶Ajbt>jbn~�Bjbm�y�jDzÑ ��ndsMkFm�sck´�Kjb�M±.j>yA�| j>m�s�²k5zÑ o?zj>�c�| B��Aj>m&j�n��?��zsct>lSo&zscnKsMt�sMy�zsc�&�&l�jbyж&zsM�&zsMo~o~sMy

m]�Mj>�.¶Vqr k� ¡ � Ø

· � � · � � ¢ ���Ä��� � � Ê ¨�� � � ¢ � ¡ ª � ¡5¢ � �#� Ø· ���M· ��� ¢ � � � � ¡ á � � £

° �;�?µ | tvndsM��n | t�z| � ¡U¢ � ��Î>� Ï u �'q| �A�fqr ySj�nplB±dqr �M�Mw�zj>m]�Djbk?�Aj�s� ¡ � · ���M·

  � ¢ � �Ä� ¨�� � � � ¢ � � ª  �&· � �c·kFlÓ±.j ¶Aj>t5zj>oAj>k�sM��z| �m�sMk'� ��w?l��&j>m&ob��µ | �M�a±.j>m&m�zscySySm&sMk§sMt�scy�zsM�&�l�sMt | m | o~o&zsM� | k

� ¡ � � ¢ � ��� ¢ � � ��� ¢ � ¡ � � £�Mí

Page 21: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

GIH X\?O�W�\?OçX\ è5^ T_é63^ Z 4 JaW ^ Z>HVPVX^ h�\� ����� � J � è Xg ê PçX^ � ° � Î�� Ï u � zscy�ndscy�zsMm | oa�'q| �A�Bj � �«�'q| �A�fqr ySj�npl�±dqr �M�Mw�zj>m]��zjbnÌs��Ùn | �~t>l{z| �±dqr �M�Mw�zj>m]��zj>m&jbk©m&jbwMj>tbtqr k'�²° � � � ãA� � ã Ç ��űUqr �D�Mw�zj>m]��zj>��n�zj>kcjbnÌs�� ãK³ | m]np�5j>ySlºn | �At>l{z| ¶&zsMm�sMk� | m&�ç¶ r k'�°xn | w#zsM�&�lQsMkF��sMmÌs�n | �At>l{z| �F±Uqr �D�Mw�zjbmF�?�AjX� �ºµ&jby��Dj�n~nºlSmk#zsc�&��s d ¶Aj>y q| y�zjboAn�scySkVsMyS�KsMtbt r k'�

°�tvu � �?�5j>y�l�zsMy�npsMy�zscm | o#��q| �~�Äzjbk B��~jbm&jbn �?k�zj>³&y�jbnpjbl�n�s� � �   � ¢ � � Ê   � � ×  �d � ¢ � � � Ê  Jd

jb�M�Djbm&ySj�npjbk�kfzjb³Bjbt>lSk'�B�j ¶Ajbt>t�qr k�kFljb�M���©Î � Ï u � zscy�ndscy�zsMm | o���q| �~�5j#jbySoM�| µ�zsM� | �æ�j>�~lÓw#zsMy�n.¶&zscnfs�k5zÑ o?zjb�M�| B��~jbm&jbn �?��zsMtblSowMj>k]n | �psMlÓwMscy

� � �   � ¢ � � � �   � � ×   � � ¢� � � � � �7  � � Ê   � � � � � × �7  �   � × �7  � � � � ��Õ×   � � d � £

��t�zsMy�npsMy�s�n | �At>l�z| ã Ç �§³ | m]np�5j>ySl�zjb�An�zj>k�zj>�AjIsd�� ãA��� ¨1� � � ãA�}á � � � � ãA� ¢ � � � � � ãA� ª· � � � ãA��á � � � � ãA�ç· � £ � � � �D�

kFlÓ±.j ¶Aj>t5zj>o�n�m]�Djb�.¶Vqr k'�° � �1á � � �Ä�   � � � q| oAo~tbjb±Uqr �D�1zjbo~�f�| y}sM��z| �}z| sMmÒsÀk5zÑ o?zj>�c�| B'�Aj>m&j�n��?�}zsMt>l�o~µ | tKndsM��n | t�z| wDjbk]Þn | �~��j>t��| kMj�n �5zj>y�o~tbj>�F�r sMt_sMy�zsc�&�&lYjbyж&zsM�&zsMo~o~sMy���j>�Dµ&scn'zsM� | t>ml� � Ø

· � � · � � ¢ � � Ø· � � á � � � · � � á � � � ¢ ©���$á � £

.�j>�c��qr mkÌjb�M�ÒzÑ w�µ | o~oAt�o~t>jb�~l�mFn�³&sM�psM�Òzjbn~j>�Aj>t>j�n~n�� Î>� Ï u � zsMyÓndsMy{zsMm | o��'q| �~�Äzjbn � · � �3·_� Ø �U��Iq| �DtMzÑ n~oMqr m&kÀjb�M�) Ç �À³&sM�psM�Òzjbn~j>�?zj>�An�zj>kcjbn{�K°�t� �¾� ð � Ê �ñ�� Ç ��óalSm]npjb�AwcsMy�y r � | m¦n~j>kFlSm]npocqr ksctbnvs�� Î � Ï u�±Uqr �D�Mw�zjbmF�Fn>��sMµ | y1s�� ��? � �@?vÇ ����±Uqr �D�Mw�zjbmF��zjb�An�zjbkÌ��j>�Mj>�M�Mj>t>l�k�s$�å�'q| �A�Bj ©zj>o�× ? ³ | m]n | k?�&sMm�wMjbnAn�oAlS� r y�z| o>zÑ k>¶~sMlSm&sMk�s�µ�s�¶Ay�zsMoAo~t]q| �;zjbwMj>y3� ��o~tbj>�AlSm]nºlS�]sMt� | o�� ��? �f� ñ ¨ � � ?� ¢ � � �× ? � ª ñ�zj>oÖí 7 � �@? � 7�� � �� ����� XG�OQO Xg W�X\?T ��� � Y AR�/ /�7 K-�2 1 D - Y -�2�2)K3�MVM 3 ySlS�

� �"� �@? �ñ ? ñ �Éñ d�� ?�çñ AC ;0; N - Y AR�/ / K- ; G

° � � ��� z°�y�y zÑ n'zsMo�l�o~��j>�Ajbn�zjb�Bjbm�sMt_sMy�zsM��&l�kFlжAjbySj>m]n�zj>oAn�n~j>µ&j�n7¶cqr k'�� ��� � XG�OQO½Xg W�X\FT � � � Î � Ï u � K3�M P 3�M�K3:2 C ; /BAC'DFE - � K3�M#1)K3 �Q3 E -�2�2 - 7Q3�2 3 NKu � -�/�1 = T ; K� T ��K3 E 3�23:T T C'D K- ;���; 3�T<3�T6T C'D � X�3 d �Öí P -�M ��- ; AR�MSG

� Ø

Page 22: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

� � ������������ ��v����%��� ��½����������������� � ��!Ä"$��%�'��������B)�*O��=�+.=?6 E 6Õ- � /�= G 0 =?+U@� +.= � =F9�68= � G: 9 � E � � +.-F< � 2 ��E1C-&4'- GIHKJMOSJD`ÌP �YP�e�JcZ �7JDO 3LBOSJcW�eB\FZ>\?`ÖXJVWbJcZ>JcTvJDO �^ X\?OQO Xg W�X\?TU\

°xµ&l�³Bjb�A±.jby~qr ySj�npjbk�n'zsM�A�M�]sMy{zsMo�zsVn}sMt�u��,j r kFySl��&j>oAt>lYn�zj>�~�5j>m§w�zjb�Dj>tbtqr k'�'sM��j>y��?m&jbkK�lS��j>m&tbl�z| ¶&zsM�~s±.jby�n~jbw�zj>ocqr m&k,o~tbj>�~l�m]nº±.jbm&m�zsMy�y���� � ���t>jbm_±.j½¶Aj>t>j�np�5j>mIs�7 j>�M��u ��� ¡ �?�5j>ySl&q| o~oAt>j�±Uqr �D���| m]�BzÑ y�n�µ�sMyS�KsMt�n?¶Aj>y q| y3�{sM�Kjby��Mjbn�s#n | w#zsM�&�lQsMkF��sMm³&sM�psM�Òzjbn~j>��ndsM��n | ��zsMm]�?m�sMk�± | � r m&k�m&jbwMj>t>ml3�f°�wDjbkFn | �A±dqr �M�Mw�zj>m]�Djbk?kcj>y�kVsM³ �>o | yQscn | opscm�sMy�kcscyS�KsMt>m&l± | ��¶ r kasMt Ø �f±.j ¶Ajbt>j�n�w�zj>�1zj>m¦�5jbwMj>tbjbnpj�n~n�¶Aj>y q| y�zjbo~j>kcjbn>� ����� � J� è Xg ê PVX^ � °�t�G1Î�7 Ï u � �&l �Bj>�Aj>m �bl�zsMy�µ�scn�z| y�j>kfzj>³5j>t5zj>oA�M�| y;sMtbn�� | m&�V¶ r k'�5µ | �M�«jb�M��o~lS�KsjbySjb�Kl�µlS³5j>�A±.jbyAqr y�jbnpj�nvzÑ �}ySj�sMt�u � j r kFy�lS�&jbo~t>l�n�zj>�A�Bjbm���µ�s�npj>y ¶Aj>ocqr y���zsÅsMt_sMy{zsM�&�&lBk�zj�n�±.jby�n�zjbnpjby3Î� Ø ��i�jbn~o~t��| ySjb�Dj>o ��� �@?�¡d¢{£{£{£�¢A? �&� ¡ � Ç 7 j>oAjbn�zj>mÀsct�GIySjbk�zjb³Bjbt5zjbo%UDs � | �&l �?�©zsVnp�~l¯Û�zsMm�sMkÅ�pscm&��¶&zsc�ps±.j>mm�zsMySy�� ���6G � �1���»� Ê Ø �� �D��°�t�GIwDjbk]n | ��±Uqr �D�Mw�zj>m]��j>�M�§µ | ��j | � | �E&&t>� r o�n�sM�Ìs�7 npsM�An | ��zsMm]� zj>o�sMtG � 7,��� ð G � �;��ñ � Ç 7xó$³ | mFn~µ�sMy��asMt�kYq| t]q| nAn{�°�t=G¦ySjbk�zj>³5j>t5zjbo©zscy�ndscy#ySjMzÑ ��n�o~l��as¦jbySjb�Kl�µlS³5j>�A±.jbyAqr y�jbnpjbmÒsMt§u � �F�BjbySl�� � G � 7,� q| oAo~t>j�±Uqr �D���|scyQsMkFt{sVn | n§zjb�An7¶Vqr k'� ��� � � J � è Xg ê PVX^ � ���M��u � �?�5j>ySl�� scyQsMkFt{sVn | nIo~l��as«j>y�j>��l µlS³5j>�A±.jbyAqr y�jbnpmj>kÀm&j�wDjbtqr m&k'��µ�s�wcsMm

| y��]scmCG�Î 7 � u��&� ¡ Ï u�� �&l �Bj>�Aj>m �bl�zsMy�µ�scn�z| ySj>kfzjb³Bjbt5zj>ob�}sM��j>yÓ�´j>y�j>�Dj�n�npjbo~tÌs¾±.jbm]npl ��Ø �U� � �D�±.jby�n�zjbnpjbySjbk?mj>k'�}zj>oºsM��j>y��?�Aj�±.j>m&m�zscySy G � 7§���� �°�t�G_wDj>k]n | �A±dqr �D�cw�zjbm]�?n�sMt� µ&lS³5j>��±.j>yAqr y�jbn#j>�M�?lSka³�sM�psc��zj�npj>�Aj>o}jby~�| zsMy�y zÑ n'zsMo&zsMm�sMk� lSy�ySjbnAwDjv³�sc�psM��zjbn~j>�~jbt5zjbo?zj>mj>k��º� | m&�V¶ r k5�

� JMéçhUJMé��'H1XJcT � °Ön | w#zsM��&lQsMkF��scm«s���q| w?lS�o?zj>��kcj>�w�zj5zj>�An�s ; =V> 3 ¶Aj>y�t�| n�ySj>�cnçq| �&�o~t]q| �#j>y�µ�sM�M�{¶ r k'��°�{q| w?lS��zÑ n�zjboÂzj>�A�&j>kfzjb�Bjbm s X = � - D Y -�M�AR�M�- P j>y�m&jbwMj>t5zjbo�µ&j>yÓ�Dj�n~n��c�]sMkF�pscm�sMySkVsMy��asMtbt r k¦�asç¶A�8s Y -�M�AR M�- PjbySm&j�wDjbt5zj>o�n�l�o>�� JMéçhUJMé��'H1XJcT � .�j>�c��qr kÖzjbo~tb�~jM�;µ | �M�ÂsM��j>mmF�?l��Bjbm8s«±.j>m]npl��&j/&�m�zÑ �bl�z| k�scyQsM³?¶&zscm8sMt�j>y�j>��lfµ&l�³Bjb��Þ±.jbyAqr y�jbnpj�nÅsMt§u � n�zj>�A�Bjbm$zjb�An~j>yS��j>tbt�qr k'� sMkFk | ��s¾o>zÑ kF�BjbySl�jb�M�?o~t>jb�F�r zÑ w± | �]sMy��©zscn�kVsM³?¶ r k8w?l�o~o~t>s�

� JMéçhUJMé��'H1XJcT � ���M�=7 �©u ��� ¡ m]�BzÑ yÓn}ndsM��n | ��zsMm]� | m�y�j>�M�Mj>mÌsM� | nAn�j>�M�=��Î 7 Ï u�l �Bj>�~jbm �>l{zsMySµ&scn'z| ±dqr �D�cw�zjbm]�D��i1jbkFlSm]npocqr k,sMt

G ��?�¡U¢{£{£>£b¢A? ��� ¡ ��� ?�¡ = ¡ × £{£{£ × ? ��� ¡ = �&� ¡ ×�� �@?�¡p¢{£{£{£b¢A? �&� ¡ � = �q| o~o~tbjb±dqr �D�1zj>o~oAj>yazjb�AnpjbyS��j>tbjbn~n;G1Î67 Ï u��Ò�l �Bj>�~jbm �>l{zsMySµ&scn'z| ySj>kfzjb³Bjbt5zj>o�n{�°�t G�j>�c�8oAlS�KsÌj>y�j>��lµlS³5j>�A±.jbyAqr y�jbnpj�n�zÑ �#ySjM�&sM�Kjby��MjbnºsMt��¦±Uqr �D�Mw�zj>m]�,�M��zsc± ¶&zsMm&sMkam&j�wDjbtqr m&k'�� JMéçhUJMé��'H1XJcT � °�t,l�mFwMj>�At§ySjbk�zjb³Bjbt5zjboÅn�zjbn~j>y�zjbm&j>k±.j>y�µ�sMo~tbm�zsMy{zsMo�zscwcsMy �&l�t | m]�BzÑ npsMm&l ySj>µjbnÅsMt§sMy�zsM��&lkFl ¶Aj>ySjbm]n�zj>o�n{����j>yÓ�Djbn�s�k�zj>oc�| �&�&l�j>kF�Bjbm¦m&jb�þ± | � r mka±.j>y�µ�sMo~tbm�zsMy�m&l3�+ -�/�1 -�2�365 C P0P 3 N G�ÎH7 Ï u � 7H-�T P C'D Y AR�/ /�7JK-�2 1 � 3 > -�M(1 -�/�1�-�M�- >h= X = � - D Y -�M�AR Me- P - P K� D M�-HG� - P ; N C M�-0/ - ; � Ç 7 - ; - P K-�2L7Q3�2 C M#1�3:2 C M#1�3:2 u ��� ¡ � E -�M = î P 3 DFP C > K3�2 1 � X C /�1 Y -�2�2)K3�MVM� Ç î ¢ î � 7 � K- ; 3 N G � î�� -�M�- > =ZY -�M�AR M�- P -�/�1 E -�- ; = T -�/:1 7Q3�M KC ; K- DFP K-�T�R 5 =�� - D -�2 ��= K3�MVX�3 P KCY AR�/ /�7 K-�2 1 / D K3 Y ��K3 7Q3�MSG

�D�

Page 23: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

G �äJMOE3LYOSJVW�JcW_OSJ&Xg Z�X^ [JcN&W ^ Z � 3L5é�éF[�XJ è �ÉX\]W�eY\FZ>\?`ÖXJVW�JMZ>JcH1XJMTUJô�j>�M�Mj>m sM� | nAnÅj>�M� G1Î 7 Ï u � �l �Bj>�~jbm �>l{zsMySµ&scn'z| y�j>kfzj>³5j>t5zj>ob��sM��j>yÓ�8jb�M� � j>y�j>��l#µ&l�³Bjb��Þ

±.jbyAqr y�jbnpj�nKzÑ ��ySjc�,°�tKu ��� ¡ �?�BjbySl�� m]�BzÑ y�n�ndsc�An | �©zsMm]� | m8y�j>�M�Mj>mÒsM��wMs�j>�M� | yÓ�]sMm��¦Î�� Ï u ��� ¡�l �Bj>�~jbm �>l{zsMySµ&scn'z| y�j>k�zj>³5j>t5zj>o>�5sM��j>yÓ�?�~jv�~jbm&�&jbySkMjbt>l�k«sMt�sMy�zsc�&�&lYn r yQsç¶A� | m&o�zsc� | kFkVsMy3Î� Ø ��i�jbn~o~t��| ySjb�Dj>o � Ç �æj>oAjbn�zjbmÀs�� ySj>kfzjb³Bjbt5zj>o%UDs � | �&l¯�?��zscn~�~lÓÛ�zsMm&sMkÅ�psMm&�{¶&zsM�~s�±.j>m&m�zscySy� � ��� � �;���»� Ê Ø �� �D��°�� ySjbk�zjb³Bjbt5zjbo�l�mV¶Aj>k]n{zÑ w zjbo� � ��� ��7Ën~j>y ¶Aj>ocqr yä�

UMj>y q| yжAj�cSC5Î� Ï uæs����l �Bj>�~jbm �>l{zsMySµ&scn'z| y�j>k�zj>³5j>t5zj>o E �Fj>�&l�kak |?| �A�&lSm�zsVndsç�F±dqr �M�Mw�zj>m]��zjbn��E � ØD¢>£{£{£�¢ � Ê Ø �����fw?lS�j>m&ob�5µ | �M�,sct[fGI³�sM� �>l�zscySlSo#�&j>�Al�w#zsMy�n.¶~sMl�n#kFl�ySj>µjbn}±.j ¶Ajbt>m&l�s< ­ fG ��?�¡U¢{£>£{£>¢A? ��� ¡ ��� �&� ¡�

C ® ¡ <­ cSC ��?�¡U¢>£{£{£>¢ ? ��� ¡ � � < C G � � ��?�¡U¢>£{£{£>¢ ? ��� ¡ �A� � i � Ø �

��? � ØD¢>£{£{£>¢ � Ê Ø �§sMySsMkF��sMm��æ°àySjbk�zjb³Bjbt5zjbo~j>k UDs � | �&l �?�©zsVnp�~l¯ÛsMlSm�sckÒsMySkVsMy��asct�zsMo&zscwcsMyºsÒ±.jbm]nplq| o~o~tbjb±dqr �D�1zj>oAn�s� fG ��?�¡p¢{£{£{£b¢A? �&� ¡ ��� � G � � �@?�¡d¢{£{£{£b¢A? �&� ¡ �A� � ��� �@?�¡U¢{£{£{£b¢A? �&� ¡ � � i � �D�

�@?�¡U¢{£{£{£b¢A? �&� ¡ � Ç � j>�M�Mj>m&y�jbnAnpj>y�l�o}�Kjb��n r �V¶ r k«sM�&m&lä�°àySjMzÑ ��ndsMk sMyQsc³?¶&zsMm m]�?lSy�w#zsMm]wcsMy{z| ��µ | �M�©sMt fG©� G')�� y�j>kfzj>³5j>t5zj>o�o~tblSm]n�zjbmÖsMt � j>y�j>��l

µlS³5j>�A±.jbyAqr y�jbnpmj>k�j>�M�,³&sM�psM�Òzjbn~j>�Aj>o�j>yA�| zsMySy3zÑ n�zsMops� �"! � � J� è Xg ê PVX^ � °�tLfG�Î�� Ï u ���&l �Bj>�Aj>m �bl�zsMy�µ�scn�z| ySjbk�zj>³5j>t5zjbo~�c�| yvsMt�n§� | m&�V¶ r k'��µ | �c�©sMt GwMj>k]n | �A±Uqr �D�Mw�zjbmF���¸zsMyÓndsMy�lvzscn~³�sM�psc��zj�npj>�Aj>t5zj>o?zj�wDjby�¶{q| nAn�y�zj�np�~jc�° � ySj>kfzjb³Bjbt5zj>o�nvs�³�sM�~sM��zjbnpjb�An~�psMm&oAtb± | �A�©zs �>l�z| ±Uqr �D�Mw�zj>m]��zj>m&jbk«m&jbwMj>t>t�qr k'� �,+ � � J � è Xg ê PVX^ � °�t � j>ySjb�Klf±.jbyAqr y�jbnpj�n�ySjMzÑ ��z| G§wMj>k]n | �A±Uqr �D�Mw�zjbmF��� zsMy�npsMySl�zscnp³�sc�psM��zjbn~j>�~jbt5zjbo?zjbnl���zsMm]�BzÑ n�zsMoAnpsM�An�z| m�sMkK� | m&�V¶ r k'�&µ�s�npj�npoAt�| y�j>�Djbo � Ç �æjbo~j�n�zj>m�±.j>m&m}zsMySyÕÈFâ>ã �� � �1��ì©í�

°���j>mmF�?l��Bjbm È?â>ã ��� � �;��òþí npjbyжAjboMqr y ��zsM�~��j>yÓ� � Ç �}�?�ps���sMkFk | ��sctåzscnp³&sM�psM�Òzjbn~j>�Aj>t5zjboAnl���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| m&sMkKmjbwDjbt>t�qr k5�� JMéçhUJMé��'H1XJcT � °�tvu �&� ¡ �?�5j>y�l�³�sM�~sM��zjbnpjb�AnpsM�An | �©zscmF� | k¾q| o~o~tbjb±dqr �D�Y�| o?zjb�;zj>m&jbk,kBq| wDj�npkMjbtbn�zj>�5j>m�sMtjbySjb�KlY±.jbyAqr y�jbnpj�n}ySjMzÑ �&z| wDj>k]n | �A±dqr �D�cw�zjbm]�»zscnp³�sc�psM��zjbn~j>�~jbt5zjbo~jvwcsM�M�,l���zsMm]�BzÑ n�zsMoAnpsM�An�z| ��wcsM�M�§³Bjb�&lS�l���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| �� JMéçhUJMé��'H1XJcT � ô�j>�M�Mj>m&jbk G�Î67 Ï u�� zjbo fG�Î�� Ï u�� | y��DsMm8�l �Bj>�~jbm �>l{zsMySµ&scn'z| ySjbk�zj>³5j>t5zjbo~jbk5�sc�Kjby��Djbk r �c�]sMm�scm&m�sMk¦sMt � j>y�j>��l1µ&l�³Bjb�A±.jby~qr ySj�npm&jbk�s§³�sM�psc��zj�npj>�Aj>o�jby~�| zscySy3zÑ n'zsMo~sMl3�ºi�j>kFlSm]npocqr kÀs��� G � ¡ )�fG�y�j>kfzj>³5j>t5zj>o�n{��sc�Kjby��«j>�c�̵ | �Kj | � | �E&�tb� r o�n�sc�¾sMt�u ��� ¡ n�zjb��� zj>o 7 ndsc�An | �©zsMm]�DsMlkBq| tDq| n~n>��B'jby�wMjbnD�| �&lSkasMt_sÅy�zj>m]�Djb�Dj>oºk�zj>�A�#zj>ob�Bµ | �M�«s��¦Î� Ï u ��� ¡ ySjbk�zj>³5j>t5zjboºwcsç¶ | mT � � | oAtbn�zsMy��ºzr �?jc�§° npsMm]n'zsM�A�M���©zsco | �&lSk¦±{zjby�zj�w�zjb�Bjbm´�Ksç¶A�Õ�lSt | mF�BzÑ ndsMm&l�± | �{¶ r k'�;µ | �M�Âs��ÉwDjbk]Þn | �A±dqr �M�Mw�zj>m]�,�&l �Yjb�~j>m �>l�zscySµ�scn�z| ��n | w�zsc�&��zs�±.j>m&m�zscySy � � ��� � �;���»� Ê Ø n~jbnpoAt�| y�j>�Djbo � Ç �æj>o~j�n�zjbm��

°´±.j>m]n~lSj>k�o~tbj>�~l�m]n�sMt � j>ySjb�KlV±.j>yAqr ySjbn~jbnYySjMzÑ ��z| wMj>k]n | �A±Uqr �D�Mw�zjbmF�Mj>kvjb�M�?�©zsMo_zsVnp³�sM�~sM��zjbnpjb�~jbt5zj>oAj>kfzj>m]n¶{q| m&m&j>k_y�zj�np�AjM�;°��Kjbm&m]�?lS�5j>m�jb�M�]sMt | m | oAtbn'zscy��?��s�o | � | y ¶ r k�sMt � sMt | m�³�sM�psc��zj�npj>�Aj>o1j>y~�| zsMySy3zÑ n'zscopsMlÓn{�sc�Kjby��Djbkal���zsMm]�BzÑ n�zsMoAnpsM�An�z| zscn~³�sM�psc��zj�npj>�Aj>t5zj>o~oAj>y�sM�}z| �&m�sMkKj>�M�?��zsMo~�}z| y3�sMkFk | ��sMt � ±.jbyAqr y�jbnpj�n#y�jDzÑ ��z|wMj>k]n | �A±Uqr �D�Mw�zjbmF�Mj>kF�f�| y1�Kl�m&��q| oAo~tbjvk�zj�n | o~tbn�zsMy��Fn�kVsM³ r m&k'�

� �

Page 24: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

G TbPS`Ì\=�YP�e�JcZ �7JDO 3LBOSJcW�JcNæXJMZbW�JMOQ`ÌJcH1XJcTUJ �G � 3L5é�éF[�XJ è �ÂZ>JcéLBOçX\FZ{P�T8XJcZbWYXJMN�XJ è JMN�P è [JcZ>H�NºXJMe�XJcZ>J�[ ^�è \]W�N ^ HfX^ WYXJcW�JMO ��� � � J � è Xg ê PVX^ � ���M��u � �?�5j>y�l � sMySsMkFt{scn | nÀoAlS�Ks µ&l�³Bjb�A±.jby~qr ySj�npm&jbk � | m&� r m&k'�_µ�s sMt �n~jbn~o~t�| ySj>�Mj>o�³ | m]n7¶&zsMm�sckÂwcsMm | yÓ�]sMm�� m]�BzÑ y�n�kYq| �~m]�Djbt>jbn~j¦sctau � �?�5j>m���µ | �M�8s�� �

� �Kj�npo~tbjbnjb�M�«o~l��as�jbySjb�KlBµ&lS³5j>��±.j>yAqr y�jbn>� �� � � J� è Xg ê PVX^ � °�t�u � �?�5j>ySl��Ém]�BzÑ yÓn�µ�sMy��asMt | m,ySjb�M�Djbm¦sc�wcs�jb�M� ��Î�� Ï u �&l �Yjb�~jbm �>l{zsMySµ�sVn'z|±dqr �M�Mw�zj>m]�D�K°�t9���?m&jbk¾sMt - Ç � ³ | mFn~��sMm�wDj�n~nI�D�psM�lSj>mo?zj>m8s � � FÈ� � - ��� ¬ � � ® ¡ < � � � - � � = �wMj>k]n | �An�zj>�An.¶cqr k'����M��� Ç � � ����oAt�zsM� | n�sMt����Aj>� r y�zsM�AlSoazj>��n�zjbk�zjbm&j>k�� | m&� r mk5�'µ�sÅ��zsM�A�Kjby�� - Ç �H� ¡ � �b��j>o~j�n�zjbm±.jbm&m�zsMy�y � �&FÈ� � - ��ý�]��� �����h* XJVWbJMO � � > -�2�2 1 = E -�2b3 � Ç � � �_� ; N K3 > -�/�1 D -0/�R M K3 D =?; K- D�P K-�T -"3 N'� Y AR�/ /�7JK-�2 1 2 -�T � 3�T6T C'D3 N � � � � ¡ � �b� 3�M 3�T N 3 P -�/�1 ;0=V> 3@X = � - D Y -�M�AR M�- P 3 Nvu � -'R T6M = 5�- ; N = P K- DHE -�2 GG �YPSe�JMZ �äJMOE3LYO�JcW8XJMZ>P è W �^ W�JMZbJDP�Y\ èB^ Z{`xX\�OSPSTIJcé �'T5XJcé?[�JMN&W ^ Z{`ÌJcH �^

ô�j>�M�Mj>m�sc� | n~n1sMt�u � �F�Bjbm�j>�c�� j>y�j>��lDµ&l�³Bjb�A±.j>yAqr ySj�n�zj>o1sMm&m�sckvj>�M��G�Î>7 Ï u � ³�sM�psc��zj�npj>�Aj>ojby~�| zscySy3zÑ n'zsMo~s�i�j>kFlSm]npocqr m&k�s97 �åu �&� ¡ ³&sM�psM�Òzjbn~j>��ndsM��n | ��zsMm]�?��sMm�j>�M� �ÌÎ>� Ï 7 ��u ��� ¡ o~l��as��'q| �~�Äzj�n{�° � k |�| �~�&l�m�zscnds��F±Uqr �D�Mw�zj>m]�Mj>lºySj>�c�Dj>mj>k s� ­ Î:� Ï u ��? � ØD¢>£{£{£�¢ � Ê Ø �Å�&l �Yjb�~jbm �>l{zsMySµ�sVn'z|±dqr �M�Mw�zj>m]�Djbk5�Ä��o~tbj>�~l�m]n�s �xySjbk�zj>³5j>t5zjbo�kFlÓ±.j ¶Aj>tbµ&jbnM�| s � � ãA���x¬ ��� ¡­¯® ¡ ­ � ãA� = ­ � ã Ç ���#± | �~��zsM��scm�� ����� � J � è Xg ê PVX^ � ° ��� G ) �»�&l �Yjb�~j>m �>l�zscySµ�scn�z| ySjbk�zj>³5j>t5zjboAn�sMt � � G_³�sM�~sM��zj�npjb�~j>t5zj>oA�f�| y�m]�Djb�AnU�±.jbyAqr y�jbnplB�'q| �A�#zj ¶Fzjbm&j>k,� | m&�V¶ r k5�°���j>mmF�?l��Bjbm �5ã Ç �D�F�~jv±.j>m&m}zsMySy� C � ãA��� C ��E � ØM¢{£{£{£b¢@? Ê ØD¢@? × ØD¢{£{£>£>¢ � Ê Ø � zj>o ­ � ãA���åãU�?sMkFk | ��s��´� G ) � ±.jby~qr ySj�npl?�'q| �A�#zjbn�sct � ±.jby~qr ySj�n G�³�sM�~sM��zj�npjb�~j>t5zj>o?zj>µ&jbtºnpsM�An | t�z| ? �?j>�&l�k³&sM�psM�Òzjbn~j>��w | m�scySm�sMkam&jbwMj>t>t�qr k'�� JMéçhUJMé��'H1XJcT � °Ö±.j>yAqr ySj�npl5�'q| �~�5j�jbySm&j�wDj>t5zj>o�n�sctvlSm&� | k | y ¶~s��µ | �M�,s �Ò��G ) �»³}zsMy��#zsU¶~s��~sç¶�nds�wcsMmsct � ±.j>yAqr ySj�npj>mY��fw?lS�&jbm&o>�'µ | �c�,s'��y�j>k�zj>³5j>t5zj>o�ã3�?�BjbySl � ã Ç �����&j>�Al�w#zsMyÓn7¶&zsM�~s�±.j>m&m�zscySy�s

� � � ãA��� �&� ¡� ­¯® ¡ �­ � ãA� � < ­ G � � � ãA�~� � i � � �q| o~o~tbjb±dqr �D�1zj>o>� � � � � J � è Xg ê PVX^ � i1jbk?l�m]npocqr kÕsMt � j>y�j>��lĵ&l�³Bjb�A±.jby~qr ySj�n�j>�M� / � G � - � � - Ç 7,��³ | m]n.¶&zscn{��°�tu � �?�BjbySl>< ­ G � - � ��? � ØD¢{£>£{£b¢ � Ê Ø ��wDj>k]n | � | k zsMyÓndsMy�Dj>m&jb��zsMyÓn�� ù � y�lSm&j�zsM�~l�o�sMyÓnpj>�AjbnÄsMt � ±.j>yAqr ySj�n/ �?�5j>y�l�ySl�m&j�zsM�AlSoKzjb�~l�mFnM�| npjb�?zjbm&j>k,� | m&�V¶ r k5�°�t�nfsct u � �F�BjbySl � � Ê Ø � �F�&lS��j>m&tbl�z| o�sVÁKm�sMy�n~j>�Ajbn{�csM��j>yÓ�Âzscnpµ&sMyQsM�Is / ³ | m]n | mÂzj>o�³�zsM�Aµ r t{sM� | os < ­ G � - � ��? � ØD¢{£>£{£>¢ � Ê Ø �ÄwDj>k]n | � | kFkVsMy�sMt� ±.j>yAqr y�jbn / �?�5j>y�l�zj>�AlSm]nD�| o>zÑ k>¶&zsMm�sckKm&j�wDjbt>t�qr k'�� JMéçhUJMé��'H1XJcT � °�� ù � y�lSm&j�zsM�~l�o�zj>�AlSm]nD�| n�zj>��jbySjb�Kjbl�n�sMt�� µ&lS³5j>��±.j>yAqr y�jbn / ³ | m]n~�BjbySl;zj>�AlSm]nD�| wDj>k]n | �psMl¯Þm&sMk§� | m&�V¶ r k'�

i�jbnpoAt�| y�j>�Djbo � Ç 7˳&sM�psM�Òzjbn~j>��jbo~j�n�zj>m«wDjb�M��qr k,sMt�n�sMt�u � �?�5j>y�l�� � �1�Äj>�M�?o?zj>�MwMj>k]n | ��n{�5sM��j>yÓ��Aj>m&�j>ySkcj>tblSk«sMt_sMy{zsM�&�&l5n r ySsç¶A� | m&o&zsM� | kFkVsMyäÎ� I

Page 25: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

� Ø ��°�t � � �1�#��j>�M�| ySj>�Mj>o�s � � ��� � � zjb�~lSm]nM�| n�zjb�~�~jc�� �D�Õ°�t�u � �?�BjbySl>< ¡ G � �1� ¢>£{£{£>¢ < ��� ¡ G � �1� ¢ � � �;�f��zsMtblSo s�n�zjb� n~j>�~�Òzj>oAt>j�npj>o | �AlSj>m]n�zs �>l{z| ¶&zscn�kfzj>³]w?lSoAj>y�l3�¼½y��Å��z| � | maj>�c� � Î 7 Ï u � ySjbk�zjb³Bjbt5zjboAn�m]�Djb�Mqr m&k'�&sc�Kjby��?�M�| y'kBq| m&m]�Mj>m«�Bj�ySjbµ&jbn#y�zscn~m&l3�]µ | �M��l �Bj>�~jbm �>l{zsMySµ&scn'z| � � � � � J � è Xg ê PçX^ � °�t � y�j>kfzj>³5j>t5zj>o�nÀsMt � ±.j>yAqr ySjbn9G8³�sc�psM��zjbn~j>�~jbt5zjbo?zj>µj>tÕnpsM�An | t�z| m | �~��zsMySl�ojb�M�?o?zjb�MwDjbkFn | �~��j>t��| m&jbkÌ� | m&�V¶ r k5�°ön | w#zsM�&�&lSsMkF��sMm�j>�c� | y��DsMm�µ&lS³5j>��±.j>yAqr y�jbn¦ySlSmj�zsM�AlSo»zjb�~lSm]nM�| npjb�~j>lÓn¾npjbk?l�m]n7¶cqr k'�ÅsM��j>y��»j>yA�| zsMy�ywcsMySsM��j>y��Ko~l��as�±dqr �M�Mw�zj>m]�,�~jb� r y{zsM�~l�oazj>��n�zj>kfzjbm&j>k«lSm]wMj>�~t_k�zj>³5j>kfzj>m]n>�ô�j>�M�Mj>m8sM� | nAn_sMt�u � �?�5j>ySl � m]�BzÑ yÓn�µ�sMy��asMt | m¾j>�c�F��Î�� Ï uÉ�l �Bj>�~jbm �>l{zsMySµ&scn'z| ±dqr �D�cw�zjbm]�D�i�j>�c��qr k§±.jby3��µ | �M�«s � Ç � � ����o~t�zsM�þj>�c�,�~jb� r y�zsc�~lSoazjb�An�zj>kMjIsMt����Fm&j>k'� ����� XG�OQO½Xg W�X\FT � � N � � � � ¡ � �b� X = � - D Y -�M�AR M�- P E K3 D > -�M(1 Æ � C 2 P �Q3 - ; - P K-�2�3 � � � M = 2 - K3 D =?;K- D = 2 P C P K- D > - D C Me-�/ - ; 3 � � FÈ�� �äÆ � 7'-�T P C'D0D 3 G

GIHKJMOST �^ � �^ `�J èBè �5PSTBXJMé?JMN ����� � � J� è Xg ê PVX^ � ô�j>�M�Mj>m´sc� | n~n�j>�M�G�Î 7 Ï u����&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| ySj>kfzjb³Bjbt5zj>ob�;sc�Kjby��Ìj>�M� �jbySjb�KlBµ&lS³5j>��±.j>yAqr y�jbn~jbn�zÑ �#y�jM��°�t � ±.j>yAqr ySj�n�GI³�sM�psc��zj�npj>�Aj>t5zj>o?zjbµ&j>t�ndsc�An | t�z| j>y�oM�| ±U�| ��j>m&m]�?lSo?zj>�DjbkMjbms� ­ C ��?�¡U¢{£{£>£>¢A? �&� ¡ ���¿¨ < ­ G ��?�¡U¢{£>£{£>¢A? ��� ¡ � ¢ < C/G �@?�¡d¢{£{£{£�¢A? �&� ¡ � ªq| o~o~tbjb±dqr �D�1zj>o~oAj>y��Kjb�Dµ�scn�zsM� | t | nAn � ­ C5Î�7 Ï u ��?~¢ E � ØD¢{£{£>£>¢ � Ê Ø �ıdqr �M�Mw�zj>m]�DjbkMjbnÌzj>��n7¶cqr k'�

°�t�jbySoM�| ±U�| ��j>mmF�?l�o?zj>�Mj>k � Ç 7˵&j>yÓ�DjbmÌwDj�n~n�±Uqr �D�Mw�zjbmF��zjb�An�zjbkMjblS�f�| y�kfzj>³5j>tbt�qr k«sJ � �1�v� ��

�� ¡½¡b� �;� £>£{£ � ¡ � � �1���� � � � ������ ¡>� �;� £>£{£ ��� � � �1�

��� oAt>lS����jbnp�AlSk r ov��zscnp�AlÓÛ | n>��sMµ | y � �$� Ê ØM£ .�j>�M��qr k»zjbo~t>�AjM�

µ | �M�«sMt�G_wDj>k]n | �A±dqr �D�cw�zjbm]��UDs � | �l��©zscn~�~l¯Û}zsVwMscyB±.j>mm�zsMySy�sJ � �;��� B � G � �;� D Ý � � G � �1�

q| o~o~tbjb±dqr �D�1zj>o>�UMj>y q| yжAj��I�YÈ � u �&� ¡ ��sÀwcsMy�z| o�j>ySjb�Ò�r � � � Ê Ø ��á � � Ê Ø �3�?j>o���zscnp�AlÓÛ | k¾wDjbk]n | ��npj>�?zjbn>��sM��j>yÓ�l�t | � | ��±;sct�u � �&� ¡ � � n�zj>�~�Aj>yä����oAt>jb�~lSm]n�sMt�j>y�oM�| ±U�| ��j>m&m]�?lSo?zj>�DjbkF�f�| y��&j!&&m&l�zsMy�m&lBn r � r mk,j>�M�

J Î 7 Ï �_��È � u ��� ¡ �#ySj>kfzjb³Bjbt5zj>o�n{�'��j>y��?m&jbk§k |?| �A�&lSm�zsVndsç�F±dqr �M�Mw�zj>m]�Djbl��&l �Bj>�Aj>m �bl�zsMy�µ�scn�z| sMk'�ÜÌl�mFn«l�o~��j>�Ajbnpjbo>�vÈFâ>ãDJ � �;�Òì í � � Ç 7,�KnpjbyжAjboMqr y3�Éi�j>kFl�mFn~ocqr kås'J � �;�§��zscnp�AlÓÛ � � �1�l�m]wDj>�At5zj�n{�Äô�jb�M�Djbm&j>k«s$# ­ C � �1� ��?A¢LE � ØD¢>£{£{£>¢ � Ê Ø �#o~t�zsc� | kKj>tbj>m��©zscn~�~l¯ÛajbySjb�Kjbl3�f¼½yÓ�§��z| � | m«s# ­ C Î 7 Ï u �&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| ±dqr �D�cw�zjbm]�Dj>kFµ&jbt�¶ r n r m&k'�'�Kjby��Mj>kFkMjby�±.jbm&m�zsMy�y ¬ �&� ¡� ® ¡ #

­ � � � C����­C ¢scµ | y� ­C s�¶&z| ySlSoA�Kjb�An �v� | m&j �dkcj>�½�?o~tblS����z| y r ���

°�t�nºo~t | k#zsco�� | m&��scm&l3��µ | �M�§sMt�j>ySoc�| ±U�| �Kjbm&m]�?lSo?zjb�Dj>k,µ&scn'zsM� | t>t�zsMk§��j>��s�±.j>yAqr ySjbn#�BjbySoc�|�Mj | ��jbn~�~l�zsd¶&zscn{�F��l�wDjbyYjbt>j>kFkcj>y�ySjbµ&jbn�k?lÓ±.j ¶Ajbt>m&l�jb�M��npjbn~o~t��| ySjb�Dj>oazjb�~lSm]nM�| wDjbkFn | ��µ | o~o~t�zscn§zj>o}kfzj�n}j>�M�]Þsct | m�³ | mFn~��sMm«wDj�n~n�zj>�~l�m]nD�| wMj>k]n | ��µ�s�¶Ay�zsMoAo~t]q| �;zjbn>�°�t�G � - � � - Ç 7,��³ | m]np�5j>ySl � � �� � � zjb�~lSm]nM�| n�zjb�~m&jbkanpjb�~��zj>o~tbjbn~j>oº��zsMt>l�o�zscnºkfzj>³5j>tblSk,s�< ­ G � - ���? � ØM¢{£{£{£b¢ � Ê Ø ��wDj>k]n | � | k'�1sM��j>y��Mj>kÀjb�M�?�Bjbm´sMtÅsc� | n~n�³ | m]n | m²zscn~�Kjbmf�| ³�sM�~sM��zj�npjb�Aw | m�sMyQsckzjb�~lSm]nM�| l3��.�j>�M��qr k�s��8��¬ �&� ¡­¯® ¡   ­ � < ­ G � - � zjbo ¤ �æ¬ ��� ¡C ® ¡ ¥ C � < C G � - �§zj>�AlSm]nD�| wMj>k]n | � | kVscn>����tbj>koAkVsMy�zsM�AlSo}oAt | �At{scn�zscnºs¨M� ¢ ¤åª � ��� ¡� ­ ® ¡

��� ¡�C ® ¡ �

­ C � - �   ­ ¥ C

� �

Page 26: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

kFlÓ±.j ¶Aj>t5zj>o�sM�V¶~s���j>��1°��Kjbm&m]��l��Bjbmv� zj>o ¤ k�qr y q| mF��q| tbm&j>k ��]n'z| yä�Äs}kfzjbn�zjb�~l�mFnM�| wDjbk]n | ����µ�s�¶Ay�zsMoAo~t]q| �;zj>�Ajn~j>y ¶Aj>ocqr y�s� | o���� ¬ ��� ¡­ ® ¡ ¬ �&� ¡C ® ¡ � ­ C � - �   ­ ¥ C

B ¬ �&� ¡­¯® ¡ ¬ �&� ¡C ® ¡ � ­ C � - �   ­   C D �� � B ¬ ��� ¡­ ® ¡ ¬ �&� ¡C ® ¡ � ­ C � - � ¥ ­ ¥ C D ��q| o~o~tbjb±dqr �D�1zj>o>�i�j>kFlSm]npocqr m&k¦jb�M� ��� G ) ��Î � Ï u � ±.j>yAqr ySjbn~l��'q| �~�Äzjbn{�v° � i � � ��jb�M�Djbm&ySj�nvkBq| wDj�npkMjbtbn�zj>�5j>mÒs�j>�~lÓw#zsMy�nAwDj>k]n | ��µ | o~oAt�zsM�~sű.jbm&m�zsMy�y

· � � � ãA�ç· � � �&� ¡� ­¯® ¡�&� ¡�C ® ¡ �

­ C � � � ãA�~� �­ � ãA� �C � ãA� � ã Ç ��� £° ��� Ø ��iazj�npj>y��f�| y�sM�}z| �&lSk'�&µ | �M�«s;�vñ#� ¢����å� � ¢���� � ���#�'q| �A�Bjb��sM�psc�ÂzÑ w?µ | o~oAt�zsM�~s�l��]sMt$ � �_ñ(� ¢���� ��� � ^` � ¬ �&� ¡­¯® ¡ ¬ �&� ¡C ® ¡ � ­ C � � � ãA�~� �­ � ãA� �C � ãA�}ÈDã £G �YPSe�JMZ �äJMOE3LYO�JcWIPSH ^ `�JcW�Z{P�X\][]\?O è �JcZbW_NºXJMe�J

ô�j>�M�Mj>m sM� | nAnÅj>�M� G1Î 7 Ï u � �l �Bj>�~jbm �>l{zsMySµ&scn'z| y�j>kfzj>³5j>t5zj>ob��sM��j>yÓ�8jb�M� � j>y�j>��l#µ&l�³Bjb��Þ±.jbyAqr y�jbnpj�n,zÑ ��ySjc�Õi�j>kFlSm]n~oMqr m&kÒsct§u ��j r kFySl��&j>oAt>l#n�zj>�~�5j>m�jb�M� ;ÎBu�� Ï u���lSt | �Kj�np�Al�zscn>�fsM��j>yÓ�jbySjb�Djbn�npjbo~t_sMt ��Ø � Ø � �«q| o~oAt>j�±Uqr �D�1zj>oAm&j>k'�

.�j>�c��qr k8sMt �GÂ�àÍ )�G¦y�j>kfzj>³5j>t5zj>o�n{� �fw�l��&j>mo>�#µ | �M��sMt �G�Î:7 Ï u � j>�M� | y��]scm �l �Bj>�~jbm�Þ�bl�zsMy�µ�scn�z| y�j>kfzj>³5j>t5zj>ob��sM��j>y���sMt �� �xÍ � �Ö��jbySj>��l&±.j>yAqr ySj�npm&jbk�sMtºj>�M�?lSk�³&sM�psM�Òzjbn~j>�Aj>o j>y~�| zsMySy3zÑ n'zscops�° < ­ �GÌ� ÔC)�< ­ G ��? � ØD¢{£>£{£>¢ � Ê Ø �Kq| oAo~tbjb±Uqr �D�1zjbo�kBq| wDj�npkcj>tbn�zj>�5j>m npjbn~o~t��| ySjb�Dj>o � Ç 7 j>o~j�n�zjbms«ySl�m&j�zsc�~lSoÌzj>�~l�m]nD�| n~j>�~jbkF�~j�±.j>m&m}zsMySy ���� ��� � �� � Ô � �

� ��� � �x�U�K°�tÅj>ySoc�| ±U�| �Kjbm&m]�?lSo?zjb�Dj>k¦�Kjb�M�Djbt>m&jbk5�wcsM�c��l�o�lS�]sMt��� ­ C�� � ­ C ��?~¢ E � ØD¢{£{£{£�¢ � Ê Ø �U�°���j>mmF�?l��Bjbm«sÅÍ�l�t | ��jbn~�~lQs�lS�&zsMm]�BzÑ n'zscoAndsc�An'z| �sMkFk | ��sMt �G�zj>oG_³�sM�~sM��zjbnpjb�~jbt5zj>oAj>kFµ&jbt_npsM�An | t�z|m | �~��zsMySl�o�j>�c��o?zj>�MwMj>k]n | �~��j>t�| k?�Aj�npjbyжAjboMqr y �� �ÖÔ ) �Â� �s�sKÍæj>�M�«l���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| lSt | ��jbnp�AlQs�sck?k | ��±.j>m&m�zscySy �� � Ê Ô ) �Õ�

GIH X\]W�eY\FZb\�`ÖXJcW�JMZbJMH1XJMT è XJDO �7JDOSO�XJMe �^ JMOST �^ � �^ `�J èBè �5PSTBXJMé?JMN°�t � j>ySjb�Kl�µ&l�³Bjb�A±.jby~qr ySj�npj�n���j>�Mµ�scn'zsc� | t�z| G�Î67 Ï u �ÌwDjbkFn | �A±dqr �M�Mw�zj>m]�?m&j>k¦wMj>�M��qr k¦s i � � �.�j/&�m�zÑ �>l�z| ��sMm«ySjMzÑ �An fG�� G�) � zscn~³�sM�psc��zj�npj>�Aj>t5zj>o?zj�n{�° � i � Ø �Õq| o~o~tbjb±dqr �D�1zj>o~�;�| y�kBq| wMjbnpkcj>tblSk'��µ | �c�xsMt � ySl�m&j�zsM�AlSo©zj>�AlSm]nD�| npj>�Aj>l_m&j>� ±Uqr �D�Djbm&j>k�s

³&sM�psM�Òzjbn~j>�Aj>t5zjboÀ��j>�Mw#zsMySsMo~t�n'zsMo&zscn'z| yä����l�wDjbyv±.j>m&m�zscySy � � ��� � � ��� � � ��� ��� � � npjbn~o~t��| ySjb�Dj>o � Ç �

jbo~j�n�zj>mY�IB | m]n | oAm�sMk�n��r m&l�kK±.j>y�µ�zÑ w?m&l�s'&&�M�DjbyS��jbnºsM�A�ps�&µ | �M�«s�ySl�m&j�zsc�~lSoKzj>�~l�m]nD�| n�zj>�~�5j>mÌs< ¡ G � � � �1�~� ¢>£{£{£>¢ < ��� ¡ G � � � �;�A�´zjbo�< ¡ fG � �1� ¢>£{£{£>¢ < ��� ¡ fG � �1�Y��zsMtblSo | k�kBq| t]q| n~npl�zscnp��j>m&j�npj�n � ��zsMo�o~t�z| wcsMysÅ��zsMtblSo�np�pscm&o~t�± | �~��zs �>l{z| nU� s �� � �1��UDs � | �&l¯�?��zscn~�~lÓÛ¦zÑ �.¶~s�ySjc��s�s�� zsMy�npsMySl�zscnp³&sM�psM�Òzjbn~j>�Aj>t5zjbo�l���zsMm]�BzÑ n�zsMoAnpsM�An�z| ��sMkFk | ��s�m | �~��zsMySl�o�j>�c��o?zj>�MwMj>k]n | �~��j>t�| k?�Ajl��]sMt f�à� � ) �_��°��Kjbm&m]��l��Bjbm�sMtÒzsVnp³�sM�~sM��zjbnpjb�~jbt5zj>o�lS�&zsMm]�BzÑ n'zsMo�w#zsMy�n�z| �5sMkFk | ��sMt f�à� Ê � )�

q| o~o~tbjb±dqr �D�1zj>o�npj>y ¶Aj>ocqr y3����M�?o~tbj>�F�r o~t�zsM� | y�zsMoAopsMy�sc��z| �&l�k5�µ | �M�«sct � ±.j>yAqr ySj�n fG�³�sc�psM��zjbn~j>�~jbt5zjbo?zj>µj>t�ndsM��n | t�z|f� ­ C�Î � Ï u ��?~¢ E � ØD¢{£>£{£>¢ � Ê Ø �#j>ySoc�| ±U�| �Kjbm&m]�?lSo?zjb�Dj>kF�Aj_±.jbm&m�zsMy�y

f� ­ C}�¿¨ < ­ fG ¢ < CFfG ª � �&� ¡�� ® ¡

�&� ¡� ® ¡ � �) � � < ­ c � � < C8c

£

� i

Page 27: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

��wcsM�&�~scnpl�k r o��©zscn~�~l¯Û | kcsVn}sMy�kcscyS�KsMtbwcs�s�±.j>m]n~l q| o~oAt>jb±dqr �M�;zj>o�n�sfJ � �1��� B ��� � �;� D Ý � J � � � �1�~� � �� � �1� � i � I �� � Ç ���ºsMySsMkF��sMm«lSo�±.jby zÑ �~µ�scn.¶ r k'�G N ^ `ÌeB\?N&W �7JDO 3LBOSJcW�RB\?Zb\ 4 �7JDOSTUH�Xg è J

.�j>�c��qr kIsMt � jbySjb�KlFµ&l�³Bjb�A±.j>yAqr ySj�npm&jbkvj>�M� G�Î 7 Ï u � ³&sM�psM�Òzjbn~j>�Aj>ofjby~�| zsMy�y zÑ n'zsMo&zscn{��ô�jb�M�DjbmKîjb�M� | yÓ�]sMmXU | �~��sMmF�?��zjb�~µ&j�nD�| t�zsM�An�npsM�An | �©zscmF�M�çsM��j>y����?zjbo~tbµ�sMyS�KsMt>s}s�7�³&sM�psM�Òzjbn~j>��ndsM��n | ��zsMm]�?m�sMk'�°�t�G � î�� � � scyQsMkFt{sVn | nºsMt � j>�M�§k | ��³�sMk]n�±.j>yAqr ySjbn~��sM�~sM�?¶&zsMm�sck�� | m&�ç¶ r k'� ��� � ��� J � è Xg ê PçX^ � °�t�G � î��#±.j>yAqr ySjbn~��sM�~sM�a±.j>y�o~tMzÑ mÄzjbmÌsct( � G � î��A��� \ £{£{£ \�� ¹ ÈFâ>ãPJ ��?�¡U¢{£>£{£>¢A? ��� ¡ �}È ?�¡5£>£{£ È ? �&� ¡

³ | tbl�n{zÑ w§oAt�zsM� | n§zj>��n7¶Vqr k'�� JMéçhUJMé��'H1XJcT � °�t¾l�mFn~j>�D�&zsMy�n~�psMmo~tb± | �~��zs �>l{z| jb�M�Djbm&ySj�n�zj>�;�| yÌzj>o«s � i � I �¦q| oAo~tbjb±Uqr �D�1zjbo~�f�| yvsM��z| �lSk'�µ | �M�«s�±.j>yAqr ySj�np��sM�~sM�§±.j>y�o~tMzÑ m&jvm&jb�Ù±Uqr �D�Ås�±.j>yAqr ySjbn#³�sM�~sM��zj�npjb�~j>t5zj>o?zj>m&jbk���j>�cw�zscyQsMoAtbn'zsco�zscn�z| y3�G N ^ `ÌeB\?N&W��BPSe�JMZ �7JDO 3LYOSJVW��äJMOSTUH�Xg è J

ô�j>�M�Mj>mIsMt � j>�M��k | ��³�sMk]n>�&q| oAo~tbjb±Uqr �D���| µlS³5j>�A±.jbyAqr y�jbn5u � �?�5j>m���°Òk | ��³�sMk]n~o�zsM��kBq| wDj�npkcj>tbn�zj>�5j>msct � o~l��as�±.jbyAqr y�jbn}m&jb�þySjbµ&jbnºjbySj>��l3�i�j>�M��qr k�±.j>y3�1µ | �M�8sMt�u ��� ¡ �?�BjbySl�7 U | �A��sMm?�F��zjb�~µ&j�nD�| t�zsM�AnIndsM��n | ��zsMm]� | m8�Kjb�]sM�&µ�sVn'z| jb�M�

| y��]scm GIÎ 7 Ï u � �&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| ySj>kfzjb³Bjbt5zj>ob�Ysc�Kjby��?�~j�n~j>yжAjbocqr y�m&j>kKsMt_sMy�zsc�&�&lY±.jby�n�zj�npjbySj>k'Î� Ø ��°�t�GIwDjbk]n | ��±Uqr �D�Mw�zj>m]��k�zj>³5j_��j>�Mj>�M�Mj>t>l�k,sMt� ±.j>yAqr y�jbnAnpj>yä�?wcsM�c��l�o�±.j>m&m}zsMySyFG � 7,��� � �� �D� °��Kjbm&m]��l��Bjbm«sMt�G>�Fn#ySj>oAt'�r k5zÑ n.¶cqr kÅs;7þt�zsc�An#npsM�An | ��zsMm]� �Ö�5j>ySoAj ¶Fzjb�~jM�&sMkFk | �}sct�G�ñ �»wDj>k]n | ��Þ±dqr �M�Mw�zj>m]�,j>�c�«÷ào~lS�Ks�j>y�j>��lYµ&l�³Bjb�A±.j>yAqr ySj�npjbnvzÑ ��y�jM���q| m&m]�DjbmÀ�5j>y�zsVnpµ�scn�z| �&µ | �M��sMt_÷ jbySjb�Kl�±.jby~qr ySj�n�u��?�?�5j>y�l�ySjbt�zsM�&zsMops�sMt � k | �K³&sMk]nº±.jbyAqr y�jbnpj�nsc�V¶~s�i�j>kFlSm]npocqr k�s � ­ C�� ¨ < ­ G ¢ < C G ª ± | �A� r y{zscwcsMy;��j>�]sc� | n~n�� ­ C Ç � � 7,� ��?~¢ E � ØD¢{£>£{£>¢ � Ê Ø �

�l �Bj>�~jbm �>l{zsMySµ&scn'z| ±Uqr �D�Mw�zj>m]�DjbkMj�nÅzjbo�sMt§zsMy�npsMy r kv��j>�Mµ�scn'zsc� | t | n~n JåÎ�7 Ï �_��È � u �&� ¡ ��ySj>kfzjb³Bjbt5zj>o�n{� ��� � ��� J � è Xg ê PçX^ � °�t� k | �K³�sckFn}±.jbyAqr y�jbn�±.jbySo~tMzÑ mÄzjbm�sMt( � �x��� \ £{£{£ \

�¹ ÈFâ>ãDJ ��?�¡p¢{£{£{£b¢A? �&� ¡ �}È ?�¡5£{£{£ È ? �&� ¡

³ | tbl�n{zÑ w§oAt�zsM� | n§zj>��n7¶Vqr k'�� JMéçhUJMé��'H1XJcT � °�t � k | ��³�sMk]n�±.jby~qr ySj�n�±.j>ySoAtDzÑ m&j�n~j>�~�Òzj>oAt>j�npj>oAj>m«m&jb� ±Uqr �D�IsMt�GvwDjbk]n | ��±Uqr �D�Mw�zj>m]���j>�cw�zscyQsMoAtbn'zsco�zscn�z| y3�� JMéçhUJMé��'H1XJcT � i�j>kFlSm]npocqr kasct�u � �?�5j>ySl�7 � � Ê � � ¢ � � � á � Ê � ¢ � � npsM�An | ��zsMm]� | m,sctbn#sMt G�Î27 Ï u ��l �Bj>�~jbm �>l{zsMySµ&scn'z| wMj>k]n | ��±Uqr �D�Mw�zj>m]�Fn{�5sM��j>yÓ�?�~j�±.j>mm�zsMySy

G �@?�¡d¢A? �U��� � | o ?�¡ � | o ? � = ¡ × � | o ?�¡ oAlSm ? � = �;× Ko~lSm ?�¡ = � � aì í]� £�fw?lS�j>m&ob�&µ | �M��sct GF�?sM�Kjby��?m&jbk�kfzjb³Bj�j>�M���o r �DsM�Bzr �8�'q| ���±.jbyAqr y�jbn�u � �?��sMmY��jbySj>�Mjbn npj>oAtvs_±.jbm]npl±.jby�n�zjbnpjbySjbk?mj>k'�À° i � Ø ��� .�j/&�m�zÑ �>l�z| sMyQsc³?¶&zsMmÕsMt �xo~t�±�zj>�~s̱.j>y�o~tMzÑ mÄzj>�Aj§sMt ( � ����� I �8 � zj>��n�zjbkMjbnkVsM³F¶ r k'�

� �

Page 28: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

� H ^ `ÌJVWbZ>PSN'LBT_OSJcNºXJMe�JMH�XJcTÕXJcZbW�JDOS`ÌJMH�XJcT�J�NºXJcW��BPSe�JMZ �7JDO 3LYOSJVW�NH3^ H53^ W�W°�t�u �&� ¡ �?�5j>y�l � ¢ � q| o~oAt>j�±Uqr �D���| m]�BzÑ yÓnaµ�scyS�KsMt | k | m�y�j>�M�Mj>m&jbk�sc�wcs´sMt G�Î � Ï u � zj>o

�G�Î � Ï u � �&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| y�j>kfzj>³5j>t5zj>oAj>k'��sM��j>yÓ�Dj>k�sMt � zj>o �� jbySj>��l±.jby~qr ySj�npjbkMjbn�zÑ �.¶&zsMk�ySjM�1°�t� ¢ �� ±.jby~qr ySj�npjbk$G ¢ �G_³�sM�~sM��zjbnpjb�~jbt5zj>oAj>l�µ&j>tvndsc�An | t�z| j>y�oM�| ±U�| �Kjbm&m]��l�o?zjb�Dj>k§y�j>�M�Mj>m&jbk�� ­ C�Î � Ï uzjbo �� ­ C�Î � Ï u ��?~¢ E � ØD¢{£{£{£�¢ � Ê Ø � £i�j>�M��qr k¦±.jby3��µ | �M�Âsct � ³&sM�psM�Òzjbn~j>��ndsM��n | ��zsMm]� | mÂsM��wMs,wcsMmÂj>�M� � Î � Ï u �&� ¡ �l �Bj>�~jbm�Þ

�bl�zsMy�µ�scn�z| ySj>kfzjb³Bjbt5zj>ob�5sM�Kjby��a�~j>m�&j>y�kMj>tblSk«sMt�scy�zsM�&�lYn r ySsç¶A� | mo�zsM� | k?kVsMyäÎ� Ø ��°�� ySjbk�zjb³Bjbt5zjbo�l�mV¶Aj>k]n{zÑ w zjbo�±.jbm&m�zsMy�y � � ����� ���� �D��i�jbn~o~t��| ySjb�Dj>o � Ç �æ³ | mFn~��sMmÌs ��� � �;��UDs � | �&l¯�F�©zscn~�~l¯Ûa�~sMm&�{¶~s � Ê Ø �°�t �G�) �C)�G�� ¡ Î � Ï �� ySj>kfzjb³Bjbt5zj>o�j>t_j>oAjbnp�5j>m¦j>�M� | yÓ�]sMm�µ | ��j | � | ��±;��j>�c±.j>ySjby�n~jbn�zjboAn�sM�s�kfzj�n}±.j>yAqr y�jbnij | mFn.¶~sMlYkBq| tDq| n~n>��sMµ | yY��zsM�A�Kjby�� � �?�Ajvj>oM�| oAlS�KsI�'q| �A�Äzj>mj>k,s�kfzj>³5jvj>�c�§o~l��asI�'q| �~�5jsct �� ±.j>yAqr ySj�npj>mY�

.�jb�M��qr m&k�j>�M� �ÌÎ � ¢���� Ï �Öo~lS�KsI��q| �~�Äzj�n}sMt �x³�sc�psM��zjbn~j>�AnpsM�An | ��zsMm]���&sMm���sM�Kjby��?m&jbk���j>�c±.j>ySjbys � �F�~j�j>oc�| �Ò� G6) �Ò±.j>yAqr ySj�npl?�'q| �~�5jM�1°�t �� ±.j>yAqr ySjbn~�~jÄj>oM�| ���� �G ) � ) � o~lS�Ks��'q| �A�Äzjbn�n~j>kFlSm]n.¶cqr ks'� k�zjb³&�'q| �A�Äzj ¶Fzjbm&j>k'� ���&! � � J � è Xg ê PçX^ � °�t�n§� | m&�V¶ r k'�}µ | �M� s8³�sM�~sM��zj�npjb�Andsc�An | �©zsMm]� | k�kYq| t]q| nAnpl� �&lжAjbk �bl�z| jb�M�l�t | ��jbn~�~l�k r oÄy�j>k�zj>³5j>t5zj>oAn}sM�§sMt � zj>o#sMt �� ±.j>yAqr y�jbn~j>k�kBq| tDq| n~n>�µ�s_��zsM�~��j>yÓ��ÖÎ � ¢���� Ï �»�'q| �~�5jjbo~j�n�zj>mÀs'�Ò��G) � zj>o ��Ò� �G) � ) ��±.j>yAqr y�jbn~lY�'q| �A�Äzj>kÀzÑ w?µ | oAo~t{scl��Kjb�Dj>�M�Mj>tbm&j>k'� ��� �&* XJcW�JMO � � � E = ��-�T ��= KC = N C > - P�D = T R ; M�-�T K- � - N K- ; P M6K- P - ; K� P 3 N� K- ; �� X = � - D Y -�M�AR M�- P -�T TJAC N AC P0P3:T T C'D K- ; ��; 3�T 3�T6T CHD � X�3 3 Y -�M�AR M�- P -�T -�M ; C Y C > -�2�2 1 =V; K-�/ - = 2 -�T > K3 P D =�� 3 = 7Q3�M�K- ; 3 ��� 3 � C E = �> K3 P�D =�� K3 7Q3�M Y -�2�2)K3�MVM 3J � �1��� B ��� � �;� D Ý � �J � � � �1�~� � �� � �1�

AC ;0; N - Y AR�/ /&K- ; P - P ; N C M�-0/ - ; � Ç � - ; - P K-�2 G� JMéçhUJMé��'H1XJcT � ° ±.j>m]npl�j>kÅsMySsM³?¶&zsMmÅy�j>µ&j�n�zjb�AnpjbyS��j>tbm&l�j>�M��±.jbyAqr y�jbnºq| m�asM��zsM�ps�n�q| �An�zj>mf�| lSt | �Kj�np�~l�k r oy�j>kfzj>³5j>t5zj>oAj>l�n�lSob�� N 3 N C 2 C ; � 3 D 3 > K- P - D - N K- ;Z> KC 5 ; N - D K-�2 -�T 3�MVT�3�M > 3 N K3 ; 3°�t fG§� �G ) �ÀÎ � Ï u � wDj>k]n | �A±dqr �D�cw�zjbm]��o~t>l�m]n�zj>m8sMt �� jbySj>��lf±.j>yAqr ySj�nvj>�M�À³�sM�psc��zj�npj>�Aj>o_j>yA�| ÞzscySy3zÑ n'zsMo&zscn}sc�V¶~s� � °�t)fGIwDjbk]n | ��±Uqr �D�Mw�zj>m]�Fn�sct �G � zsMy�npsMySl�zscn~³�sM�psc��zj�npj>�Aj>t5zj>o?zj�wDjby1kVsM³?¶ r k5� �° � i � I �8q| oAo~t>j�±Uqr �D�1zjbo¾o~t>jb�~l�mFnÂsct fG$zjbo �G ³�sM�psc��zj�npj>�Aj>t5zj>o~jbkxj>y�oM�| ±U�| �Kjbm&m]��l�o?zjb�Dj>l��f�| y�m]�Mj>�Ank]wcsM�&�~scnpl�k r o��©zscn~�~l¯Û | k?kVsMyB±.j>m&m�zscySy

fJ � �1��� B ��� � �;� D Ý � �J � � � �1�~� � �� � �1��}zsM�~��j>yÓ� � Ç �}�?�ps��

°���j>mmF�?l��BjbmÌsct �G�) � )G�� ¡ � fG)�G�� ¡ �&lжAjbk �>l{z| n}wDjbo~oAtqr k«sMt� zj>oºsMt �� ±.j>yAqr y�jbn~j>k�kYq| t]q| nAn{�sck?k | ��sMt�GÕzj>o�fG�³�sM�~sM��zjbnpjb�~jbt5zj>oAj>kÌsMySkVsMy��asct�zsMo~sMk | �#sMt_j>�M�?��zsMo~µ | t_�Aj>m&�j>y�nº±.j>yAqr ySj�nplY³ | m]n | k?m&sMksct | m | o ��?�¡U¢{£>£{£>¢A? ��� ¡ � Ç � ³�sM�psc��zj�npj>�?zj>��n�zj>kcj>k ±.j>ySjbySm&jbk��Kjb�&� ��q| m&m]�Mj>må�Bjby�zscn~µ�scn�z| ��µ | �c�©sn~j>kFlSm]n~jbn~nIySjbk�zj>³5j>t5zjbo�sMt � ±.j>yAqr y�jbn~�M�| y;sMt ��å�F�~j�sM�&��scm8sMt�j>oAjbn~�Bjbm8ySjbo~t�lSt | �Kj�np�AlQs��µ&sÌsMt�j>y�oM�|±U�| �Kjbm&m]��l�o?zjb�Dj>kF�~j�±.j>mm�zsMySy�� ­ C}� f� ­ C ��?A¢LE � ØD¢>£{£{£>¢ � Ê Ø ���

� �

Page 29: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

� ) *�� /�= G 0 =?+U@� +.= � =?9Ò41@2 GçÚ @� +.= �H <>=?+7+7=?6L�ÄC=?E>=G èB^ Z{`xX\�O�é63^ Z 4 3LYOSJVW´XJcZbW�JDOQ`�JMH1XJMTUJ

°�t � �©u���o~l��as�j>ySjb�Kl?µ&l�³Bjb�A±.j>yAqr ySj�npjbn}zÑ �.¶~s�ySj�s�7 �©u���� ¡ m]�BzÑ yÓnfndsc�An | �©zsMm]� | m´zj>��npj>y��Kjbt>j�n~nG�Î 7 Ï u��«wDj>k]n | �A±dqr �D�cw�zjbm]�D� ��q| �MtDzÑ n~ocqr kKsMt� �?m&jbk,j>tbj>m GI³�sM�~sM��zj�npjb�~j>o�jby~�| zsMy�y zÑ n'zsMo&zscn{�° / � G � - � � - Ç 7,��±.jby~qr ySj�npl�³ | m]np��sMm¾wMj>�M��qr m&kÀjb�M� ¤ Ç � ù � � ¤ ý� �B��zjb�~l�mFnM�| wDjbk]n | ��n{�

B�j½¶Aj>t>t�qr kÌjbtbnvkFl;sMt;Gų�sM� �>l�zscySlSo��&jb�~lÓw�zscy�n7¶~scl�wcsMy�s ¤ �²¬ ��� ¡­ ® ¡ ¥ ­ � < ­ G � - ��± | �~��zsM��sMmY��i�j>kFlSm]npocqr ksctbn�s / ³ | m]n | mÒzscnpµ�scyQsM��z| �´o>zÑ k | n{�Dsc�Kjby��Dj�n�s ¤ ¢ � � - ��wDj>k]n | � | k�±.j>oAtDzÑ npj>mj>k�kFlä�1°©���?�&l��Kjbm&t>l{z| o�»scÁ�m�sMyÓn�zj>�Am&j>kKwDj>�c��qr k«s ¤ ¢ � � - �ÄwMj>k]n | � | k�zsMyÓndsMyB�Kjb�Dµ�scn�zsM� | t | nAn�lS��zscmF�BzÑ n'zsMo&zscn{�

#jvy�j>µ&j�n�y�zscn~m&l3�?µ | �M�§s / ³ | mFn~m�sMkawcsMm | yÓ�]sMm§u��F�?�BjbySl�m]�BzÑ yÓn}kBq| �~m]�Mj>t>j�npjc�YsM��j>yÓ�KsMt � � �scyQsMkFt{sVnp��z| yÄj>�M� � j>�M�?oAt>j>�F�r zÑ wDj�n��Kj�npoAt§kFl3��° ���FnÅsct � ±.j>yAqr ySjbn ¤ zjb�~lSm]nM�| lS�&zsMm]��µ | tKnpsM�An | t�z|m | �~��zsMyS��jbn~o~tbjbn_zÑ w�zjbm&j>k§mjbwDjbt>t�qr k5�.�j>�c��qr mkÌs97 �xu ��� ¡ ³�sM�~sM��zj�npjb�Andsc�An | �©zsMm]�?��sMmÌj>�M� | yÓ�]sMm �ÌÎ>� Ï 7 � í Ç ��ºj>�M�?o~tbj>�F�r�Aj>� r y�zsM�AlSo}�'q| �A�Äzjbn>��sMµ | y � � í]��� - zj>oºs;����G) �x��q| �~�Äzjb�~j�� � ����� � ¢ � � � íD��� ¤ npjbyжAj>ocqr y3�

������� � J� è Xg ê PVX^ � ° � � G') � ±.j>yAqr ySj�nplº�'q| �~�Äzjbn«sMt � ±.jby~qr ySj�n / ³ | m]np�5j>y�l�m | �~��zsMyS��jbn~o~t>j�n��q| �~�Äzj½¶?zj>m&jbk,� | m&�V¶ r k'�° �KÎ � Ï � �©u �«�Aj>� r y�zsM�AlSo#��q| �~�5jvsÅ���?�lS��j>m&tbl�z| o ��scÁ�m«sMy�n�zj>�~�5j>m�zscy�ndscy�zsMm | oÄn{zÑ ³ r oBzr �f°�tl���zsMm]�BzÑ n | n~n� o>zÑ kF��sMm�s���k5zÑ o?zj>�c�| B��~j>mjbn��F��zsMt>l�o�zscn�k�zjb³Bjbt�| wMj>k]n | �A�Kjbt�| k¦ySj>�c�Dj>mj>k � ¡ Î>� Ï u��

zjbo � �5Î>� Ï u � �º�fw?lS�&jbm&o>�Bµ | �M��s�� � í]�}³ | mFn~�BjbySl�B��~jbm&jbn �FwDjbkFn | � | k?�~sK±.j>mm�zsMySy � ¡b� í]�#� ¡��� � ¤zjbo � � � íD��� � � - ���

°xn | w#zsM�&�lQsMkF��sMmÌsMt�GIwMj>k]n | �A±Uqr �D�Mw�zjbmF�«��zsMo | �&�Aj>m&� �r ³�sc� �>l{zsMySl�o#�&jb�~lÓw�zscy�n7¶~scl�n}s < �­� C G��?~¢LE � ØD¢{£{£{£b¢ � Ê Ø ��oAt>lS����z| y r �K�KsMy�± | �{¶ r k�¶Ajby½q| y�m&l3�1°*���©zsco | �&lSk��&j>�Al�w#zsMyÓn7¶&zsM�~s�s

� � � � ãA��� �&� ¡� ­¯® ¡ � �­ � ãA� � < ­ G � � � ãA�A��×�&� ¡� ­¯® ¡

�&� ¡�C ® ¡ �­ � ãA� � �C � ãA� � <

�­� C G � � � ãA�A�

kFlÓ±.j ¶Aj>t5zj>o�nvkVsM³?¶ r k'��° � I � Ø ��jb�M�Djbm&y~�| o?zj>�ao~t>jb�~l�mFn�s$�;�?m&sMk�saíKµj>y��Mj>m¦wMjbnAn�j>yA�| ¶Aj>y�j>oº�'q| �A�fqr ySj�n�zj>�Aj±.jbm&m�zsMy�y�s� � í]��� ¨�� � � � í]� ¢ � � � í]� ª· � � � í]��· � � ¬ ��� ¡­¯® ¡ ¬ ��� ¡C ® ¡ ¨ < �­ � C G � - � ¢ � � - � ª � ¥ ­ � ¥ C· ¤ · � � � � Ø �

q| o~o~tbjb±dqr �D�1zj>o>������ � � J � è Xg ê PVX^ � °�t9GawMj>k]n | ��±Uqr �D�Mw�zj>m]�¿zsMyÓndsMy�y�jDzÑ ��n� ±.j>yAqr ySj�n / � G � - � � - Ç 7,�v³ | m]n.¶&zsM��sMmn~j>kFlSm]n~oMqr m&k,jb�M� ¤ � ¤ ý�]�5�azj>�AlSm]nD�| wMj>k]n | �An{�'sc�Kjby��?�~j�n~j>yжAjbocqr y ¤ �»¬ �&� ¡­¯® ¡ ¥ ­ � < ­ G � - �U��°

� ù � ¤ ��� ¬ �&� ¡­¯® ¡ ¬ �&� ¡C ® ¡ ¨ < �­ � C G � - � ¢ � � - � ª � ¥ ­ � ¥ C· ¤ · �oAt�zsM� | n�s�±.j>yAqr ySjbn ¤ zj>�AlSm]nD�| l���zsMm]�?µ | t�npsM�An | t�z| m | �~��zsMyS�'q| �~�;qr y�jbn�zjbm&j>k�� | m&�V¶ r kas / ³ | m]n~��sMm��� JMéçhUJMé��'H1XJcT � ° � � � Ø ��q| oAo~t>j�±Uqr �D�1zjbo~�f�| y#sM��z| �}z| sMm´sÌm | �~��zsMy��'q| �~�;qr ySjbn��Kjb�Dj>�M�Mj>tblSkÂs¦��j>�M±.jbySj>yA�|m | �~��zsMyS��jbn~o~tbjbn�zÑ wajby~�| ¶AjbySj>o#�'q| �~�;qr ySjbn�zjbwMj>y3�f����f�| y�³5j>�&l��ÅsMtvkBq| wDj�npkcj>t>l�k'�5µ | �M�,sctvlS��zscmF�BzÑ n'zsMo�ndsM��n'z|zsVnp³�sM�~sM��zjbnpjb�~jbt5zj>o�m&jb�»w#zsMyÓn | t�ndscn.¶~s��Kjb��s�m | �~��zsMy��'q| �~�;qr ySjbn�zj>��n�zjbk�zj�n{��°���j>m&m]�?lS�5j>m_lS�&zsMm]�BzÑ n'zscoAw#zsMy�n�z|zsVnp³�sM�~sM��zjbnpjb�~jbt5zj>o�n�µ�s�¶�n r m&k�w�zjb�D�~jc�sMkFk | ��s�m | �A�©zsMy��'q| �A�fqr ySj�nIzj>��n�zjbkMj�s � Ê Ø �3�?o~tbj>�~jbo?zjb�~j�w#zsMyÓn | tblSk'�

� �

Page 30: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

GÙ` X\FT ^ R�PSN � �^ `�J èBè �5PSTBXJMé?JMN���"! � � J � è Xg ê PVX^ � ° � ­ C�� ¨ < �­

� C G ¢ � ª ± | �~� r y�zscwcsMy5m]�Dj>��n � ­ C�Î 7 Ï u ��?A¢�E � ØD¢{£{£>£b¢ � Ê Ø ��l �Bj>�~jbm �>l{zsMySµ&scn'z| ±Uqr �D�Mw�zj>m]�Mj>kMj�n sMt � ±.j>yAqr y�jbn G�³�sM�~sM��zj�npjb�~j>t5zj>o?zj>µ&jbt�ndsM��n | t�z| ��zsMo | �&lSk�±U�| �Kjbm&m]�?lÓÞo?zj>�DjbkFm&j>k«m&j�wDj>tbt�qr k'�

i�jbnpoAt�| y�j>�Djbo � Ç 7 j>oAjbn�zjbm¦¶Aj>y½q| yжAj � � �1�IsMt�nÅsMt � � Ê Ø �3�?j>�&�Aj>m&� �r k]wcsM�&�~scnpl�k r o���zscnp�AlÓÛ | n>�sc�Kjby�� ? �Fj>�&l�kÂo | ��zsc��sMmÒs E �?jb�&lSk¾jbySjb� zjb³&³Bjbm � ­ C � �1���¦¼½yÓ���©z| � | m´sÌ�©zsMo | �lSkÀ±U�| ��j>m&m]�?lSo?zj>�Djbkzscy�ndscy�j>�M� � Î 7 Ï �_��È � u ��� ¡ �#�&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| y�j>kfzj>³5j>t5zj>o�n�m]�Djb�cqr m&k'�° ±.j>yAqr ySjbn / ��G � - � � - Ç 7§��³ | m]n7¶&zsc��sMmÅwDj>�c��qr mk�jb�M� ¤ �Ö¬ ��� ¡­ ® ¡ ¥ ­ � < ­ G � - ��zjb�~l�mFnM�| wDjbk]n | ��n{�° ¤ zjb�~lSm]nM�| lS�&zsMm]��µ | t�ndsM��n | t�z| m | �~��zsMy��'q| �~�;qr ySjbn~jbn#sMt�j>y�oM�| zj>o��©zsco | �&lSkűU�| ��j>m&m]�?lSo?zj>�Djbk?�;�| y�s

� ù � ¤ ��� ¬ �&� ¡­¯® ¡ ¬ ��� ¡C ® ¡ � ­ C � ¡�¢>£{£{£>¢ ��� ¡ � � ¥ ­ � ¥ C¬ �&� ¡­¯® ¡ ¬ �&� ¡C ® ¡ � ­ C � ¡�¢{£{£{£�¢ �&� ¡ � � ¥ ­ � ¥ C� � � �D�

kFlÓ±.j ¶Aj>t5zj>oAo~j>y�ySjbµ&jbn���j>�DsM�&m&lä�� JDLBT è PSJMZ�WBXJVWbJMOSJ

i�j>kFlSm]npocqr m&k¾s 7à³�sM�~sM��zjbnpjb�AnpsM�An | �©zscmF�?��scm¾j>�M� �ÌÎ�� Ï 7 �Aj>� r y�zsM�AlSo��'q| �~�Äzj�n¾zj>o�sMt zsMyÓndsMySs��j>�Mµ�scn'zsc� | t | n~n��Ò��G�) �Ö±.j>yAqr ySj�npl5�'q| �~�Äzjbn{�i�j>�M��qr kű.j>y3��µ | �M�as � � ãA� � ã Ç ���#³ | m]np�5j>ySl�� � � ãA�ÄlS�&zsMm]�ºzr m | �~��zsMyS��q| �~�;qr y�jbn~�~j � � ����� � � � � ãA�~�vý�Öín~j>y ¶Aj>ocqr yä�f��tºj>o~j�np�5j>m,s � � � ��� � � � � ãA�~��� ¡

��� ����� � � ¨2� � � � ãA� ¢ � � � � ãA�~� ª q| oAo~tbjb±Uqr �D�1zjbo~�f�| yBsc��z| �&l�k5�Fµ | �M�� � � ãA�azj>o�� � � � ãA��ySl�m&j�zsM�AlSo~sMm§±Uqr �D�Dj�npySjbm&j>k'�?npj>µ}zscn�±.j>m&m�zscySy � � ãA��ì©íþs � �?� � �#jb�M�Djbm&ySj�n�oAt>jb�~lSm]n>�ÜÌlÓwDj>y � � � � ãA���Ç � � � ��� � �}s�É�'q| �A�Bj � ãA�űU�| m | �~��zsMySl�o�j>�c��o?zj>�MwMj>k]n | �ps´o~jb� ySj>µjbnKj>ySjb�Kj�s

� � � ��� � zjb�~lSm]nM�| n�zjb�~m&jbk5�&wcsM�c��l�o_¨ � ãA� ¢ � � � � ãA�~� ª ý�Öí�npjbyжAj>ocqr y3�° � �?� I �vq| o~o~tbjb±dqr �D�1zj>o~�;�| y�sM��z| ��z| sMm,±.j>m&m�zscySy�sMt¨2� � � � ãA� ¢ � � � � ãA�~� ª �  B� ãA� � � � � ãA� �?¨ � ãA� ¢ � � � � ãA�~� ª

jb�M�Djbm&y~�| o?zj>���°�±.j>m]npl�j>kF�f�| y���zsM��kYq| wDjbn~kMjbt>lSk¦sMtIsMy�zsM��&l�kFlжAj>y�j>m]n�zjbo>�YsM�Kjby��Mjbn�sKoAt{sMkFlS� | �&sMy | ����sMmÜ�j r oAm&lSjb�ºn�zjbnpjbySjbk�zjbm]n�npsM�AnpsMm�sMkKo~t�zsc� | mY���������* XJcW�JDO�� � > -�2�2 1 = E -�2�3 ��� G�) � -�/:1 C M#1�3:2 Y -�M AR M�- P = /�AC'DHE - � 3�X C M 7Q3�M 3 > -�M#1 ã Ç � ��D -� � ����� � � � � ãA�A��ý��í P -�M ��- ; AR M � 3�T6T C'D 3 � ã � E -�M = /BAC'DHE AR�M�- P - ; - > M�-�X - P�� K- ;ZY -�2 2[K3�MVM

� � ãA� � ب � ãA� ¢ � � � � ãA�A� ª � � ����� � � � � ãA�A� £

G �äJMOE3LYOSJVWbP1[JcN&W ^ Z>`ÌJMH �^ PSZ�X\ è �5`�J è WbP;RBJMZ{PÓ[�X\?O�Wäh�\°�t � �©u � o~l��as�j>ySjb�Kl?µ&l�³Bjb�A±.j>yAqr ySj�npjbn}zÑ �.¶~s�ySj�s�7 �©u ��� ¡ m]�BzÑ yÓnfndsc�An | �©zsMm]� | m´zj>��npj>y��Kjbt>j�n~nG�Î 7 Ï u��«wDj>k]n | �A±dqr �D�cw�zjbm]�D� ��q| �MtDzÑ n~ocqr kKsMt� �?m&jbk,j>tbj>m GI³�sM�~sM��zj�npjb�~j>o�jby~�| zsMy�y zÑ n'zsMo&zscn{�ô�j>�M�Mj>m�sM� | n~n,jb�M� �8ÎH7 Ï u � �l �Bj>�~jbm �>l{zsMySµ&scn'z| ySj>kfzjb³Bjbt5zj>ob�¿°��Kjbm&m]��l��Bjbm npjbn~o~t��| ySjb�Dj>o� Ç 7$j>oAjbn�zj>m§sct � � �1�1wDjbk]n | ��n�zr �c��npjbkFlSm]n7¶Vqr k'�]��lSm]n�j>�c�'G � �1�;kMj>tb�f�| ³ | m]nYzr lS��zscmF�BzÑ n | nAnfo~t>sMkVsMo~t�n{�sck?k | ��sMt����FnºsMt�G>�?�Aj>y�³�sc�psM��zjbn~j>�~jbt>j�n~n � µlS³5j>�A±.jbyAqr y�jbn#±.jbyAqr y�jbnpl5wDjbkFn | �~��j>t��| ¶Fzj>mj>k,m&j�wDj>tbt�qr k'�°�t�G�ySjbk�zj>³5j>t5zjbo�< ­ G ��? � ØD¢{£{£{£b¢ � Ê Ø �}³�sM� �bl�zsMy�lSo}�&jb�~lÓw�zscy�n7¶~scl | y��DsMm�±.jbyAqr y�jbnplBwDj>k]n | �~��j>t��| kMj�nkfzjb³Bjbt>m&jbk5�BsM��j>yÓ�Dj>k§s�±.jbyAqr y�jbn�n~jbnpoAt�| y�j>�Djbo�³ | m]n.¶&zsM��sMm��Dj>mj>��zscyж&zsMkasÅy�lSm&j�zsM�~l�oKzj>�AlSm]nD�| npj>�Ajbn>�

� í

Page 31: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

ô�j>�M�Mj>m¾sc� | n~n�j>�c� �´Î�7 Ï u��̱.j>yAqr y�jbn~lYwDjbkFn | �~��j>t��| zj>o�s / �*G � - � � - Ç 7,��³ | m]n~��sMmÀjb�M�¤ Ç � ù � zjb�~lSm]nM�| wDjbkFn | �>��°*7 ³�sM�~sM��zjbnpjb�AnpsM�An | �©zscmF�?��scm«wMj>�M��qr m&k,jb�M� | yÓ�]sMm�xÎ � Ï 7 �©u �&� ¡ o~lS�Ks��'q| �A�Äzjbn>�'sMµ | yBwMscyQsM��j>yÓ��ã " Ç �M�?�~jvnpjbyжAjboMqr y � � ã " ��� - zj>o� G�) ��� � � ã " � � ¤ ��i�j>kFlSm]n~oMqr k«sMt���) �ÌÎ � Ï u � �&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| y�j>kfzj>³5j>t5zj>o�n{����,+ � � J� è Xg ê PVX^ � °�t � � )��� � � ã " � �&j>�Al�w#zsMy�nAwDjbkFn | �An�sct � ±.j>yAqr y�jbn~lºwMj>k]n | �A�Kjbt�| ¤ o~tbj>�~l�m]npll���zsMm]�?��j>m]npl��&jb�~lÓw�zscy�n7¶&zscm�sMk�� | m�V¶ r k'�°xn | w#zsM�&�lQsMkF��sMm�j>tbj>m�wDj>k]n | �~�~sKs$7 � ��¶Aj>y q| y�zjboAnºsMy�kcscyS�KsMt>t r k'�� JMéçhUJMé��'H1XJcT � ° ¤ �x¬ ��� ¡C ® ¡ ¥ C � < C/G � - �1wDj>k]n | ��o~t>jb�~l�mFn~l>7 � �xlS�&zsMm]����j>m]n~l�&jb�~l�w#zsMyÓn�m&jb� ±Uqr �D��s³&sM�psM�Òzjbn~j>��ndsM��n | ��zsMm]�?��sMm,wDj�n~n �Ö�'q| �A�Bjv��j>�cw�zscyQsMoAtbn'zsco�zscn�z| y3���KlÓwDj>yYs��&j!&�m�zÑ �>l{z| sMyQsM³F¶&zsMm§npjbyжAj>ocqr ys

7 � �æ� ��� ¡� ­ ® ¡ ¥ ­ � < ­ � � - �q| o~o~tbjb±dqr �D�1zj>o>���o~tbj>�~l�m]n��}zsM�~��j>yÓ� = Ç u�o~t�zsM� zj>o}� ¢ ¤ Ç � ù � wDjbk]n | � | k,�KjbySy�jbn~n#±.j>m&m�zscySy

7���� �æ� = � 7�� � ¢ 7�� � � �æ� 7�� �å×C7 � � £

G�� JDP è é�\FZ�WbJ è 2�OSJMN�XJce�JMH1XJMT ��\�` X\FT ^ R�P�N¾\�O�\?e � ^ Z{`�\i�j>kFlSm]npocqr k�sMt �ÖÎ 7 Ï u � m | �~��zsMy�lSo�jb�M�?o?zj>�cwDj>k]n | �~��j>t��| n{�f��lSm]n_sMt�GawMj>k]n | �A±Uqr �D�Mw�zjbm&m]�Dj>y³&sM�psM�Òzjbn~j>�Aj>t>j�n~n � µ&lS³5j>��±.j>yAqr ySjbnfj>�M��±.jby~qr ySj�nplwMj>k]n | �~��j>t�| ¶?zjbn{� ��m&m&jbk�s / ��G � - ��³ | m]np�&sMmKwMjbn~nl���zsMm]�?��j>m]npl��&jb�~lÓw�zscy�n7¶~sclS�ps_npj>y ¶Aj>ocqr y�sMt_sMy{zsM�&�&lYn~�~lÓw�l{zsMySl�o�q| o~o~tbjb±dqr �D�1zj>o>�

������� XG�OSO½Xg W�X\FT � � - P ; N C M�-0/ - ; ¤ Ç � ù � - ; - P K-�2 3 7 � � = D K3:2 1 > -�2 P = 5�- D = 7ZK3�M P E -�2�2 -h7Q3�2 3 � ù �K- D = 2 P C P K- DHE -�2 G° � � Ø � z°�y�y zÑ n'zsMo�l�o~��j>�Ajbn�zjb�Bjbmåzjb�An~j>yS��j>tbm&l�y�j>µ&j�nºsMt_sMy�zsM��&lY± | �]scyS�Kscn{�

����� � � J � è Xg ê PçX^ � °�t ( ù � ¤ ��� Ê 7 � � � ¤ Ç � ù �x��q| oAo~tbjb±Uqr �D�1zjboxzscy�ndscy���j>�Dµ�sVn'zsM� | t | nAn( ù Î�� ù � Ï � ù � ySl�m&j�zsM�AlSo�ySjbk�zj>³5j>t5zjboAn�sMt9GawMj>k]n | �A±Uqr �D�Mw�zjbm&m]�Dj>y#³�sM�psc��zj�npj>�Aj>tbjbn~n � ±.j>yAqr ySj�n/ �?�5j>y�l��8jblSm&�DsM�An~j>m?�?y�j>kfzj>³5j>t5zj>o?zj>m&jbkÌ� | m&�V¶ r k5�� JMéçhUJMé��'H1XJcT � °�jby~�| �&�l��&j!&�m�zÑ �>l{z| o~t>jb�~l�mFnºnpj�npoAt�| y�j>�Djbo ¤ � ¬ ��� ¡C ® ¡ ¥ C � < C/G � - � zj>�AlSm]nD�| wMj>k]n | �~�psn~j>y ¶Aj>ocqr y

( ù � ¤ ��� Ê �&�¡� ­¯® ¡ ¥ ­ � < ­ � � �1� £

� JMéçhUJMé��'H1XJcT � °åk�zj>oM�| �&�&lSjbk�o | �&zsMm��asç¶A�Å��j>�D� r ndscn.¶ r k'�Fµ | �M��l���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| zscn~³�sM�~sM��zj�npjb�~j>t5zj>ojbo~j�n�zj>mÀsMt¾zj>�~l�m]nD�| n�zj>�~jbm«�&j!&�ml�zsMyÓn�8jblSm&�DsM�An~j>m?�?y�j>kfzj>³5j>t5zj>o�s � Ê Ø � �?oAt>j>�Aj>o?zj>�~j_w#zsMy�n | t>l�k5����� � � J � è Xg ê PçX^ � °�t�G�zscy�ndscy�³&sM�psM�Òzjbn~j>�Aj>t>j�n~n � ±.j>yAqr y�jbn / ³ | m]n~�BjbySl��©zsco | �&lSk«sMySsM³± | �A�©zsd¶&zsMm�sMkmjbwDjbt>t�qr kÌs � � ù � � ¢ ¤ ��� ¨ ( ù � �;� ¢ ¤,ª � � ¢ ¤ Ç � ù �Ö�q| o~o~tbjb±dqr �D�1zj>o~oAj>y��Kjb�Dµ�scn�zsM� | t | nAn � � ù Î � ù � á�� ù � Ï u �&lSy�lSm&j�zsM�~l�o ± | �~��zscn{�

�Ø

Page 32: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

��q| m&m]�Djbm¦l��]sMt | ySµ�scn�z| �?µ | �M�«sct �ÖÎ>7 Ï u���ySj>kfzjb³Bjbt5zj>oAo~jby�n~j>yжAjbocqr y�sct¨ < �­ � C G ¢ � ª �¿¨ Ê < ­ � ¢ < C G ª � � � � �

jb�M�Djbm&y~�| o?zj>��#��m&m&jbk,lSoA�Kjb�~jbn�zj>�5j>m�ySj�wDj>tbjbn~µ&jbnM�| �'µ | �M�§n~jbn~o~t�| ySj>�Mj>o»��� ¬ �&� ¡­¯® ¡   ­ � < ­ G � - � ¢¤ �»¬ �&� ¡C ® ¡ ¥ C � < C G � - �#wDjbkFn | � | k?�~s�±.jbm&m�zsMy�y

� � ù � � ¢ ¤ �f� ��� ¡� ­ ® ¡��� ¡�C ® ¡ � ­ C � - � �   ­ � ¥ C £ � � � I �

° � � � I �#k?lÓ±.j ¶Ajbt5zj>oA�f�| y���zsM��kBq| wDj�npkcj>t>l�k�sMt_sMy�zsM��&lYkFlжAjbySjbmFn�zj>ob������ � XG�OQO Xg W�X\?T ��� � � ù > K3 ; C 5 = TO3�M 3 � Y C'D > 3[-�/�1 ; N =?> > - P�D = T R ; E = M = 2 - K3 D =?; Y C'D > 3 G

ÜÌlÓwDj>y�s'� � ù sMySsM³± | �A�as�oAt>l��K��jbn~�~lSk r ob��s � � i � .�j!&�m�zÑ �>l{z| ��z| yYsM��z| �lSk§s�kBq| wMjbnpkcj>t��| zsMySy3zÑ n'zsco>����"! � XG�OQO Xg W�X\?T ��� N ( ù�� - = 2 / 3 D�P -�2 � M�-�T K- � - N K- ; AC 2 365 �'R 2 /�K3�M P G� JMéçhUJMé��'H1XJcT � .�j>�M��qr k zjbo~tb�~jM�'µ | �M�«s � � � �D��zjbo � � � I �#j>�M�Mj>m&yA�| o?zjb�Dj>k«oAt>j>�AlSm]n�±.j>m&m}zsMySy

� � ù � ¤ ¢ ¤ �;�$· ¤ · � ��� ù � ¤ � ¢scµ | y ¤ Ç � ù � zjbo ¤ ý� ����° � � � � ��q| o~oAt>j�±Uqr �D�1zj>oAm&j>k,³5j>�lS��sMt�sMy{zsM�&�&l���zscn~�~lÓÛ�jb�M�Djbm&ySj�n�±.j>y�j>y���j>�

� � �;��� Ê � � � � �1�~� Ý � � G � �1� ¢��j>yÓ�?�Bjbm¦sMt�G ¢ � wMj>k]n | �A±Uqr �D�Mw�zjbmF�Mj>k�UDs � | �&l¯�F�©zscn~�~l¯Û&sclBoAt>j>�Aj>³5j>y�m&j>k'�G�� JDP è é�\FZ�WbJ è 2�OSJMN�XJce�JMH1XJMTI` X\]W�Z{P�4'T�Hç\?O1[]\?OVX^ OSJ&Xg Z�X\FTU\

°�t$G§wDjbk]n | ��±Uqr �D�Mw�zj>m&m]�Mj>y#y�jDzÑ �An � ±.j>yAqr ySj�nvj>�M� / � G � �1��³ | m]n~�BjbySl�zj>�AlSm]nD�| n~j>�?zj>mÕwMj>�M��qr kÂsMt( ù Î � ù � Ï � ù � �Õj>lSm�]sM�An~j>m?�FySj>kfzjb³Bjbt5zj>o�n{��°�t

( ù � < ­ G � �1�~��� ��� ¡� ® ¡ F ­ � �1� � < G � �1�

kFlÓ±.j ¶Aj>t5zj>oA�Bjbm¦o~tbj>�Aj>³&yA�| F ­ � �;� � $ ¢P? � ØD¢{£{£{£b¢ � Ê Ø �#jb�M��qr n~npµ&scn'z| kK�Kjb�Dµ�scn�zsM� | t>m�sMkKj>�M�� � Ê Ø �1á � � Ê Ø � �?jbo * � �;���©zsVnp�~l¯Û | n{�FsM��j>yÓ�Ås � ù � zjb�~l�mFnM�| n�zjb��< ¡ G � �;� ¢�£{£{£�¢ < �&� ¡ G � �;����zsMtblSo&zsM�psmÄzj>tbwMj§zÑ �.¶~s�ySj�sct ( ù q| m&sM�V¶ r m&��zsMy�n}y�lSm&j�zsM�~l�o�ySjbk�zj>³5j>t5zjboAn>�°þ�&j/&�m�zÑ �>l�z| oAt>jb�~lSm]n~l � � ù � < ­ G � �1� ¢ <�C G � �1�~�Å� ¨ ( ù � < ­ G � �1�~� ¢ <�C G � �1� ª j>�M�Mj>m&yA�| o?zjb��zjbo � � � I �

kBq| wMjbn~kMj>t�n�zjb�BjbmÀ±.j>m&m�zscySy � ­ C � �;��� ��� ¡� ® ¡ F ­ � �1� � � C � �1�

��?~¢LE � ØD¢{£{£{£b¢ � Ê Ø �U� ����f�| y���zsM��kBq| wDj�npkcj>t>l�k'�Bµ | �M�«s�ySjMzÑ ��z| ��zscn~�~lÓÛ��~s* � �1��� � � �;� � � � �1� � � � � �

n~j>y ¶Aj>ocqr yä�&sMµ | y � � �;��s�J � �;����zscnp�AlÓÛalSm]wDjb�~t5zjbn�¶Aj>y½q| ySl3�� �

Page 33: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

G �YPSe�JMZ �äJMOE3LYO�JcW_\FR ^ W�W�e ^�è W 4 JMOQP � �^ é 3^ Z 4 3LBOSJcW�JMP�XJMT�� �^ PSZ�X\ è �\?P ��� LBOSJMZ_WYXJVW�JDOSJ������� � J� è Xg ê PVX^ � °�t ( ù Î � ù � Ï � ù � �Õj>lSm�]sM�An~j>m?�FySj>kfzjb³Bjbt5zj>oÅo~sç¶&zscn�zjb�An�zj>kMjbl�nKs¦±.j>yAqr ySj�n /³ | m]np�5j>y�lY±U�| �'q| �~�;qr y�jbnpjblSm&jbkK� | m&�ç¶ r k'�°�o~sç¶&zscnAwDj>k]n | � | kFµ | t�ndsM��n | t�z| � ù �©�?�BjbySl?lS�&zsMm]� | kVscnfs�±.j>yAqr ySjbn / �?�5j>ySlF±U�| l���zsMm]�]sclSm�sMkvmjbwDjbt>t�qr k5�

ô�j>�M�Mj>m�� ­ ��? � ØD¢{£>£{£>¢ � Ê Ø �;jb�M� | y��DsMm | �An | m | �A�©zscy�n���zsMtblSops�s � ù �©�?m&j>k'�?sM��j>yÓ��mj>k_j>ySjb�Kjblsct ( ù �Õj>l�m&�]sM��npj>mF�?ySjbk�zjb³Bjbt5zjbo�o~sç¶&zscn~wMj>k]n | �psMlä�Ä��oAt>j>�AlSm]nº±.j>m&m�zscySy( ù � � ­ ��� � ­ �{� ­ ��? � ØD¢{£{£>£b¢ � Ê Ø � ¢

scµ | y�s � ­ o~t�zsM� | k«sMt� ±.jby~qr ySj�n�±U�| �'q| �~�;qr y�jbnpjbl�s / ³ | m]np�&sMm��#° � ­ � � ù � � ­ �Öj>�M�Mj>m&yA�| o?zjb�§sM�A�psµ�zÑ w{¶~sI±.j>yBs�&��c�Dj>y��Kj�n{��µ | �M�KsI±U�| ��q| �~�;qr y�jbn ��j>�Djb�M�Djbt>lSkKsI±U�| l���zsMm]�?µ | tºndsM��n | t�z| m | �A�©zscyS��jbnpoAt>j�n�zÑ wjby~�| ¶AjbySjbo}�'q| �~�;qr ySjbn�zjbwMj>y3�° � ù � zjb�~l�mFnM�| n�zjb�~�5j>mvwDjb�M��qr m&kvj>�c� ¤ j>�c��o?zj>�MwMj>k]n | �An{�M�Kjby��?m&j>k�s�� ­ j>�M�?o?zj>�MwMj>k]n | �A�psMyF�5j>t�zsM�AnoAt]q| �MjKy�j>�M�Mj>m � ­ �K��o~t>jb�~l�mFn�s � ­ � í 7 � ­ 7 ����oAt]q| �D�Aj � | o � ­ � ¨ ¤ ¢ � ­ ª n~j>yжAjbocqr yä��°�t�sMy�zsM��&lq| o~o~tbjb±dqr �D�1zj>o>�1�Kjby��MjbnI� r ySjb��n�zjbn~j>ySjbk�zj>m]n�oAt | k]ndsck¾j>��y zÑ n~j>m&lä��sM�A��z| y�oAt�z| yä�1µ | �M��s«m | �~��zsMy��'q| �~�;qr ySjbnzjb�An�zjbk�zjbnº�Kl�k�zj>m]n�ySjbµ&jbnºµ�scn�zsM� | t>m&l�s�±U�| �'q| �~�;qr y�jbnpjbkKl�o~��j>�Ajbn�zjb�Bjbm��

����� �&* XJcW�JMO � � ¤ K- D = 2 P C = D K3�2 1 X C N P 3 DFP C N KC 2 C'D > K3:M?/BAC'DFE AR M�- P D - Y -�2 2[K3�MVM� ù � ¤ ��� ��� ¡� ­¯® ¡ � ­ � � | o � � ­ £

��� � � � J� è Xg ê PVX^ � °*���*G�) �ÌÎ>� Ï u � �~jb� r y{zsM�~l�o�±.jbyAqr y�jbnplY�'q| �~�Äzjbn�sMt � ��q| �~�;qr y�jbn~l�w | m�scy�zsMm�sck� | m&�ç¶ r k'�µ�sIn~jbn~o~t�| ySj>�Mj>o#ã Ç �Kjbo~jbn�zj>m � � � ãA� ops�¶&zscn~wMj>k]n | �~s�sMt ( � ����� �Õj>l�m&�]sM��npj>mF�?ySjbk�zjb³Bjbt5zjbo~m&jbk5�

G T�H ^ Z>Hç\]W�é63^ Z 4 3LYOSJVW´XJcT_\«NH3^ H1XJMeYé 3^ Z 4 3LBOSJcWIJMé�� �7JDO 3LBOSJcWbP;e ^�è W 4 \ è°x±U�| �'q| �~�;qr y�jbnpjbk§sMySsM³?¶&zsMm§n | w�zsc�&�&l�± | �]scyS�KsMkVscn#n r � r m&k�zjb�An~j>yS��j>tbm&l3�

��� � � � J� è Xg ê PVX^ � ° � ù � � �&� ¡­¯® ¡ � ­ o~t�zsM� | n�sMt � ±.jbyAqr y�jbn / ³ | m]np�5j>y�l#oAt | �At{scn~�'q| �~�;qr ySjbn�zj>m&jbkmjbwDjbt>t�qr k5�°�t� ±.j>yAqr ySjbn / �F�BjbySl�kBq| t5zj>³&�'q| �~�;qr y�jbn�zjbm¦s ü ù � ¡��� ¡ ¬ ��� ¡­¯® ¡ � ­ o~t�zsc� | n,zjb�An7¶Vqr k'�

° � � � � �#jb�M�Djbm&y~�| o?zj>�KkBq| wDj�npkMjbtbn�zj>�5j>m�s / � G � �1�#³ | mFn~�BjbySl�oAt | �At{scn~�'q| �A�fqr ySj�np�~j� ù � ÈFâ>ã � � �1�

ÈFâ>ãPJ � �1� � � � i �n~j>y ¶Aj>ocqr yä�²��o~tbj>�AlSm]nÀs�� � ÈFâ>ã �

ÈFâ>ãPJ kFlÓ±.j ¶Aj>t5zj>oAo~j>y¾zj>��npjbyS��j>t>j�n~n � Î�7 Ï uà�&l �Yjb�~jbm �>l{zsMySµ�sVn'z|±dqr �M�Mw�zj>m]��zÑ �ö~s�ySj�sÅo~t | �~t>scnp�'q| �~�;qr y�jbnpj�n�s�±.j>yAqr y�jbn#³ | m]n.¶~sMlS��scm��.�j!&&m&l�zsMy�m&l�n r �V¶ r kÒs ü Î�7 Ï uþ±dqr �M�Mw�zj>m]�FnÅlSo>� sMµ | y�s üÒ� �;� � � Ç 7,�_±dqr �D�cw�zjbm]��zjb�An�zj>k��j>�Mj>�M�Mj>t>l�k«sMt�G � �1��³ | m]n~�BjbySl�kBq| t5zj>³&�'q| �~�;qr y�jbn~n~j>yä�

� �

Page 34: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

G é63^ Z 4 3LYOSJVW>P � ^ é�\?OQ`�\?N�X\]WbeB\FZ>\?`ÖXJVWbJcZ>JcH1XJcTUT�JMO1TUHçJD` 4 J è PfP è []\FZ>PS\ èBê PçX\�hU\���&�&l���sMt � j>ySjb�Klº±.j>yAqr ySjbn~jbn~m&j>k j>�M����q| �MtDzÑ n~jbnAn G»Î�7 Ï u�� ³�sc�psM��zjbn~j>�~jbo,j>yA�| zsMySy3zÑ n�zsMo�zscnscySkVsMyS�KsMt�n r k'�ºi1jbk?l�m]npocqr k�� | oAn�sMt G�wDjbk]n | ��±Uqr �D�Mw�zj>m]�?m&j>kÀs i � � ��.�j/&�m�zÑ �bl�z| �&sMm¦y�jDzÑ ��n fG�� G ) �

zsVnp³�sM�~sM��zjbnpjb�~jbt5zj>o?zjbn>�°�tºë�o~tblS����z| y r �²wMj>�M�Mj�±.j>yBsMt ØD¢ Ê Ø zjb�An�zj>kMjbkMj�nºsMo~tbj>�~l�m]n{�µ | �M�KsMt¾zsVnp³�sM�~sM��zjbnpjb�~jbt5zj>oÄlS�&zsMm]�BzÑ Þn�zsMoAnpsM�An�z| wMsc�M��l���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| ��ÜÌl�m]n�lSo~��j>�Ajbn~j>o>�csMt fG�zj>oHG#³�sc�psM��zjbn~j>�~jbt5zjbo~j>kFµ&jbt#�Aj>m&�&jby�n�m | �~��zsMySl�ojb�M�?o?zjb�MwDjbkFn | �~��j>t��| kF�~j f� �åë � � ) � n~j>y ¶Aj>ocqr yä�����&�f�| y�sct | m&m&sMy�sM��z| �lSk'�&µ | �c�§±.j>m&m}zsMySy�s

< ­ f�Ë��ë�� ���¡�

C ® ¡ <­ cSC � < C � ) �

��? � ØD¢{£{£>£b¢ � Ê Ø �¿q| oAo~t>j�±Uqr �D�1zjbo>�.�j>�c��qr kÀsMt � ±.j>yAqr ySj�n fG§³�sM�~sM��zjbnpjb�~jbt5zj>o?zj>m&jbkÕ��j>�c±.j>ySjby~�| f( ù Î � ù � Ï � ù � �8jblSm&�DsM�An~j>m?�

y�j>kfzj>³5j>t5zj>o�nvs / � fG � �1� � � Ç 7,��³ | mFn~��sMm���° ±.j>m]n~l�j>�M�Mj>m&yA�| o?zjb�asMySsM³?¶&zsMm«�5j>y{zscnpµ�sVn'z| ��µ | �M�f( ù B < ­ fG � �;� D ��ë � �&�

¡�C ® ¡ <

­ cSC � �1� � ( ù B < C/G � c �~� �;�A� D �åë�� ( ù B < ­ fG � �1� Dn~j>y ¶Aj>ocqr yä�f¼½y��a�©z| � | m«lS�]sct | yÓn r kKs�kBq| wMjbn~kMj>t��| kFlжAjbySj>m]n�zj>oAn>����,+ � XG�OQO Xg W�X\FT � � 3)3 N K3 P�� 3 D 3 > K- P - D - N K- ; = D K3�2 1 K� P K3 ; P 3 D�P KC � 3:T T C'D 3 / Ç � � C 2 P0E -�M = � - = 2 /63 DFP -�2 �Me-�T K- � - N K- ; -�T D - = / 3 N f( ù � ( ù G

� > -�2 2 1 = E -�2 3 N K3 P�� 3 D 3 > K- P - D - N K- ; = D K3�2 1 K� P K3 ; 7ZK3�M P KC � 3�T6T C'D Y -�2�2)K3�MVM f( ù � Ê ( ù £° � � I � z°�ySy3zÑ n'zsMoA��z| yBkYq| wDjbn~kMjbt>lSk'�Bµ | �M��sÅo~t | �~t>scnp�'q| �~�;qr y�jbnpj�nºySjMzÑ ��z| f� Î � Ï u¸zj>o���Î�7 Ï u

±dqr �M�Mw�zj>m]�Djbk?kcj>y f� � � ëM� ��� ¡ � � ) � � � � � �n~j>y ¶Aj>ocqr yä� ° fü Ç � � ���Òzjbo ü Ç � � 7,��kYq| t5zj>³�'q| �~�;qr ySjbn~l}±Uqr �D�Mw�zj>m]�Djbk jbo~j�n�zj>�5j>m ³5j>�lS�Õ±.jbm&m�zsMy�yfü �åë � ü ) ���� JMéçhUJMé��'H1XJcT � ��q| m&m]�Djbm,ySj�wDj>tbjbn~µ&jbnM�| �µ | �M��sMt fGÀzj>o�Gv³&sM�psM�Òzjbn~j>�Aj>t5zjbo~jbk?mj>ka��j>�c±.j>ySjby~�| ��zsMo | �&lSk±U�| �Kjbm&m]��l�o?zjb�Dj>k,��zscnp�AlÓÛsMl��psIl��]sMt�s

f� � �1� �åë � B ��� � �1� D Ý � � � � � �1�~� � �� � �1�q| o~o~tbjb±dqr �D�1zj>o>�GIHKJMOSJD`ÌP �7JDO 3LBOSJcW_LB` 4 PQOQP�N'LBT�e ^è WähU\�P������� � � J� è Xg ê PVX^ � °�t � jbySjb�Kl�µ&l�³Bjb�A±.jby~qr ySj�n�jb�M� / ³ | m]n.¶&zscn r ���&lSy�lSk r o�³ | m]npm&sMk«m&jbwMj>t>t�qr k'�Yµ�swcsMySsM��j>y�� � o~t�zsM���asMyBs / �?�BjbySl �8jblSm&�]sc�Anpjbm?�?y�j>k�zj>³5j>t5zj>o~�Aj�±.j>m&m}zsMySy ( ù � � � ? È'�&sMµ | y ? ÈÅs � ù �zjb�~lSm]nM�| n�zjb�}lS�&jbmFn~lSk r o�ySjbk�zj>³5j>t5zjbo?zj�n�¶Ajby½q| y�l3�° � ý��í�jbo~j�np�5j>m�s / �Fn�o~t�±�zj>�AlSk r o�³ | mFn~m�sMk'� � �Öí}j>oAjbn�zjbm�³5j>�&l��}³&yQscm�zsM�~l�o�³ | m]npm�sMk�� | m&�V¶ r k'�

����� � XG�OQO Xg W�X\FT � � 3 3 N � �æu � -�M�- > = X = � - D Y -�M�AR�M�- P AC ; ; N - ; �.C 2 P �Q34R > E = M = T R ; � 3�T6T C'D 3 N �7Q3:/:14-�/�1 /BAC > E Y -�M�AR Me- P 2 -�T 783'/�1 � -�5 = / -�/�1<X = � - D ; K� T62�3�T 3 Y -�M�AR Me- P 5 3 D 3 E �Q3 G� I

Page 35: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

G����5Z{PSTdW ^ �1JMO"2 �MXJMOSJaTUHVPQ` 4 X^ OQLB` ^ N���\«RYJMZ>P�[�X\ ê PVX^ T � ^ Z>`,LYOçX\FN.�j>�c��qr kasMt � j>ySjb�KlBµ&lS³5j>��±.j>yAqr ySjbn~m&j>kÅjb�M�§�{q| �DtMzÑ npj�n~n�G�Î27 Ï u � ³�sM�psc��zj�npj>�Aj>o}jby~�| zsMy�y zÑ n'zsMo&zscn{�i�jbn~o~t��| ySjb�Dj>o � Ç 7Ëj>oAjbn�zj>mÀsct < ¡ G � �;� ¢�£{£{£>¢ < �&� ¡ G � �1� ¢ � � �1�åwDjbk]n | � | k«j>�M�,�}zsMt>l�o�zscn�sM�V¶&zsck§sMtu��«wDjbk]n | ��n�zj>�Am&j>k'�5�Kjby��MjbnºsMt�G�³&sM�psM�Òzjbn~j>�Aj>t5zjbo~µ&jbt��Aj>m&�&jby�n N�s r oAo �F±{zjbySj���zsMtblSo~m&sMk§� | m&� r mk5�° n | w�zsc�&�&lQsck?�&sMm�sMySkVsMy��asct>m&l�± | �{¶ r kåsMt � � � Ê Ø ¶Ajby½q| y�zj>o�n{� i�j>kFl�mFn~ocqr k�lSoA��zj�nÀsMt G

��zsMo | �&�Aj>m&� �r ³�sM� �bl�zsMy�lSo��&j>�Al�w#zsMyÓn7¶~sMlÓn{� ����l�y�w#zsMm]wcsMy�z| ��µ | �M�©s�< �­ � C G � �;� � � Ç 7,��wDjbk]n | �Am�sMk s�� ��� � �öySl�m&j�zsc�~lSo�zj>�~l�m]nD�| n�zj>�~�Aj��Kjb�M�| y�j>�Djbo�k | �K³ | m&jbm&o~jÅzj>³&³5j>m � ­ C � �;� � � �1� £ B�j ¶Ajbt>t�qr k�kFlFs(< �­ � C G � �1�wMj>k]n | �An�sMt�< ¡ G � �;� ¢_£>£{£>¢ < �&� ¡ G � �1� ¢ � � �1�#��zsMt>l�oAwMj>k]n | � | k,ySl�m&j�zsc�~lSo#k | ���lSm�zs �bl�z| ¶~sck�zjbm]n{Î

< �­ � C G � �1����� ® ¡ �

­ C � �;� � < G � �;��× � ­ C � �;� � � � �1� £

����� � ��� J � è Xg ê PçX^ � ° ±.jbmFn~l�kFl�±.j ¶Ajbt5zjbo�sMySsM³?¶&zsMm©zjb�AnpjbyS��j>tbjbn~n�

­ C Î 7 Ï u ��?A¢LED¢ $f� ØM¢{£{£{£�¢@� ��±dqr �M�Mw�zj>m]�DjbkMjbn_sct � ±.j>yAqr ySj�n(G�³&sM�psM�Òzjbn~j>�Aj>t5zjbo?zjbµ&j>t�npsM�An | t�z|�#µ�~lSo�n | �Bj>y �F±{zj>y�jvo~tblS����z| y r � | kFm�sMk�m&jbwMj>tbtqr k'�

°àkBq| wDj�npkcj>t�| zsMySy3zÑ n�zsMoKoAt>jb�~lSm]n�s �#µ&�AlSo�n | �Bj>y �?o~tblS����z| y r � | kVscnasMt¾j>y�oM�| ±U�| ��j>m&m]�?lSo?zj>�Djbk���zsM���j>�Mµ�scn'zsc� | tbt�zsMk'����� � XG�OQO Xg W�X\?T ���� X D =V; P C � -�M � Y K-�Me- ; N =V> E KC MSR > C T D 3 Y -�2�2[K3:MVM 3

­ C � Ø

���� ® ¡

# �� < ­ � C � ×C< C � � ­ Ê < � � ­ C�� £ � � � � �

AC ;0; N - Y AR�/ /&K- ; G°x±.j>yAqr ySj�npj>y���zjbySj�np�5j>m§�&jb�~lÓw�zs �>l�z| o�± | �~� r y�zsMkFm�sMk�o~t | kÄzsMoºm&jbwMj>t>ml;s

< �­� C Gv�

�� ® ¡ �

­ C � <

G}× � ­ C � � ¢ � � � � �

< � �� ��

�® ¡

�� ® ¡ # �

� � � < �

G � � � Ø í]�jb�M�Djbm&ySj�npjbkMjbn ��?~¢ ED¢ �Ö� ØD¢>£{£{£>¢ � ���x��tbj>kFkMjbyvkcsc³ �bo | ySscnp��scm �5zjbySoAt>j>�F�r �Kjb��¶Ajb�M�Djbt>m;qr mk5��µ | �M�s � � � Ø í]�§q| oAo~tbjb±Uqr �D�1zjbo~jbk´s �Õj>lSm�]sM�An~j>m?�FySj>kfzjb³Bjbt5zj>oAj>kF�~j«w | m�scnpk | t�z| � � � � �Ij>�M�Mj>m&y�jbn~m&j>k8±.jbySjbySm&jbk��j>��G10�\?LBTUT 2&JMé��J è OSJVWbJcN���\ � \?P è \FZ>RYP"2 � ^ RB\FHVHVP"2&JMé���J è O�JcW�JMN

°�t#sMy�zsc�&�&lQsck�o | �&zsMm�sctbn;w?l�t>o~��zsMyж r k'�cµ | �M�_sMt � ±.j>yAqr ySj�npjbn�y�jDzÑ �&z| GIÎ 7 Ï u � y�j>kfzj>³5j>t5zj>ofjbySoc�| zj>o��zsMo | �&l�k�±U�| ��j>m&m]�?lSo?zj>�DjblBkBq| tDq| n~n#�Kl�y��Djbm¾q| oAo~tbjb±Uqr �D�1zjbo~jbk©zsMy�ySm�sck�±.j>m&mY����mm&j>k �5zj>y ¶&zsM��z| y�n~j>kFlSm]npocqr ksct�G_wDjbkFn | �A±dqr �M�Mw�zj>m]�«µ�sM�A�asM��~j>m�Ä�r ³�sM� �bl�zsMy�lSoÄ�&j>�Al�w#zsMy�n.¶~sMl�n>�;° � � � � �#j>�M�Mj>m&y�jbn�sMyQsc³?¶&zsMm,m]�Djb�.¶cqr ks< �­� C � � G���< � � <

�­� C Gc���

�� ® ¡ < � �

­ C � <

G}×

�� ® ¡ �

­ C � < �

� � G}× < � � ­ C � �$× � ­ C � < � �

� �

Page 36: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

kFlÓ±.j ¶Aj>t5zj>o�n{� ° � � � � ��� � � � Ø í]�Ä�&jb�~lÓw�zs �>l�z| o�± | �~� r y{zsMkVscn#±.jbySµ�sco~t>m}zsMy�wcs�s

< � � < �­ � C GM����� ® ¡ < � �

­ C �@<

G&×

�� ® ¡ �

­ C�� ��

�® ¡ � �

� �@< �

G�× � � � ���Ä×$< � � ­ C � � Ê � ­ C � ���® ¡

�� ® ¡ # �

� � �@< �

G��q| o~o~tbjb±dqr �D�1zj>o~µj>t�¶ r n r m&k'� ��m&m&jbk�zscnp�Aj>m&�&jbt5zjbo?zjbwMj>y;s< � � < �­ � C Gc���

���® ¡ � < � � �­ C ×

�� ® ¡ �

­ C � �

� Ê � ­ C

�� ® ¡ # �

� � � � <

�G}× B < � � ­ C�×

�� ® ¡ �

­ C � � D � �

jb�M�Djbm&ySj�npj�n�kVsM³?¶ r k5�B�j ¶Ajbt>t�qr k�� | oAn�kFl]sF< � � < �­ � C Gc� Ê < C � <

�­� � Gc��wDj>k]n | �An1sF< �

G � ��� ØM¢{£{£{£b¢ � Ê Ø � ¢ � ��zsMt>l�oAwMj>k]n | � | ky�lSm&j�zsM�~l�o�k | ���&l�m�zs �bl�z| ¶~sMkfzjbmFn>� � ��t r ���©zs �bl�z| oºlSm�&j�Û�k�zj>m]nW$�µ&j>yÓ�DjbnAn�sMy�kcscyS�KsMt>t r k c�Fn{� �À°¿wDjb�M�Djbo³&sM� �bl�zsMy�lSo��&jb�~l�w#zsMyÓndsMk�sMyQsM³�n r ySsç¶A� | m&o&zsM�]s�kBq| wDj�npkMjbtbn�zj>�5j>mÅj>tbj>m�kFl�±.j½¶Aj>t5zj>o~�5j>mÅsct�q| oAo~tbj>ofjb�M��qr n~npµ&scn'z|zjb�An�zjbkMjIí��¼½y��a�©z| � | m,npjbyжAj>ocqr ySm&jbk§sMt�scy�zsM�&�l�q| o~oAt>j�±Uqr �D�1zj>oAj>k

< � � �­ C Ê < C � �­ � ×��` ® ¡ B � `­ C � �` � Ê � `­ � � �` C D �

��` ® ¡ # ��` B � ­ C � ` � Ê � ­ � � ` C D ¢ � � � ØDØ �

< � � ­ C Ê < C � ­ � ×��` ® ¡ B � `­ C � ` � Ê � `­ � � ` C D �»í £ � � � Ø �D�

� � � £ÓØDØ � AC ;0; N - Y AR:/ /&K- ; -�T6- P�� 3�R ;0; � -�/�1 -�2�Me- P -�T62 -�T � 3 N � � £ÓØ �D� AC ;0; N - Y AR:/ /&K- ; -�T6- P � -85 = /�� 3 = 2�3 D 5 = � C 5�3 N8N = � -�/�1 -�2�Me- P -�T62 -�T 2 -H7'- N�N AR T G

° � � � ØDØ �5j>�c�Dj>mySjbn���scy | yS�&sMy�zsMm�o~tbj>�~jb³&y~�| kFl�±.j½¶Aj>t5zjbo~oAj>yFsMt 7x³�sc�psM��zjbn~j>�AnpsM�An | ��zsMm]� | mv�&j/&�m&l{zsMyж r ksct �� C ­ � < � � �­ C Ê <�C � �­ � ×

��` ® ¡ B � `­ C � �` � Ê � `­ � � �` C D � � � Ø � �

�l �Bj>�~jbm �>l{zsMySµ&scn'z| ±dqr �D�cw�zjbm]�Dj>kcjbn ��?A¢ ED¢ � ¢ ��� ØD¢{£>£{£b¢ � � £ ��t>jbkMj>mKk5zÑ w�qr y�s�7 �©u � npsM�An | ��zsMm]� | m�5zj>y�o~t>jb�F�r ��zjb�Òzjb�An~j>yS��j>tbm&l�sMt

� C ­�

���® ¡

�� C ­ � �

wcsMy{z| o�±Uqr �D�Mw�zjbmF�Mj>kcjbn�l�o>�� JMéçhUJMé��'H1XJcT � ° � � i � z°�ySy3zÑ n�zsMo#kBq| wMjbnpkcj>t�n�zj>�5j>m�s�±.jbmFnº�&j/&�m&l{zsMy�n

� C ­Ç � � 7,�

��?~¢LED¢ � ¢ $�� ØM¢{£{£{£�¢ � Ê Ø �ıdqr �D�cw�zjbm]�Dj>kcjbn�sMtvj>y�oM�| ±U�| �Kjbm&m]�?lSo?zjb�Dj>ka�©zsM�}��j>�Mµ�scn'zsc� | tbt�zsMk'�° � � � ØDØ ��jb�M�Djbm&ySj�na��lSm�&k�zjbn | yS�&sMy�zscn�o~t | � | tbt r kÒ��j>�Õs�� �

±Uqr �D�Mw�zj>mmF�Mj>yä�#�Ksç¶A� µ&sç¶�npo r m&kw�zjb�D�~jaq| o~oAt>jb�Dt5zjboAnvsct �IlSm&�j�Û��~j�mÄzjbtbwDj Ø �FnM�| y � �?l��&����t�zsMyÓndsMy�sMt

� C ­�

���® ¡

��` ® ¡ � �

# ��` B � ­ C � ` � Ê � ­ � � ` C D

jb�M�Djbm&y~�| o?zj>�Mµ&j>t}¶ r n r mk5�ÄÜÌlÓwDjby�±.j>m&m�zscySy8¬ ��® ¡ �

# ��` � � ` �&s�N�s r oAo �?jb�M�Djbm&ySj�npj>kF�f�| y1sct

� C ­� � ­ C � � Ê � ­ � �

C � � � Ø I �

� i

Page 37: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

q| o~o~tbjb±dqr �D�1zj>o~jbk�sM��z| �&m&sMk'�G � ^ Z{`�\�2&eYZ ^%4 O�XJD`�\ ��� ^�èYè JcW�WYXJcW�JDOSJ

ô�j>�M�Mj>m7Ëj>�c�Õq| o~oAt>jb±dqr �M���| m]�BzÑ yÓn�µ�sMyS�KsMt�n}u ��� ¡ �?�Bjbm��#i�j>kFl�mFn~ocqr m&k | yÓ�]sMm � ­ C ¢&� ­ CvÎ 7 Ï u�l �Bj>�~jbm �>l{zsMySµ&scn'z| ±Uqr �D�Mw�zj>m]�Mj>kMj�n ��?A¢GE � ØM¢{£{£{£�¢ � Ê Ø � ¢ �Kjby��DjbkF�~jI±.jbm&m�zsMy�y � ­ C�� � C ­ zj>o � ­ Cv� � C ­ �° � ­ CI±dqr �D�cw�zjbm]�Dj>kÀn~j>�M�Mj>m&jbk¾j>y�j>�Dj�n�sct | m¾±.j>yÓn�zj�npj>y�m&j>k¦lSo>��µ | �M�Àn~jbnpoAt�| y�j>�Djbo � Ç 7à³ | mFn~��sMmÂsMtzscy�ndscy r k,��j>�Dµ&scn'zsM� | t | n~n J � �1�ºoAt>lS����jbnp�AlSk r o��©zscn~�~l¯Û�j>�M�̳ | t>l�n{zÑ w«�&j/&�m&lÓnvk]wMsc�&�pscn~lSk r o�± | �~��zscnsc�Ì��j>��sMt�u �&� ¡ wDjbkFn | �An�zj>�Aj>m��

B�j>yÓnpj>µjbnD�| sIkfzjb�~�Äzjbo>�&µ | �M���Kl�m&�&j>m�j>oAjbn~�Bjbm§wMscm?�?j | y��DsMm9G�Î27 Ï u � �&l �Yjb�~jbm �>l{zsMySµ�sVn'z| wDjbk]Þn | �A±dqr �M�Mw�zj>m]�D�sM�Kjby��Åjb�M��j>ySjb�Kl'±.j>yAqr ySj�npjbn�zÑ �Äy�jM��zj>o}sM��j>yÓ�?m#zj>y5sMt�j>ySoc�| zj>o#�©zsMo | �lSk�±U�| ��j>m&m]�?lSo?zj>�Djbk��j>�Mj>�M�Mj>t>mj>k�sMt_sM� | nAn � ­ C ¢ � ­ C�±dqr �D�cw�zjbm]�Dj>kFkcj>y3��fw?lS�&jbm&o>�Dµ | �M�IsMt�zÑ �M�v±.j>yÓnpj�n~nfkfzjb�~�Äzjbo~�~j�s�w#zsMyQsMoAt#m&jb�]scn{zÑ w5�Mµ&l�o~tbj>m�jb�M�I±.jbyAqr y�jbn�j>y�oM�| zjbof��zsMo | �&lSk±U�| �Kjbm&m]��l�o?zjb�Dj>lÓwDj>y�kVsM³ �>o | yQsVnp��sMmÅs8N�s r oAo �?jb�M�Djbm&ySj�npjbk?mj>kÒzj>o�s_ܦsMl�m�sM�A�&l¯� � | ��sMtbt>l �?j>�M�Mj>m&y�jbn~j>kDÞmj>k,lSo�n~j>y ¶Aj>ocqr y�m&lAqr kÅkcj>ySyä�° ³&� | �&y�zjb�as�kVsM³ �>o&zsMm�s � ­ C ±Uqr �D�Mw�zj>m]�DjbkF�f�| y�k�zj>³&tbjbnAn-J � �1� � � Ç 7,�;��zscnp�AlÓÛ | k_lSm]wDjb�An'zscy�zsMo&zscwcsMyzjb�AnpjbyS��j>tbtqr kÅs;# ­ C ±Uqr �D�Mw�zjbmF�Mj>kcjbn{�Ä��tbn#kYq| wDjbnM�| j>m§wMj>�M��qr kKs � � � � �f± | �A� r yQsÕzsMyÓndsMy'��j>�Dµ�sVn'zsM� | t | nAn�

­ C Î 7 Ï u ±Uqr �D�Mw�zj>m]�DjbkMj�n{�¦¼½yÓ���©z| � | m8jbySy�j>mf�| �AlStbm&lfn r �ç¶ r kÂs � � � ØDØ ��� � � � Ø �D��q| o~oAt>jb±dqr �M�;zj>oAj>kn~j>y ¶Aj>ocqr y�zj>o?zjbn{�Å°$³�sM� �>l{zsMySl�o��&l �Yjb�~j>m �>l�zscySj>�c�Dj>mySjbn �?�~jbm&�&oAt>j>�Aj>k�j>y���zjbySj�n�zjbnvsMySkVsMy��asMt�wcsa�&lSt | m]�BzÑ ndscm&ly�j>µ&j�n�sMt�sMy�zsc�&�&l�sMySsM³]wDj�nD�| j>�~jb�&��zj>m]�Fn{�'�Kjby��Mjbn | mm&jbn�n�zj�npjbySj>kfzjbmFnºndsc�An r m&k§o~t�zsM� | m�����"! � * XJVW�JDO � � -0/�1 AR T Y -�M � X C /�1_3 N 365 C P0P � ­ C ¢9� ­ C Y AR�/ /�7 K-�2 1�-�T D -47 C 2�3 P T C N KC 3�2 P -�M ��- ; AR�MV2 -�T3 � � £ÓØDØ � ¢¾� � £�Ø �M� -0/�1�-�2�M�- P -�T G � N - ; - P�E -�2 P - P ; N C M�-0/ - ; - Ç 7 � C 2 P 7Q3�M 3 > -�M(1 � � � � 7,�2 1 K� M P AC ;0; N - Y AR:/ / C TJAC'D 2 1�- N - P K-�2 > -�/<Me-�X - P 36562 = -0/�1 C M(1�3�2 G�Î � Ï u�� 5 = � - D -�2 ��= K3�M?X�3 P KC 7'-�T��P C'D Y AR�/ /�7 K-�2 1 P � 3 > -�M#1�-�/:1 -�Me- >h= X = � - D Y -�M�AR Me- P - P K� D Me- K- ; 3 N G � X - N P 3 DFP C N KC -�M ; C K- ; > K3 ; C 5 = TY C > -�2�2 1 =?; K-0/ -�T > -�/ -0/�1�- N 2 -�T<3 � ­ CDñ � ¢ � ­ C]ñ � Y AR�/ / 7 K-�2 1�-�T6T6-�MSG

� > -�2 2 1 = E -�2 3 N �G�Î � Ï u�� 7'-�T P C'D Y AR�/ /�7 K-�2 1 =?;&> -�/ C M�5WK3 ; K3 P T K- � - N = 3 Y -�M 365�3 P 2�3�T � 3�T6T C'D7Q3�2 C M(1�3�2 Í ÎMu�� Ï u � = D K3�2 1 K� P K3 ; P 3 DFP KC = N C > - P D = 3 � X C /�1 Y -�2�2[K3�M?M �G��ÖÍC)(G £

� �

Page 38: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

�B)�*�� /�= G 03=�+U@� +7= � 4'=F2 � = �' 9 � E�41@2 GçÚ C= G �äJMOE3LYOSJVWbP;é63^ Z 4 JÀXg [�� ^ TUTUH;X\ è \FNÕ[]\FZ{P�X\ ê PçX^ hU\

���&�5j>m s¦±.j ¶Ajbt>jbn~�Bjbm jbySoc�| k�zj>m]nKsMt>t>sMy}sMt,scyQsM³]wDj�nD�| k�zj>�A�#zj>oAo~j>y}± | �MyQsMy�k | t r m&k'��µ | �M�8sÀµ&l�³Bjb��Þ±.jbyAqr y�jbn�kfzj�n�sM� | nAn�³ | m]n.¶&zscn q| oAo~t>jbkBq| nD�| ±.j>yAqr ySjbn~lc��q| �~�Äzjbk�kBq| t�qr yM��lSk�zj>m]n�ySjbµ&jbn�k?lÓw#zsMyQsMoAtbnpsMm&l]s���lSm&l��©zscySlSozÑ w�µ | o~oAt;zr o�zsc��zr n{� ° ³&� | �&y�zj>�Ks�n�zsM�~�M�DsMy�zscops¾o | �&zsMm©scySkVsMyS�KsMtbm&l#± | �{¶ r kÒsMt�j>yA�| t�| k�zjbnK±.j ¶Aj>tbjbn~�Bjbm�5jbwMj>tbjbnpj�n~n�± | �]sMy��asMkVscnKzj>o;¶Aj>y q| y�zjbo~jbkMjbn>�ô�j>�M�Mj>m�sM� | nAnÅj>�M� � � u � o~l��asÀjbySjb�Kl�µlS³5j>�A±.jbyAqr y�jbn{�fsM�Kjby��Mjbn�s+7 �Ëu �&� ¡ m]�BzÑ yÓn�ndsc��Þn | ��zsMm]� | m zjb�An~j>yS��j>tbjbnAn9G Î�7 Ï u�� wMj>k]n | �A±Uqr �D�Mw�zjbmF�ÖzÑ �Åy�jM� �Iq| �DtMzÑ npocqr kÒsMt � �?m&jbk jbt>jbm G³&sM�psM�Òzjbn~j>�Aj>o�j>yA�| zsMySy3zÑ n�zsMo�zscn>�°xnçq| �Àq| �Ao?zjb�Òzjb�~�&jbk�zj>�5j>m¾s�n | w#zsM�&�lQsMkF��sMmÌsMy�kcscyS�KsMt>t r kasct � �å� Ê Ø ¶Aj>y q| y�zjboAn>�°��Kjbm&m]�?lS�5j>m8wMj>o~oAt�qr kÕsMtÅu � n�zjb�_wcsMySsM��j>y�� �Ö� ��?�¡U¢{£{£>£>¢A? �#�Àzjbo�� � � � ¡U¢{£>£{£>¢ � ���vj>ySjb�Kjbl�n>�

sck?k | �º�Kjb��zsMySySsM³ | ��zscoÄoAt>j>�AlSm]n � � ¢ ����¶Aj>y q| yжAj�sMt�u � � n�zj>� ��?�¡U¢{£{£>£>¢A? � ¢ � ¡�¢{£>£{£>¢ � �#�å³ | m]n.¶&zscn{�°�t GI³�sc�psM��zjbn~j>�~jbt5zjbo~µ&jbt�npsM�An | t�z| jbySoc�| ±U�| �Kjbm&m]�?lSo?zjb�Dj>k�±.j>y�µ�sMo~tbm�zsMy{zsMo�zscwcsMy'wDjbt>jbo~ocqr k,�5jvsMt�n�sMtõ²Î 7 á¦u � Ï ux± | y��Fn | m | o�±dqr �D�cw�zjbm]�?n>�'sM��j>y��?�Aj�±.j>m&m�zscySy

õ � � ¢ ����� � ¬ �­¯® ¡ ¬ �C ® ¡ � ­ C ��?�¡U¢{£{£>£>¢A? �#� � � ­ � �LC � � � Ø ��}zsM�~��j>yÓ� � Ç 7 zj>o�� Ç u � jbo~jbn�zj>m���ô_zscn~µ�scn�z| ��µ | �M��sK� � �Fw#zsMyÓn | t�z| o�õ¿±Uqr �D�Mw�zj>m]��sMtvu � � �?�5j>y�l7Ëá � u ��� ð:��óc��m]�BzÑ yÓn�µ�sMy��asMt�±.j>ySj�n~n��&l �Yjb�~jbm �>l{zsMySµ�sVn'z| �

° 7 ³�sc�psM��zjbn~j>�AnpsM�An | ��zsMm]���&sMm�npjbkFlSm]npocqr m&k�jb�M��� Î[� ¢���� Ï 7 � u � �~j>� r y�zsM�AlSoKo~lS�Ks��q| �~�Äzj�n{��° �Ö� G )���±.j>yAqr ySj�npl��'q| �~�5j�w�zj>�D³ | m]n7¶~scl�ySj>�c�Dj>mj>k / � � � ���zjbo Æ � � ��� ���Å�fw?lS�&jbm&o>�µ | �M�«s;�©��q| �~�5j�zÑ w?µ | oAo~t�zsc�ps $ � �Ä��� \_^` õ � � � ãA� ¢ � � � ãA�~�&ÈDãn~j>y ¶Aj>ocqr yä��°�t�sMy{zsM�&�&lSsMk�o | �&zsMm�j>�M��o~t�qr kFo?zj>�Djbo � �&j�m&j>�»j>y�zj>�Do?zj>�Djbod�Y±.j>yÓn�zjbn~j>yÓn�sM� r m&kvsM�A�ps#w | m�scnpk | t�z| sMm��µ | �M�as / ¢�Æ ³ | mFn | kVscn�q| o~o~tbj>kBq| nM�| ±.jbyAqr y�jbnpl���q| �~�ÄzjbkKkBq| tDq| n~n�s�� �Kl�m&lS��zsMy�lSo}zÑ w�µ | o~oAt;zr o�zsc��zr ySjb�M�Djbm��

.�j>�c��qr mk§j>�M� | y��DsMm Î � ¢���� Ï u � �&l �Yjb�~jbm �>l{zsMySµ�sVn'z| ySjbk�zj>³5j>t5zjboAn>�Bsc�Kjby��?�~jvl��]sMt � ?��� � zjbo� � � ���]��� ����l�y�w#zsMmÅ��q| �DtDzÑ npj>ml&ySjbµ&jbn�jb�M� | yÓ�]sMm�ë�³ | tbl�n{zÑ wIo~t�zsc� | n>�]µ | �M��npjbn~o~t��| ySjb�Dj>oã Ç � ¢���� zjbo d�Ç�� Ê ë ¢ ëD��j>oAjbn�zjbmÅsMt�� � ãA��× d � ãA�̳ | mFn;j>y�j>��j�s�7 npsM�An | ��zsMm]��m&sMk'�1i�j>kFlSm]n~oMqr kIsMtbns � Î � ¢���� á � Ê ë ¢ ëM� Ï 7 �l �Bj>�~jbm �>l{zsMySµ&scn'z| wDjbk]n | ��±Uqr �D�Mw�zj>m]�Fn{�FsM��j>yÓ�Dj�nfs � � ã ¢0d ��� � � ãA�b× d � ãA�jb�M�Djbm&ySj�n�zÑ ��ySjc�� ����� � J� è Xg ê PVX^ � ° � ySj>kfzjb³Bjbt5zj>o�nÌs 7 ³&sM�psM�Òzjbn~j>��ndsM��n | ��zsMm]�?��sMm©wDj�n~n�²��q| �~�5j�$o~tbj>�~l�m]npl�jb± | �A�©zs �bl�z| ¶&zscm�sMk�� | m�V¶ r k'�

�Iq| �DtMzÑ npj�n~n d�Ç�� Ê ë ¢ ëD��zjb�An�zj>k¾j>oAjbn�zj>m´s ��� � ãA����� � ãA�f× d � ãA� � 7 ã 7 � ��q| o~o~tbjb±dqr �D�1zj>o~oAj>y��j>�Mµ�scn'zsc� | t | n~n ��� Î � ¢���� Ï 7 �'q| �A�#zjbnºsMt��8j>�M�?lSka�&jb± | �A�©zscy�n}�'q| �~�Äzj½¶?zj>m&jbk§m&jbwMj>tbtqr k'�� JMéçhUJMé��'H1XJcT � °�t�©wDjbkFn | �A±dqr �M�Mw�zj>m]�?�~jÅ�Kjb�]sM� | n~n�±.jby�n�zjbnpjbyf��lQscnAn���zsc�~��j>y�� ��� � Ê ë�ò d òxëM��jb± | �A�©zsMyÓn#��q| �~�Äzjb�~jº±.j>m&m�zscySy ��� � ���� � � ���zj>o ��� � � ��� � � � �U����t5zj>��n}o~t | kÄzsMo}sMt�n�� | m&��scm&l3�?µ | �M�Ks� �&jb± | �~��zs �>l{z| &Û�j>m�µ�sM�M�{¶~s�sMt��Ò�'q| �A�Bj�w�zjb�D³ | mFn.¶~sMl�n>����Ky3zÑ n~oMqr kÅ��zjb���Kjb�&�&µ | �M�«s ��� ySjbk�zjb³Bjbt5zjboº�j>�~lÓw#zsMy�n.¶&zsM�ps � �� � ãA� ��� � � ãA��× d � � ãA� � ã Ç � ¢���� �n~j>y ¶Aj>ocqr yä�

� �

Page 39: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

°x±.j>m]n~l��jb± | �A�©zs �bl�z| nı.j>y�µ�sMo~tbm�zsMyÓwMs�n~j>kFlSm]npocqr k,s / ¢�Æ ³ | mFn | kVscnIq| oAo~t>jbkBq| nD�|, � � G) ��� � Ê ë�ò d ò�ëM�ı.j>yAqr ySj�npl5�'q| �~�Äzj>kMj�n{� �Kzj>y�o~t>jb�F�r �Kjb��¶Ajb�M�Djbt>m&lä��µ | �M�a±.j>m&m�zscySy:, " � ���ô�j>�M�Mj>m0#��ÀÎ � Ê ë ¢ ëM� Ï u²sMt�sK±dqr �D�cw�zjbm]�D��sMµ | yfs #�� � d �º±dqr �D�cw�zjbm]��zjb�An�zj>k¦��j>�Mj>�M�Mj>t>l�kÀs , �±.jbyAqr y�jbnplB�'q| �A�BjÅzÑ w�µ | o~oAt�zscwcsMy � Ê ë�ò d ò ëM����° � � � Ø �ı | �A� r ySs�kYq| wDjbn~kMjbtbn�zjb�Bjbm�s

#�� �ed ��� \ ^` õ � � � ãA��× d � ãA� ¢ � � � ãA��× d � � ãA�A�&ÈDã � � � �D�jb�M�Djbm&ySj�npµ&jbt�¶ r n r m&k'�

�º�?l�y�w#zsMm]wcsMy�z| �cµ | �M��sM��j>m&m]�?lS�5j>mÅs / zj>o Æ ³ | m]n | kVscnºq| o~o~tbj>kBq| nM�| ±.jby~qr ySj�nplF�'q| �A�Äzj>kIkBq| t]q| nAn�m&l�m �>o| y��]scm��'sM��j>y��?m&jbk§sMt�zÑ w�µ | o~oAt{sÅk?l�o~jb�&�¦s'�ÖzÑ w?µ | oAo~t�zscm�zsMy3�'sck?k | � # �� � í]����í�n~j>y ¶Aj>ocqr yä�� ��� � � J � è Xg ê PVX^ � °-��� G6) �¦�'q| �~�Äzj�n sMt � µlS³5j>�A±.jbyAqr y�jbnfo�nds �bl | m}zsM�~l r of�'q| �A�Äzj ¶Fzjbm&j>k�m&jbwMj>t>t�qr k'�µ�s�}zsM�~��j>yÓ� | y��DsMm �Î � ¢���� Ï u �&� ¡ �l �Bj>�~jbm �>l{zsMySµ&scn'z| ySjbk�zj>³5j>t5zjbo~mÄzjby3�YsMµ | y � ?�#� �xzj>o � � �#� ���±.jbm&m�zsMy�y�s # �� � íD���Öí q| oAo~tbjb±Uqr �D�1zjbo>�� JMéçhUJMé��'H1XJcT � .�j>�M��qr k,sct � � � ãA��� <�� � � õ � � � ãA� ¢ � � � ãA�~��jb�M�Djbm&ySj�n~n~j>y��Kjb�Dµ�scn�zsM� | t | nAn� � Î � ¢���� Ï u �l �Bj>�~jbm �>l{zsMySµ&scn'z| ±dqr �M�Mw�zj>m]�Fn � �Ë� ØD¢{£>£{£b¢ � ��� i1�~sM��zÑ �>l | m�zsMySl�o | k | kF��z| y�skBq| wMjbn~kMj>t��| n�zj�npjby���j>�c± | �]sMy��asMt�zsMo�zscm�zsMy5sMt�� � ã Ç � ¢���� µ&j>yÓ�DjbmÌwDj�n~n ���� � ãA���&jb�~l�w#zsMyÓn7¶&zsM�~s�sÈÈDã B <�� � � õ � � � ãA� ¢ � � � ãA�A� D ¶Ajby½q| y�zj>o�n�sMySkVsMy��asct>t r k5�°�t��8Î � ¢���� Ï u � zjbo �Î � ¢���� Ï u � ySjbk�zj>³5j>t5zjbo~jbk�k |�| �~�&l�m�zscnds��F±Uqr �D�Mw�zj>m]�Mj>l�ySjb�M�Djbm&j>k� � zjbo�� � � ��� ØD¢{£>£{£b¢ � ���� ����� * XJcW�JMO � � N<� ¢���� = 2 P - D 7Q3�M?MSR > C 2 K- DFP -�M > - N - P P �Ö� G ) � /BAC'DHE -@3 N � -�Me- >h=.Y -�M�AR�M�- P 2 -�T-0/�1 ; P 3 ��= C 2[K3 D = R ; /BAC'DHE K- ��- 3�T6T C'D K- ; ��; 3�T<3:T T C'D � X 3 E K3 D > -�M#1 ã Ç � ¢���� - ; - P K-�2 P -�M ��- ; AR MV2 -�T 3

< � õ � � � ãA� ¢ � � � ãA�~� ÊÈÈDã B <�� � � õ � � � ãA� ¢ � � � ãA�A� D ��í � � � � �

� �K� ØD¢{£{£{£�¢@� � AC ; ; N - Y AR�/ /&K- ; -�T G��PSH ^è ��Xg W�X\?T �° � � � �M�#j>�M�Mj>m&y�jbn~n~j>y�ySjMzÑ ��n #��K±dqr �D�cw�zjbm]�«íç�?�5j>y�lY�&jb�~l�w#zsMyÓn7¶&zsM�~s�±.j>m&m}zsMySy

# �� � í]������ ® ¡

\ ^` � < � õ � � � ãA� ¢ � � � ãA�~� ��� � � ãA��× <�� � � õ � � � ãA� ¢ � � � ãA�~� ��� �� � ãA���ÄÈDã £ � � � I �

°�³�sc� �>l{zsMySl�o#l�mFn~j>�D�&zsMy�zsco}o~t{sc��zsMy��Ds�o~t>jb�~l�mFn

\_^` <�� � � õ � � � ãA� ¢ � � � ãA�A� ��� �� � ãA�&ÈDã�� <�� � � õ � � � ãA� ¢ � � � ãA�~� ��� � � ãA��� ^` Ê \_^`ÈÈDã B <�� � � õ � � � ãA� ¢ � � � ãA�A� D � � � � ãA�&È]ã

n~j>y ¶Aj>ocqr yä��ÜÌlÓwDj>y�s�±.j>m]n~l�j>�M�Mj>m&y�jbnf¶ | �&� | yS�&sMy�zsMmao~t>jb�~jb³&y~�| j>y�oM�| ndsM�Åjby�n?�r mlSk'� � � � I � �?�f�| y�s

# �� � í]������ ® ¡

\ ^` � < � õ � � � ãA� ¢ � � � ãA�~� ÊÈÈDã B <�� � � õ � � � ãA� ¢ � � � ãA�A�ED � ��� � � ãA�&ÈDã � � � � �

q| o~o~tbjb±dqr �D�1zj>o~µj>tĶ r n r m&k'����j>� m&jbµ#zj>tv�5j>y{zscnpm&lä�Fµ | �M�Ks � � � � � j>�c�Dj>my~�| o?zj>����zsM�Äl��]sMt | yж~s_sIn�zj�npjby�n{�� �

Page 40: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

� JMéçhUJMé��'H1XJcT � °ÉwcsM�~l{zs �>l{z| o~t�zsM�ÌzÑ n'zsco�j>y���zjbySjbn�zj>�5j>mÕs�o�nds �bl | m}zsM�~l r ovµ&j>yÓ�DjbkMjbn��Kjb�Dµ�scn�zsM� | t�z| � � � � �jb�M�Djbm&ySj�npjbkMjbn�sMt_� r ySjb� ��ô�sM�D�~sMm&�Djd�F±{zj>y�j��&l �Yjb�~j>m �>l�zscySj>�c�Dj>mySjbn �?�~jbm&�&oAt>j>�Am&j>k,oAt | kÄzsMoºm&j�wDj>tbm&l3�°�n | w#zsM�&�&lSsMkF��sMmÀsctO� ¢���� l�mFn~j>��wMscySy r � | m,�j!&�m&l{zsMy�n(��� G() �Ò±.jbyAqr y�jbnplY�'q| �~�Äzj>�M�| y�±.jby�n~j>o~oAt�qr k'�µ | �M��zÑ w�µ | o~oAtIoAt>j>�AlSm]n�wcsMm«³�sM�psc��zj�npj>�Aj>t�wDjM�Ä��o~tbj>�~l�m]n���zsc�~��j>y���ã Ç � ¢���� µ&j>yÓ�Dj>mÌlS�]sct

õ � � � ãA� ¢ � � � ãA�A� � Ø�£ � � � i �� ��� �&* XJcW�JMO � � N K� 7HX C ;0; N ; N - D = 2 P � 3 D 3 > K- P - D - N - P0P ��� G ) � /BAC'DHE - 3 N � -�Me- >h=�Y -�M�AR Me- P 2 -�T9-0/�1; P 3 ��= C 2)K3 D = R ; /BAC'DFE K- ��- 3�T6T CHD K- ;���; 3:T<3�T6T C'D � X�3 P - P ; N C M�-�/�- ; ã Ç � ¢���� - ; - P K-�2 Y -�2�2)K3�MVM?2�3�T 3 N

� � � � ãA��× �� ­¯® ¡��C ® ¡ �

­ C � � � ãA�~� � � �­ � ãA� � � �C � ãA���Öí � � � � �

� $�� ØD¢>£{£{£�¢@� � AC ;0; N - Y AR�/ / K- ; -�T G��PSH ^è ��Xg W�X\?T �°ö±.jbmFn~lÅk?l ¶Aj>y�j>m]n�zjboÕl��]sMt | y�zsMo&zsMµ | tÒscySkVsMyS�KsMtbt r k»s � � Ø � iazj�npj>yÓn{� B;lS�c�Dj>y�j>���5j´w�zjbwMj©s � � � i �jb�M�Djbm&y~�| o?zj>�Mjbn�sMt � �?mÄzj>y�mj>�þm�sc�M� | �� �§lSm&�j�Û«jbo~jbn�zj>mÌkYq| tbwDj�npy�j>m¦oAt�zsM� | y{zsMo~o~sMy�m]�Djb�.¶cqr k«s

< � õ � � � ãA� ¢ � � � ãA�~���Ø� ¬

�­¯® ¡ ¬ �C ® ¡ < � � ­ C � � � ãA�A� � � �­ � ãA� � � �C � ãA� ¢<�� � � õ � � � ãA� ¢ � �Ã� ãA�~���»¬ �­¯® ¡ � ­ � � � � ãA�~� � � �­ � ãA� ¢È

ÈDã B <�� � � õ � � � ãA� ¢ � � � ãA�A� D � ¬ �­¯® ¡ � ­ � � � � ãA�~� � � � �­ � ãA��× ¬ �­¯® ¡ ¬ �C ® ¡ <�C � ­ � � � � ãA�~�� � �­ � ãA� � � �C � ãA�jb�M�Djbm&ySj�npjbkMjbn>����t>jbkF�f�| y�zj>oºs � � � � �vq| oAo~t>j�±Uqr �D�1zjbo~�f�| y�sc��z| �&l�k5�µ | �M� ��³ | mFn | opscm¦sMkFk | ��ySj>oAto�nds �bl | m}zsM�~l r o}�'q| �A�#zj ¶Aj_sMt � ±.j>yAqr y�jbn~m&j>k'�µ&sÅn~j>yжAjbocqr y�m&j>kas�� ­ ® ¡ � ­ � � � � ãA�~� � � � �­ � ãA��×

�� ­ ® ¡��C ® ¡

Ø� B < ­ � C � � � � ãA�~��× < C � � ­ � � � ãA�A� Ê < � � ­ C � � � ãA�~� D � � �­ � ãA� � � �C � ãA���Öí

� �Ì� ØD¢{£{£{£�¢@� ��q| o~o~tbjb±dqr �D�1zj>o~jbk'� �?t | � | t>t r k¦��j>�,s�±.j>m]npl1j>�M�Mj>m&y�jbnpjbkMj�n�s # �� � � ãA�~��o~t�zsM� | k?kVsMyä�zjboIµ�s�¶�npo r m&k¾w�zjb�D�~jÀq| o~oAt>jb�Dt5zjboAnÅs��Âl�m&�&j�Û��AjKmÄzjbtbwMj Ø �FnM�| y � �?lS��a°���j>mmF�?l��Bjbm´sMy�kcscyS�KsMt>t r kÀs

�#µ�~lSo�n | �Bj>y �?o~tblS���}z| y r � | kF�ps_w | m�sVnpk | t�z| � � � � �ı | �~� r y�zscn>�sck?k | ��s � � � � �vq| o~oAt>j�±Uqr �D�1zj>oAj>kcjbnkVsM³F¶ r k'�� JMéçhUJMé��'H1XJcT � ° oAnps �>l | m�zsM�Al r o��'q| �A�Bj«± | �]scyS��zscn�npjb�~��zj>o~tbjbn~j>o~jbmåk?lºySjbµ&jbnKnpjb�.¶Aj>oAtbn~j>m&l�sM�~�~s8sMtjbo~j�np�~j_lSob�'sM��lSk | �}s�±.j>yAqr y�jbn~lB�'q| �~�5j_jb�M�,m]�BzÑ yÓn}lSm]npjb�AwcsMy�y r � | mawcsMmåzj>��npj>y��KjbtbwMjM�

I í

Page 41: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

G �äJMOE3LYOSJVWbP;é63^ Z 4 Ja`ÌJ è WBXJ è [�JcWUWÕXJMZ>P è W �^ O�JMé?JMT�[JcN&W ^ Z{`�JMH �^ N¾e�X\FZ ��L5Hç\�` ^ TUTYX\Fé?\° 7 ³�sc�psM��zjbn~j>�AnpsM�An | ��zsMm]���&sMm´wDjb�M��qr m&k´j>�c� � Î � Ï 7 �Ëu � oAlS�Ks¦�'q| �A�Äzjbn>�8ô�jb�M�Djbmsc� | n~n_jb�M���ÙÎ6� Ï u � �l �Bj>�~jbm �>l{zsMySµ&scn'z| y�j>kfzj>³5j>t5zj>ob�fsM��j>yÓ�DjbnÒzr �c�¦npjbkFlSm]nMqr m&k'����lSm]n�j>�M�Ào~lS�KswMj>k]n | �~��j>t�| n�sC� � G$) �à±.j>yAqr y�jbn~lv�'q| �~�5jÂ�KjbmFn�zj>m�� °�� wMj>k]n | �A�Kjbt�| n¾sMt � ±.jby~qr ySj�npµ&jbtzjb�~lSm]nM�| ySjb�Dj>oAm&j>k � lSy�ySj�n~wDjfnpsMm&�Djbm �>l{zsMySl�o~m�sck�'m&jbwMj>tbtqr k�µ�s}��zsM�A�Kjby���ã Ç �M�?�~j�±.j>mm�zsMySy�� � ãA� Ç � � � ��� � �

� �"! � � J� è Xg ê PVX^ � ° �æ� G�) �²±.j>yAqr ySj�nplf�'q| �~�5j���j>m]n�zj>m´y�j>�M�Mj>m´sc� | n~nIj>�c���özj>�AlSm]nD�| ySj>�Mj>ovwDjbk]Þn | �~��j>t��| ��°��}�Fn�³�zsc�~µ r t{sc� | oAm�sMk§� | m&�ç¶ r k'��µ�s�npjbn~o~t��| ySjb�Dj>o�ã Ç �§jbo~j�n�zj>m¦s�� � � ãA� ¢ � � � � ãA�~�wMj>k]n | � | k,y�lSm&j�zsM�~l�opsMmÂq| oAo~tbjb±Uqr �D���| j>k'�� JMéçhUJMé��'H1XJcT � ° ±.j>m]nplv�&j/&�m�zÑ �>l�z| ��z| y�sMt | mm�sMyvsM�}z| �&lSk'��µ | �c� s �f�?��j>m]npl�� zjb�~l�mFnM�| ySjb�Dj>o�wDjbk]Þn | �~��j>t��| sMkFk | �«zj>o �>o~sMk«sMkFk | �º³}zsM�~µ r t>sM� | o>��µ�s���zsM�A�Kjby��Kã Ç �§³ | m]np��scm�n~j>yжAjbocqr y

� � � ãA� Ê ¨�� � � ãA� ¢ � � � � ãA�A� ª � � � � � ãA�A��� � £ � � � � �

� JMéçhUJMé��'H1XJcT � i1jbk?l�m]npocqr k�s �;�?m&sMk¾sÌ��� � ��.�j/&�m�zÑ �>l�z| oAt>jb�~lSm]n~l f�x� ��) cËzscn~³�sM�psc��zj�npj>�Aj>t5zj>o?zj�n{�°��Kjbm&m]�?lS�5j>m s�;�?��j>m]npl�� zjb�~l�mFnM�| ySjb�Dj>o�wMj>k]n | �A�Kjbt�| ³}zsM�~µ r t>sM� | o>��sck?k | �§s f�É��j>m]n�zjbm wMjbn~nf�����) c´wMj>k]n | �~��j>t�| lSo}³�zsc�~µ r t{sc� | ob�� ����� XG�OQO½Xg W�X\FT � + -0/�1�-�2 -�T �ÙÎ � Ï u � K- ; � Î%� Ï u � K- D = 2 P C Me-�/ - ; 7'-�T P C'D > - N C T 3 ��� G�) �Y -�M�AR Me- P = /BAC'DFE - > -�2 P K-�2 G�� 3 � K- ; � � K3 D X R N 3 > C ; 7'-�T P C'D > - N C T � 3:T T C'D 3 N� � ãA���¿¨ � � ãA� ¢ � � ãA� ª � ã Ç ��� Y C'D > R M�K3 7Q3�M K- D�P -�M > - N - P0P �¾Î2� Ï u Y AR:/ /�7 K-�2 1 K3�M?M 3�2 5WKC G 3^ [JVWbN?JMHV`ÖXJ è � � � 3 3 � Y -�M�AR�M�- P = /BAC'DFE - > -�2 P K-�2 7'- P0P � K- D = 2 P C Me-�/ - ; 7H-�T P C'D > - N C � K3 D X R N 3 > C ; �3:T T C'D 3 ·���· 7Q3�M KC ;,Y AR�/ /�7JK-�2 1<T C 2 ; P 3�2 ; G

ô�j>�M�Mj>m�� Î$� Ï u � j>�c� zj>�AlSm]nD�| ySj>�Mj>oÌwMj>k]n | �~��j>t�| s�� ��j>m]n�zjbm��ài�jbnpoAt�| y�j>�DjboÀã Ç �jbo~j�n�zj>m¿s�� � ãA�«wDj>k]n | ��j>�M��zj>��npjbyS�Ò�r j>mx±.j ¶Ajbt>µ&j�nD�| kFl�s � � ����� � n�zjb�< ¡ G � � � ãA�~� ¢{£{£{£b¢ <���G � � � ãA�~��}zsMt>l�oAwDjbk]n | �~sMl�wcsMy�s

� � ãA���Ö¬ �­¯® ¡� ­ � ãA� � < ­ G) � � ãA�± | �~��zsM��sMm�����j>m�&j>y ¶cqr kaµ | tbt�zs��}�?µ&jbt_s�±.jbm]npl�j>�c�Dj>mySjbnAnpjby���j>�Mµ�scn'zsc� | t | n~n ­ Î2� Ï u��? � ØM¢{£{£{£�¢@� �I±Uqr �D�Mw�zjbmF�Mj>kcjbn{��sM�Kjby��Mj>kF�M�| yÄkBq| m&m]���r �5j>y�zsVnpm&lä��µ | �M�8�&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| sMk'�Õ°��y�j>kfzj>³5j>t5zj>o��&j>�Al�w#zsMy�n.¶&zsM�ps�sÅã Ç �§µ&j>yÓ�Djbm�±.jbm&m�zsMy�y

� � � ãA���x¬ �­ ® ¡� �­ � ãA� � < ­ G) � � ãA��× ¬ �­¯® ¡ ¬ �C ® ¡� ­ � ãA� � 6�C � ãA� � < �­ � C G�) � � ãA� £° � � � � �Kzj>o � � � � �#jb�M�Djbm&ySj�npj>k�sMySsM³?¶&zsMm,l��]sMt | ySµ�sVn'z| sMt_sMy{zsM�&�&lYk?l ¶Aj>y�j>m]n�zjbo>�� ��� � XG�OQO Xg W�X\FT � � � K- D = 2 P C Me-�/ - ; 7'-�T P C'D > - N C�� K3 D X R N 3 > C ; 3�T6T C'D K- ; ��; 3�T�3�T6T C'D � X�3 E K3 D > -�M(1ã Ç � X -�M(1�-�2 P -�M ��- ; AR M?2 -�TO3

�� ãA��×

�� ­¯® ¡��C ® ¡ �

­ C � � � ãA�A�� ­ � ãA� � �C � ãA���»í � � � � �

� $�� ØD¢>£{£{£�¢@� � AC ;0; N - Y AR�/ / K- ; -�T G

I Ø

Page 42: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

i�j>kFlSm]npocqr m&k,j>�c� ��Î � Ï 7 �åu � o~lS�Ks��'q| �A�#zjbn�zj>o�s���� G() �Ö±.j>yAqr y�jbn~lY�'q| �A�Äzjbn>�#.�jb�M��qr m&kjb�M�,ã " Ç �§³�sM�~sM��zj�npjb�?zj>��n�zjbkMjbn«zj>oºs � � ����� � � zj>�AlSm]nD�| n�zj>�A�Bjbm�w#zsMySsMo~oAt r mk,k?l�jb�M� ¤ zjb�~lSm]nM�| wDjbkFn | �An>�� �"! � * XJVWbJMO � � � /BAC'DFE - > -�2 P K-�2 �.C 2 P C ; 3�2<-0/�1 C M(1�3�2 � � K3 D X R N 3 > C ; K- D = 2 P C Me-�/ - ; 7'-�T P C'D > - N C7Q3�2 � 3 > -�M#1 D - Y -�2�2[K3�M?M � � ã " ��� ¤ GG é�J ^ RYJcWbPSN'L5TIé 3^ Z 4 XJcNæXJMZ�WbJMOQ`ÌJcH1XJMTUJ

°*7˳�sM�psc��zj�npj>��ndsM��n | ��zsMm]�?��sMm§n~j>kFlSm]npocqr m&k§jb�M� �ÖÎ � Ï 7 �©u � oAlS�Ks��'q| �~�Äzjbn{�� �,+ � � J � è Xg ê PVX^ � ° �Ò��G�) ���'q| �A�#zjbn�sct � µ&lS³5j>��±.j>yAqr y�jbn �Dj | �&j�npl�k r o��'q| �A�#zj ¶Fzjbm&j>kK� | m�V¶ r k'�&µ�s� �5Î2� Ï u � | y��DsMm zj>�~l�m]nD�| y�j>�Djbo�wDj>k]n | �~��j>t��| s'����j>m]n�zj>m��5sM��j>yÓ�a³�zsc�~µ r t{sc� | ob�� JMéçhUJMé��'H1XJcT � ° � � Ø � z°�y�y zÑ n'zsMoA��z| y5kYq| wDjbn~kMjbt>lSk'�'µ | �c�§µ�s�s;��� G ) �»±.j>yAqr ySjbn~lB��q| �~�5j��Dj | �&j�npl�k r o>�sck?k | ��s   � · � �3·�o~jb�Bjbo~o?zj>�M±dqr �D�cw�zjbm]�¦k | m&oAnpsMm&ob�

° k�zjbn�ySjb� r n'z| �&�&l��&j/&�m�zÑ �>l�z| n�&��M�DjbySjb���Bj�w�zj�wDj�j�w?lS�&jbm&o>�'µ | �c�§lS�DsMt�sMt_sMy�zsc�&�&lYkFl ¶Aj>ySjbm]n�zj>ob�� �"! � XG�OQO Xg W�X\?T � � �Ò��G�) � Y -�M�AR�M�- P = /BAC'DFE -&3:T T C'D K- ; ��; 3:TO3�T6T C'D / - C 5�- P = T R ; � X�3 E K3 D > -�M(1 ã Ç �> -�M?M�- P0P 3 � � � � ãA� ¢ � � � � ãA�A� 7'-�T P C'DQC T M = 2 - K3 D =?; 3�2 AC ;0; N - Y AR�/ / C T G� JMéçhUJMé��'H1XJcT � �s�s �¾±.j>yAqr ySjbn~lF�'q| �~�5j#oAj>�5j>oAo?zj>�c±Uqr �D�Mw�zj>m]�Dj¦zsMy�yQsMm&�}z| zj>ofs �³�zsMy��#zsU¶~s�j>�M�Ij>�M�Mj>m&jbo~�Ajjbo~l�k5�'sck?k | ��s'� � �&� �Ìj>�c�Dj>my~�| o?zj>�akBq| wMjbn~kMj>t�n�zjb�Bjbm�s'��j>�M�§�Dj | �&jbn~lSk r o��'q| �A�Bjc�

° � i � � �#jb�M�Djbm&ySj�npj�n�zjbo�s � � ��� z°�y�y zÑ n'zsMo�n#±.j>y�µ�sMoAt>m�zsMyÓwcs�kYq| m&m]�Dj>m¦lS�]sMt | ySµ&scn'z| sctIscy�zsM�&�lBn�zj�npjby3�� �,+ �<* XJVWbJMO � � �»� G ) � /BAC'DHE -"3 N � -�Me- >h= Y -�M AR M�- P 2 -�T -�/�1 /�- C 5�- P = T R ; /BAC'DFE K- ��-@3:T T C'D K- ;��; 3�T<3:T T C'D � X 3 3 � T C C'D 5 = 2[K3 P 3 � Y AR�/ /�7 K-�2 1�- = D - P - P ; N C M�-0/ - ; ã Ç � - ; - P K-�2 Y -�2�2)K3�MVM?2�3�T 3

� � � ãA��× �� ­¯® ¡

��C ® ¡ �

­ C � � � ãA�~� � �­ � ãA� � �C � ãA���»í � � � Ø í]�

� $�� ØD¢>£{£{£�¢@� � AC ;0; N - Y AR�/ / K- ; -�T G° � � ���_zjbo � � I �;iazj�npj>y�j>kF�f�| y�kBq| wMjbn~kMj>tblSk'�µ | �M��sMt � µlS³5j>�A±.jbyAqr y�jbn1oAnds �>l | m�zsM�~l r oIzjbo��Dj | �&jbn~lSk r o��q| �~�Äzjbl�wMj>y�kVsM³ �>o | yQscn~��sMm�sMt_sMy�zsc�&�&lYkFl ¶Aj>ySjbm]n�zj>o�n�npj>µjbn7¶Vqr k'�

� ��� � * XJVW�JDO � � 7 � 3 D 3 > K- P - DFP 3 DFP C > K3�2 1 E 3�2 M�-0/�1�-�2�3 5 C P0P -�/�1 �åÎ � Ï 7 �æu � D -�/�R M K3 D =?;/�AC'DHE -'G � ��� G() � Y -�M AR M�- P = /�AC'DHE -"-0/�1 ; P 3 ��= C 2[K3 D = R ; /BAC'DFE K- ��- 3 N � Y -�M�AR Me- P 2 -�T 3�T6T C'D K- ; ��; 3�T3:T T C'D � X�3 3 � K� 7HX C ;0; N 2�3:T > -0/ Y -�Me-�M�C K3 P�� 3 D 3 > K- P - D - N K- ; -W-�/�1 /�- C 5�- P = T R ; /BAC'DFE -HG

I �

Page 43: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

G �äJMOE3LYOSJVWbP;é63^ Z 4 JKé�J ^ RYJcWbPSN'L5TIé 3^ Z 4 3LBOSJcW�Jô�j>�M�Mj>m8sM� | nAn�sct � ±.j>yAqr ySjbn~j>m�j>�c� ��� G�)��ÖÎ:� Ï u � o�nds �bl | m}zsM�~l r o��'q| �~�5jM��i�j>kFlSm]npocqr ks+�;�?µ | t�ndsM��n | t�z| � � ¡

��� �� � � npsMm&�Djbm �>l{zsMySl�o�j>�M�?o?zjb�MwDjbk]n | �A�Kjbt�| n>�»°�t¦j>��&lS�Dl�jb�~j>���zjbmF�Mj>kscyQsM³?¶&zscmakBq| m&m]�DjbmÌ�5j>y{zscnpµ&scn'z| �Fµ | �M�Ks � jb�M��³�zsM�~µ r t>sM� | o#wDjbk]n | �A�Kjbt�| s�� ��j>m]n�zjbm�����t�sIn�zj>m]�zjbo�s���� ØDØ � .�j/&�m�zÑ �>l�z| l�m&� | k | yж&zsckasÅ�Dj | �&j�nplSk r o��'q| �~�;qr y�jbn#± | �]sMyS��zsMm�sckK�5jbwMj>t>j�n�zjbo?zjbn>�

i�j>kFlSm]npocqr m&k s 7 ³�sM�psc��zj�npj>��ndsM��n | ��zsMm]�?��sMmåj>�c� � Î�� Ï 7 �öu � �Aj>� r y�zsM�AlSo§�'q| �~�Äzj�n{�.�jb�M��qr k,s;��� G) ��±.j>yAqr y�jbn~lY�'q| �A�Bj � � Ø

· � � · � �ÀnpsMm&�Djbm �>l{zsMySl�o#jb�M�?o?zj>�cwDj>k]n | �~��j>t��| ¶?zjbn>�� ��� � � J � è Xg ê PVX^ � ° �Ò� G) ��±.j>yAqr y�jbn~lY�'q| �A�Bj�ã Ç �§³ | m]np�5j>ySlY�Dj | �&jbn~lSk r o��'q| �A�fqr ySj�n�zjbmÌs��� � ãA��� Ø

 Y� ãA� · � � � ãA� Ê ¨ � � � ãA� ¢ � � � � ãA�~� ª � � � � � ãA�~��·�m&j>��m&jb�]scn{zÑ w§oAt�zsM� | n§zj>��n7¶Vqr k'�°»±.jbm]npl5�&j!&�m�zÑ �>l{z| o~tbj>�AlSm]n#wMj>µ&j�n7¶cqr kKs��Ò±.j>yAqr ySjbn~l��'q| �A�Bj�³ | m]n.¶~sMlS��scm§s��Dj | �jbnpl�k r o#�'q| �~�;qr y�jbnpj�ny�jDzÑ ��z| ��� Î � Ï ux± | yÓ�Fn | m | o�±Uqr �D�Mw�zj>m]�Fn{��º�?l�y�w#zsMm]wcsMy�z| ��µ | �M�alS�]sct�sMt_sMy�zsM��&lYkFlжAjbySjbmFn�zj>ob�

� �,+ � XG�OQO Xg W�X\?T � � � � G�) � /BACHDHE - 3 N � Y -�M�AR Me- P 2 -�T -0/�1 ; P 3 ��= C 2[K3 D = R ; /BAC'DFE K- ��-�3�T6T CHD K- ; ��; 3�T3:T T C'D � X�3 Y -�2�2[K3�M?M ��� �Öí G�Iq| �DtMzÑ npocqr m&k8jb�M� Ç �´³�sM�~sM��zjbnpjb�?zjb�An�zjbkMj�n{�©° �²��q| �~�Äzj�n��Kjb�M�| y�j>�Djbo~j>m�wDj�n�zÑ npocqr kÒ�&zs�sMt �jbySjb�Kl�µ&l�³Bjb�A±.j>yAqr ySj�n� � ?�º³ | m]np�5j>y�lvzj>�~l�m]nD�| µ&lS³5j>�Ao{zÑ k>¶&zsM�~s�}°�tazÑ �M�§mF�Mj>��n�� Î%� Ï u � kfzj>³�'q| �~�Äzj>�~jn~jbn~o~t�| ySj>�Mj>o�ã Ç �§µ&jby��Mj>m«±.j>m&m�zscySy

� � ãA��� � � ãA� Ê ¨1� � ãA� Ê � � ?� ¢ � � � � ?�A� ª � � � � � ?�A� £°���o~l��as��'q| �~�5j r �c�]sMmÅm&jb� ±.j>yÓn�zj�npySjbmfqr y?�~j>� r y�zsM�AlSob�M�&j�� � � ?����� � � ��1n~j>y ¶Aj>ocqr yä�cn~j>µ�zscn�� � � ?�vý� ���� ��� � XG�OQO Xg W�X\FT � � �Ò� G�) � Y -�M�AR�M�- P = /BAC'DHE K-�2 -�T[3 N �.C 2 P�E -�M = / - C 5�- P = T R ; /BAC'DFE AR M�- P - > -�/�-�/�1 - N = T3 � 7H- P AR�M�- P = /BAC'DFE - � E -�M = /BAC'DFE AR M�- P K-H7'-�M G

I �

Page 44: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

�B)�*O� u ��� Ú =?+ 0 =?+U@� +.= � =?9Ò4'=F2568= � G: C-{<>-°�t_sMy�kVsMyS�KsMt�zsco | kÅo~tbj>��³ | mFn.¶&zsM��z| y�l��Dj>m,± | m]n | opsMk§sMt�u � �F�BjbySlYoAlS�Ks�±.j>yAqr y�jbn~j>k'� UMj>y�j>m±.j½¶Aj>tbjbnp�5j>m�jbt>j>kÌ�?zjbo~t>y�jbn~j>o�npsMm r yS��zsMm]� | t�zsMo�zsc�psakcj>�cqr y1o | �b��i�j>�A��zjbo~t>j�npjbo~j>m�±.j>y�± | �{¶ r k,µ�sMoAt>m�zsMy�m&ls i ��zjbo � �f±.j ¶Aj>tbjbn~j>kF�Bjbmxzj>��npj>y��Kjbt>j�n~n}± | �]scyS�KsMkVscnKzjbo�s�y�jbwDjbt>j�npjbnAn�q| o~o~tbjb±dqr �D�1zj>o~jbkMj�n{�°�tfu � �?�&sMm�ySj>�c�Dj>m_sM� | n~n�sct � oAlS�Ks#j>y�j>��lc±.j>yAqr ySj�n{��sM��j>yÓ�Dj�n�s�7 �©u � ³�sM�~sM��zj�npjb�Andsc�An | �©zsMm]� | mzjb�AnpjbyS��j>tbjbn~n G�Î 7 Ï u � �&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| y�j>k�zj>³5j>t5zj>oÄzÑ �fy�jM��° ³�sc�psM��zjbn~j>�~jbkFm#zj>y?�as�¶A��sMy�kcscyS�KsMt>m&l± | ��¶ r k�sct ? � ?�¡U¢a  � ? ��¶Ajby½q| y�zj>oAj>kMj�n{� °�t GÀy�j>k�zj>³5j>t5zj>o§k |�| �A�&lSm}zscndsç�]±Uqr �D�Mw�zj>m]�Djbl�ySj>�c�Dj>mj>k� ¢ �5¢�� Î�7 Ï u�����o~tbj>�~l�m]nº±.jbm&m�zsMy�y

G ��?�¢p  �f��� �@?�¢~  � = ¡ × �Y��?�¢p  � = ��× ����?�¢p  � = ��}zsM�~��j>yÓ� ��?�¢p  � Ç 7 j>oAjbn�zjbm��#ÜÌlSm]n�lSoA�Kjb�~j�npj>ob��j>�M� / � G ��?�¢p  � �~��?�¢p  � Ç 7,�#³ | m]np��scm¦s�{�?�&lS��j>mt>l�z| o � ù � zjb�~lSm]nM�| n�zjb�~m&jbkÂjb�M��npjb�~��zj>o~tbjbn~j>o���zsMtblSo&zscn�sc�V¶&zsMk�s < ¡ G ��?�¢p  � ¢ < � G ��?�¢p  ��wDjbk]Þn | � | k5�°�t� ±.j>yAqr ySjbn�G�³�sM�~sM��zjbnpjb�~jbt5zj>o?zj>µ&jbtInpsM�An | t�z| m | �A�©zscySlSoÄj>�M�?o?zj>�MwMj>k]n | �A�Kjbt�| n�sMt

� � Ø·/< ¡ G�á< � G'· < ¡ G�á< � G

kFlÓ±.j ¶Aj>t5zj>oAo~j>y�ySjbµ&jbn�y�jDzÑ �~m&lä�f°�t�j>y�oM�| zj>o���zsMo | �&l�kK±U�| ��j>m&m]�?lSo?zj>�DjbkMj�n�s� ¡½¡ �¿¨ < ¡ G ¢ < ¡ G ª ¢ � ¡ �#�¿¨ < ¡ G ¢ < �AG ª ¢ � � �#�²¨ < �AG ¢ < �AG ª ¢

�{¡½¡ �²¨ < ¡b� < ¡ Gc� ¢ � ª ¢ �>¡ �}� ¨ < ¡>� < � GM� ¢ � ª ¢ � � �}�¿¨ < � � < � Gc� ¢ � ª± | �~� r y{zsMkaµ&scn'zsM� | t>t�zsMka��j>��

.�j>�c��qr k zj>o~tb�~jc�5µ | �M�,s1J Î 7 Ï �I�YÈ � u � �åySjbk�zjb³Bjbt5zjbo~�~j�±.j>mm�zsMySy�sÈFâ>ãDJÙ� � ¡½¡ � � � Ê � � ¡ �U� � � ·6< ¡ G�á< � G�· �

q| o~o~tbjb±dqr �D�1zj>o>�G � JDL5T è PSJMZ 2&WBXJVWbJMO;é?J ^ `ÌJcW�Z>PS\?P�[�X\?O�W ^ HV\]W�\

° � � Ø �;iazj�npjby'sv±.jbyAqr y�jbnpl?�'q| �A�Bj}��q| �~�;qr y�jbn�zj>m&jbkÂzj>o�sct,zjb�~l�mFnM�| lS�&zsMm]�?µ | t#npsM�An | t�z| m | �A�©zscyS�'q| �A�fqr ySj�nAÞmj>k«s�kVsM³ �bo | ySscn'zsVn_zÑ �.¶~s�y�jM�° 7 ³�sc�psM��zjbn~j>�AnpsM�An | ��zsMm]���&sMm wMj>�M��qr m&k�jb�M� �¦Î � Ï 7 �~jb� r y{zsM�~l�o���q| �~�Äzj�n zjboasctxzsMyÓndsMySs��j>�Mµ�scn'zsc� | t | n~n��Ö� G ) � ±.jbyAqr y�jbnpl1�'q| �A�#zjbn>��i1jb�M��qr k¦±.j>yä��µ | �M�¾s � � ãA� � ã Ç ����³ | mFn~�BjbySl � � � ãA�l���zsMm]�ºzr m | �~��zsMyS��q| �~�;qr y�jbn~�~ja±.jbm&m�zsMy�y� � ����� � � � � ãA�A�Òý� í�åÜ�lSm]nal�o~��j>�Ajbnpjbo>�}jbtÌjbo~jbn~�Bjbm � � ãA�Àì ín~j>y ¶Aj>ocqr yä�;° � � Ø �;iazj�npjbySm&jbk��Kjb�M±.j>y�j>y�sMtºsMy{zsM�&�&lk?l ¶Aj>y�j>m]n�zjbo>��sM�Kjby����Dj | ��jbn~�~lSsMl�npsM�AnpsMyS�Kscnfµ | �~� | tM�

� -�T = 2 P ; AR�T 3 N P 3 N-� /BAC > E Y -�M�AR M�- P - P � 3 > -�M#1 2 -�T ; R�/63 D 3� Øñ � � ����� � � � � ãA�A�çñ � � -�2 P�D R > 3 � -85 = /3 � ��G � � � ãA�~��× Ø

� � ����� � � � � ãA�A� � � � � ãA�~� � C 2 P G � � � 2�3�T"3 ã � E -�M = ; =V> R�M�KC ; K� T �Q3 K3:M P 3:M 3 N � ; N Y K- D K3 E KC MT =?> - P ; N - P P TJAC'D > -�/ -0/�1�- N = T 3 � Y -�M�AR M�- P = /BAC'DHE - ã X -�M#1 -�2L7'- P P ;0=V> R M�KC TJAC'D K-'7H-�MSGI2I

Page 45: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

G �äJMOE3LYOSJVWbP;e ^è W ^ N ^ TUH�W�X\?O � ^ H;X\FTU\° � � � � ��q| oAo~t>j�±Uqr �D�1zjbo�sMtbnv� r ndscn.¶~s�Yµ | �M�¦sMt�u � �?�BjbySl�±.j>yAqr ySjbn~j>kFmÄzjby;s§oAt | �At{scn~�'q| �;qr y�jbnÌzjb�An�zj>kMjl���zsMm]�BzÑ n�zsMoAw#zsMyÓn'z| zscn~³�sM�~sM��zj�npjb�~j>t5zj>o�j>oAjbn�zj>mÀoAj>�Ùw#zsMyÓn�j>y~�| ¶Aj>ySj�n{�°xn | w#zsM�&�lQsMkF��sMmÌsMtvu � �?�5j>ySlB±.j>yAqr y�jbn#o~t | �~t{sVnp�'q| �A�fqr ySj�n�zj�n N_s r o~o �F�'q| �~�;qr ySjbn~m&j>kKlSo}m&j�wDjbt>t�qr k'�

� ����� � J� è Xg ê PVX^ � i1jbk?l�m]npocqr k�sMt � jbySj>��l}±.j>yAqr ySjbnKj>�c� / ³ | m]n.¶&zscn zjbo,sMt | nAndsMml � ù N_s r o~o ���q| �~�;qr y�jbn~jbn{�;° / ±.j>yAqr ySj�npl5³ | m]n | n � ù ì í�jbo~jbn�zj>m�j>y�ySlS³�nplSk r oij | mFn~m�sMk'� � ù ò�íÅjbo~j�n�zj>m�µ&l�³Bjb�~� | Þy�lSk r o}³ | m]npm�sck5� � ù �»íÅjbo~j�n�zj>m̳Bjb�&lS�ų�sc�psM� | ySl�k r o}³ | mFn~m�sMk§� | m&�ç¶ r k'��º�?l�y�w#zsMm]wcsMy�z| �>µ | �M��sMt G wMj>k]n | �A±Uqr �D�Mw�zjbm&m]�Dj>yF³�sc�psM��zjbn~j>�~jbt>j�n~n � ±.j>yAqr y�jbnB³ | mFn.¶~sMlS�&sMm�s-N_s r o~o ���q| �~�;qr y�jbn~jbn}y�jDzÑ �&z| � Î 7 Ï u ±Uqr �D�Mw�zj>m]�?�~jv±.j>mm�zsMySy

� � �>¡½¡�� � � Ê ���{¡ ��� �� ¡½¡ � � � Ê � � ¡ �U� � £ � � � Ø �

.�j>�c��qr k«sctbn�sMt * Î%7 Ï �I�YÈ � u � ��y�j>kfzj>³5j>t5zj>o�n{��sM��j>yÓ�?�~j * ��?�¢p  � � B J ��?�¢p  � D �¡� � ��?�¢p  �

n~j>y ¶Aj>ocqr yBnpjbn~o~t��| ySjb�Dj>o ��?�¢p  � Ç 7þ�KjbySy�jbn~n>��ÜÌl�mFn�l�o~��j>�Ajbnpjbo>�'s / � G ��?�¢p  �å³ | m]np�5j>y�l( ù Î � ù � Ï � ù � �Õj>l�m&�]sM��npj>mF�?ySjbk�zjb³Bjbt5zjboAnvsct * ��?�¢p  ����zscnp�AlÓÛÂzÑ �.¶~sÅySj_s9< ¡ G ��?�¢p  � ¢ < � G ��?�¢p  ��}zsMt>l�oAwDjbk]n | � | kF�psKmÄzj>tbwMjM� �Kq| m&m]�Dj>mÌj>y�ySj>m;�| �~l�t>µ&j�nD�| ��µ | �M�§±.jbm&m�zsMy�y�sMt

* � ØÈFâ>ãPJ

�� � � � �>¡½¡ Ê � ¡ � �{¡ � � � � �>¡ � Ê � ¡ � � � �� ¡½¡ �>¡ � Ê � ¡ � �{¡½¡ � ¡½¡�� � � Ê � ¡ � �>¡ �

� � � � �D�

q| o~o~tbjb±dqr �D�1zj>o>�º���&�f�| y1��zsM��kBq| wDj�npkMjbt>l�k5�Yµ | �M��sMt�G>�?�~jby�³�sM�psc��zj�npj>�Aj>tbjbn~n � ±.j>yAqr ySj�n�³ | m]n.¶~sMlS��scm¦skBq| t5zj>³&��q| �~�;qr y�jbn~jbnºySjMzÑ ��z| ü Î 7 Ï u ±Uqr �D�Mw�zj>m]�Fnºsü � � ¡½¡�� � � Ê � � ¡ � �>¡ �;× � � � �{¡½¡

� � � ¡½¡ � � � Ê � � ¡ ��� � � £ � � � � �± | �~� r y{zscwcsMyVySj>µjbn�kFl�±.j½¶Aj>t>ml]s#±U�| ��j>m&m]�?lSo?zj>�Djbk?�;�| y3�1���Ky�zj>kMjbt>t�qr m&k���zs���µ | �M��lS��zscmF�BzÑ n'zsMo�w�zscy�n'z| zscnp³�sc�psVÞ�Òzjbn~j>�Aj>t5zjbo�jbo~j�n�zj>mÀsÅkBq| t5zj>³&��q| �~�;qr y�jbn�jby~�| ¶AjbySjbn#w�zscy�n{�� ��� � � J� è Xg ê PVX^ � °�t � o~l��as�±.j>yAqr y�jbn~jbn��Kl�m&lS��zsMy�±.jbyAqr yÓjbn~m&j>k�m&jbwMj>tbtqr k'��µ�s,s§kBq| t5zjb³&�'q| �A�fqr ySj�nvsMt� q| o~oAt>jbo�³ | m]n7¶&zsM�&sMm«j>y�n?�r m&lSk'�G � �^ PSZ�X\ è � ^ N8`�JMé%�B\]W�X\FZ ^ H;X\?TU\

.�j>�c��qr k§sMt�G�Î 7 Ï u � wDj>k]n | �A±dqr �D�cw�zjbm&m]�Djby1y�jDzÑ ��n � j>ySjb�KlB±.j>yAqr y�jbn#j>�M� / ��G � ¡�¢ �U��A� ¡�¢ �U� Ç 7§��³ | m]n7¶&zscn>�§° / �?�5j>y�l � ¡�¢ � ��±U�| �'q| �~�;qr y�jbnpjbkMj�n �5zj>y�o~t>jb�F�r zr �c���Kjb�Dµ�scn�zsM� | tbm&l3��µ | �M�jby~�| ���kFl�o~t�zsc�¦zÑ n7¶ r kIs � ù ¢ ü ù zj>�An�zj>kcj>kMj�n}s � � � Ø �Izj>o � � � � ��q| o~o~tbjb±dqr �D�1zj>o~jbkÅsMyQsc³?¶&zsMm��1��tbn kBq| wDj�nD�| jbms � ¡ � � �#� � ù zjbo � ¡ × � �#�x� ü ù j>�M�Mj>m&y�jbnpjbk§��j>� | y���zsMo~sMk�zj>m]n}kVsM³?¶ r kas�k�zjbn}±U�| �'q| �~�;qr y�jbnpj�n{�i�j>kFlSm]npocqr m&k´s / ³ | mFn~��sMmÒjb�M� ���Fn�z| yÄk�qr y q| mF��q| t�| ¤ � ¥�¡ �%< ¡ G � ¡U¢ ����× ¥ � �5< � G � ¡U¢ ���

zjb�~lSm]nM�| wDjbkFn | �An>��° � � � �D�vq| oAo~t>j�±Uqr �D�1zjboAnº±.j>ySµ&sMo~tbm�zsMy�wcs��5j>y{zscnpµ�sVn'z| ��µ | �M�,l��]sMt�sMt�scy�zsM�&�l�kFlжAj>y�j>m]n�zjbo>�� ¤ ; 3 ��K3 P 7H-�T P CHD 3�3 N ( ù Î � ù � Ï � ù � � - = 2�/ 3 DFP -�2 � M�-�T K- � - N K- ; 2 -�T)3�T6T C'D K- ; ��; 3�T@3�T6T CHD �X 3 Y -�2�2[K3�M?M

������

�.¥ ��� � Ê ¥�¡½¥ � �.¥�¡ � �� ¡½¡�� ¡�¢ �U� � ¡ � � ¡d¢ ��� � � � � ¡U¢ ����>¡½¡>� ¡�¢ �d� �>¡ � � ¡�¢ �U� � � � � ¡�¢ �U�

������

�Öí £

I �

Page 46: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

ô�j>�M�Mj>m&jbk � ¡ zjbo¾��� | y��]scm | ��n | � | m�zsMy�lSo¦jb�M�?o?zjb�MwDjbkFn | � | k s � ù � y�lSm&j�zsM�~l�o�zj>�AlSm]nD�| n�zj>�A�Bjbm����j>yÓ�Djbk?�Aj�lS�DsMt ( ù � � ¡ ��� � ¡ � ¡�¢ ( ù � � ����� � �'� � £ .�j>�c��qr mk�jb�M� ¤ Ç � ù � � ¤ ý� �5��wMj>k]n | ��n{�sc�Kjby��?m&j>k�s�� ¡ �?�M�Mj>y5�Bjbt�zsM��n�o~tDq| �1zj�n1¶Aj>y q| yжAj ����� r ySj>��n�zj�npjbySj � sMt{sct�s � � ���;i§zjbnpjbyÃ��o~t>jb�~l�mFnÄnpjbyжAj>ocqr y� ù � ¤ ��� � ¡ � | o � ��× � ��oAlSm � � £ � � � I �

i�j>�M��qr ka±.jby3�&µ | �M�§l��]sMt � ù 7 íÒzj>oºs / m&j>�þ³yQsMm�zsc�~lSo#³ | m]n.¶~sKsMt � ±.j>yAqr ySj�npm&jbk5�� �"! � � J � è Xg ê PVX^ � °�t�n�� | m&�V¶ r k5�'µ | �M�Ìs ¤ Ç � ù � � ¤ ý� �B��wDjbk]n | �vsco~t>l��K³�n | n~lSk r o�lS�&zsMm]�?n�sM���j>��sMtÀzjb�~lSm]nM�| n�zjb�~�5j>m���µ�s�±.jbm&m�zsMy�yH� ù � ¤ � ��í £

�fw?lS�&jbm&o>�cµ | �M�_s�µ&lS³5j>�A� | ySlSk r o�³ | m]n�zj>�AlSm]nD�| n~j>�?zj>�5j>m�kfzj�nfsMoAt>l��K³n | npl�k r o�lS�&zsMm]��wcsMm��Ds�³�sM�~sM� | Þy�lSk r o � �&j�m&j>� ³&ySsMm�zsM�AlSop�ij | mFn,zjb�~lSm]nM�| npjb�?zj>�5j>m«³5j>�&l��Åj>�M�M�� �,+ � � J� è Xg ê PVX^ � ° 7 �Öu � ³�sM�psc��zj�npj>��ndsM��n | ��zsMm]�?��sMmÌySj>�c�Dj>mÀsM�wcsaj>�c� ��Î>� Ï 7 �Aj>� r y{zsM�~l�o��q| �~�5jM�Ì° � � G�) � ��q| �~�Äzj�n�sMt � ±.j>yAqr ySj�n�sco~t>l��K³�n | nps�w | m�sMy{zsMm�sMk¾mjbwDjbt>t�qr k5��µ�s�npjbn~o~t��| ySjb�Dj>oã Ç �,µ&j>yÓ�Djbm � � ����� � � � � ãA�~���Öí n~j>y ¶Aj>ocqr yä�G � LBe�P è 2 �MXJMOSJ,P è R�PSN�X\]W�Z{P�4

�Iq| �DtMzÑ npocqr k«sMt'G§Î>7 Ï u � y�j>kfzj>³5j>t5zj>oAo~j>y�³&sM�psM�Òzjbn~j>�Aj>t>j�n~n� jbySj>��l�±.j>yAqr y�jbn�j>�M� / ³ | m]n7¶&zscn>�i�j>�c��qr k±.j>yä��µ | �M�8s / m&jb� ³&ySsMm�zsM�AlSoI³ | m]n>�fwMsc�M�?lSo ( ù ý� ����° ���?�lS��j>m&tbl�z| o � ù � y�lSm&j�zsM�~l�ozjb�~lSm]nM�| npjb�~jbn}n~j>kFlSm]n~oMqr k§� | oAn�jb�M�ºzr n~npsMy�j>�c�,j r kFy�lS�&jbo~t>l�o>zÑ kFm�sMkalSo>�°���j>mmF�?l��BjbmawcsMyQsM��j>yÓ� ¤ Ç � ù � � ¤ ý� �B�fwDjbk]n | �A�ps�l��]sMt� ù � ¤ ��ý��í��sMkFk | �#sMySkVsMy��asMtbt r ksct� ¤ ��� Ø

ñ � ù � ¤ �çñ ¶Aj>y q| y�zjboAn{�H.�j>�c��qr k¾zj>oAt>�AjM�Fµ | �M��sMt� ¤ ���Kjb�Dj>�M�Mj>tblSkIs ¤ lS�&zsMm]�?µ | t#npsM�An | t�z|

m | �~��zsMyS��jbn~o~tbjbn_zÑ w / �?�5j>y�l�o~l�� r y{z| kBq| �?zjbm&j>k,sÅo r �]sM��zsVwMscy3��°g��Fmxzscn~µ�sMySsM��z| ¤ lS�&zsMm]�ºzr j>�c�Dj>mj>o~jbm¶Ajby½q| y ¶cqr kÕkFlºsMt�nas�k�zjbnK³ | mFn | n{�#sM��j>y��Mj>k ���Fn�z| y�¹� ¤ �In'zscw | ySo&zsM�D�~s�wcsMm&m&sMk'��°�tÕq| oAo~tbj>oO���?mzsVnpµ�sMySsM��z| jb�M�Djbm&j>oAj>mÂw�zjb�Dj>tbt�qr kÌj>yfjbtbn_s§kFlжAjby½q| y�zj>oAn>� � °�tÅsMo~tblS��³n | nplSk r o�lS�&zsMm]�?nvsM��z| j>�c�Dj>mj>o~jbmmj>� ¶Aj>y½q| y~qr m&k´kFlº³ | mFn | kVscn{� � °�t´zÑ �M��m]�Dj>��n��>jbm]np��zscySo~tblS���Kj�np�AlSk r oKsMyQsMkFt>scn | nas � ±.jbyAqr y�jbn /³ | m]np�5j>y�l:. r ³&l�m?�?l�m&�&lSkÄzscn~�~l¯Û}zscm�sMkÅm&jbwMj>tbtqr k'�

° � ù � o>zÑ k | m�wDjb�M��qr kåsMtbnÌs .�j>o �>sM�An~j>o½�F±�zj>y�jÕk |�| �A�&lSm}zscndsç�F�~j>m�&o~tbj>�An>�vsMµ | y�s kcj>t>�;�| ³ | mFn����sMt zj>yÓwDjbkFn | � | k ³5j>�lS� | y��DsMm � ¡�¢ ��� | �An | � | m�zsMy�lSo�jb�M�?o?zj>�cwDj>k]n | � | k'��sM��j>y��Mj>kÒo~sç¶&zscn~wMj>k]n | �psMlsct ( ù �Õj>l�m&�]sM��npjbm?�?ySjbk�zj>³5j>t5zjbo~m&jbk'��° � � � I ��q| oAo~t>j�±Uqr �D�1zjbo~�f�| y�kBq| mmF�Mj>mÕySjbwMj>tbjbnpµjbnD�| ��µ | �M�Àjbt>jbmk |�| �~�&l�m�zscnds��?�~jbm&�&o~tbj>�A�BjbmÀs�. r ³&l�m?�F±{zj>y�j�lSm�&lSkÄzscn~�~lÓÛ | n}s� ¡ � � × � � �

� ��� Øjb�M�Djbm&ySj�n¦zÑ �.¶~sÀy�jM����o~tbj>�~l�m]naµ�s�s / ±.j>yAqr ySjbn~lij | m]n�j>y�ySlS³�nplSk r ob��sMkFk | �Ks+. r ³&l�m?�?lSm�&lSkÄzscn~�~lÓÛÕjb�M�jbySy�lS³&oAt>lSob�Ä°��Kjbm&m]�?lS�5j>mÀs / ±.j>yAqr ySjbn~l�³ | m]n�µ&l�³Bjb�~� | ySl�k r o>��sck?k | ��s9. r ³&l�m?�?l�m&�&lSkÄzscn~�~l¯Û§kfzjbn | yÓ�]sMmµlS³5j>�~� | ySs r m&l�z| ¶~s���sM�Kjby��Mj>k¾sMoAt>l��K³n | n'zsclfkBq| t]q| oAj>k'���s�³5j>�&l��¦s / ±.j>yAqr ySj�npl;³ | m]nI³�sM�psc� | y�lSk r ob�sck?k | � . r ³&lSmF�?lSm&�lSkÄzscnp�AlÓÛ�kfzjbn�³�zsM�Aµ r t{sM� | o}jb�M�Djbm&j>o r m&l�z| ¶~s�G * �BJ ^ ZbJD`�\,JMé?Z>JMé�PQLB`

i�j>kFlSm]npocqr m&k_j>�M��u � �?�BjbySl � j>y�j>��l]±.j>yAqr y�jbn~jbn{�csM��j>yÓ�Djbn;sMt G�Î 7 �©u � Ï u � wDjbk]n | ��±Uqr �D�Mw�zj>m]�zÑ �YySjc� N_s r o~o�sMtÅzsMyÓndsMySs ±.jby�±.j>�j>t>j�n~n1sMy{zsM�&�&lVn�zjbn~j>yÓn{�çs�±.j>yAqr ySjbn~j>y���zjbySjbn5jb�M�?lSkºySj>�c± | m]n | o~sM�&��j>�Aj>�&��zj>m]��zj�Þmj>k§ndsc�An | n~nds��&j>t5zj>�An�s�yQsVnplSm � X - C'D - > 3O-0/ D -0/ = R > j>y�m&jbwMj>t5zjbo~oAj>y;l�ySy�jbn~n~jM�I i

Page 47: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

� ����� * XJcW�JDO � � � Î 7 Ï u ; N C'D N 3 P /BAC'DHE AR�M�- P � Y AR�/ /�7JK-�2 1 P 3 N G � 3 D 3 > K- P - D - N K- ; X - N P 3 DFP C N KC� ­ C ��?~¢LE � ØD¢ �D� -�M ; C Y C > -�2�2 1 =V; K-�/ -�T�-�/�1 K- DFP -�M > R -�2 > -�/ X�3 P K3 D C N�N K3�T G��PSH ^è ��Xg W�X\?T �Ü�lSm]nºlSo~��j>�Ajbn~j>o>�'s �¸±Uqr �D�Mw�zj>m]�?�~jv±.j>mm�zsMySy�s � ����� ���

��� ���q| o~oAt>jb±dqr �M�;zj>ob� °<��� ØD¢�E �Ö� ¢

? �ö� ¢ $�� Ø lSm&�&jUÛ�j>k©scySkVsMyS�KsMt�zsMo�zscwcsMy�s � � � Ø I ��j>�c�Dj>mySjbn~�f�| y�sMt¡ � � ¡ � � � � �>¡½¡ Ê � �>¡ ��� �

jb�M�Djbm&y~�| o?zj>�asM��z| �lSk'����m&m&j>k,kBq| wMjbn~kMj>t�n�zjb�Bjbm� �

¡ � � ¡

� ¡½¡ � � � Ê � � ¡ ��� � � � � � �n~j>y ¶Aj>ocqr yä� ° � � � � ��zj>o � � � Ø � �Àq| o~oAt>j�±Uqr �D�1zj>oAj>k scyQsM³?¶&zscm sMt

¡ � � ¡ Î�7 Ï u ±Uqr �D�Mw�zj>m]��sMt¾j>y�oM�|±U�| �Kjbm&m]��l�o?zjb�Dj>kÂlSoA�Kjb�~jbn�zj>�5j>mÒ��zsM�_�Kjb�Dµ�scn�zsM� | tbµ�scn�z| �̼½yÓ�Â�©z| � | mÒs � � � � �_jb�M�Djbm&ySj�n�lS�]sct | y ¶~sÌsn�zjbn~j>y�n>�

G * �BJ ^ ZbJD`�\,JMé?Z>JMé�PQLB` NH3^ [JVWbN?JMHV`ÖXJ è ��J°ÒkBq| wDj�npkcj>t�| n�zj�npj>yDsMtfu � �?�BjbySlV±.j>yAqr y�jbn~j>kºkYq| t]q| nAnplDl�t | ��jbn~�~lSk r oBySjbk�zjb³Bjbt5zjbo~j>kF�Aj�sM��j>�c��o~t�qr kFo?zj>�Mj>o±.jby�n�zjbnpjby�n>� � °�t�l�t | ��jbn~�~lSk r o#ySj>kfzjb³Bjbt5zj>o�n�s i �f±.j½¶Aj>t>j�n�w�zjb�1zj>mÌ�©zsc�§zj>�An~j>y��KjbtbnMqr k'� �

� ��� � * XJcW�JDO � � > -�2�2 1 = E -�2 3 N«u � � E -�M = � K- ; f� -�Me- >h=,Y -�M�AR�M�- P -�T T AC N AC P P > -�/ 3 56X�3 P KC -0/�1= N C > - P�D = T R ; M�-�T K- � - N K- ; � 3�T6T CHD 3 N -0/�1 > K3 ; 2�3�T > -�/ Y -�M�-�M P - P - P0P �.C 2 P C T E 3:2 3 Y -�M�AR Me- P -�T � 3 R ;0; �/�AC'DHE AR Me- P - = > -�/ -0/�1�- N 2 -�T G��PSH ^è ��Xg W�X\?T �ô�jb�M�Djbm�sMt�G¦Î 7 �$u � Ï u � wDjbk]n | ��±Uqr �D�Mw�zj>m]�8sMt � j>�M�?lSk¾³&sM�psM�Òzjbn~j>�Aj>o_j>y~�| zsMySy3zÑ n'zscops����t>jbm³&sM�psM�Òzjbn~j>�Aj>t5zjbo~�5j>m¦sMt� ³ | m]n7¶~sMl�m�sMk3N_s r o~o �F�'q| �~�;qr ySjbn�zjbn�sc�V¶~s���j>�Ås � Î27 Ï u ±Uqr �D�Mw�zj>m]�M�i�j>�M��qr kÒ±.j>yä�}µ | �M� � zj>o f� kBq| t]q| nAn,sM��wMsÂwMscm©j>�M� l�t | ��jbn~�~lSk r o���j>�M±.jbySj>yÓnpj�n�zj>ob�9.�jb�M��qr ksctbn�sMt f�å�]n�ySjMzÑ ��z| fG8Î 7 Ï u � wDj>k]n | �A±dqr �D�cw�zjbm]�?n>�#sc�Kjby��?�~ja±.jbm&m�zsMy�y�sMt§s¦n r ySsç¶A� | m&o&zsM�&��µ | �M�sctÕl�t | ��jbn~�~lSk r o«y�j>k�zj>³5j>t5zj>o¦n~jbnpoAt�| y�j>�Djbo �@?�¡d¢A? �U� Ç 7 j>oAjbn�zjbm sMtFG ��?�¡U¢A? �U� Ç � ³ | m]npµ | tÕsMtfG �@?�¡d¢A? �U� Ç f� ³ | m]n | n,�~jbm&�&j>y�l3� °àk | �&zsM�&��sMmåy�jDzÑ ��ndsMk sMySsM³?¶&zsMm jbt¾j>o~j�np�5j>m � ­ CÒ� f� ­ C npj>y¯Þ¶Ajbocqr yä�vÜÌlÓwDj>y1sai}µ&j | �~jb�asKj>�D�Aj>�Dl r �Ëo~t>jb�~l�mFnvsao~t | �~t>scnp�'q| �~�;qr y�jbn��]±Uqr �D�Mw�zj>m]�«kFl�±.j½¶Aj>t>µjbnD�| sct�j>y�oM�|±U�| �Kjbm&m]��l�o?zjb�Dj>kF�f�| yä�'s � Î27 Ï u ¢ f� Î 7 Ï u ±Uqr �D�Mw�zjbmF�Mj>k«sMt | m | o~sMk'�� JMéçhUJMé��'H1XJcT � ¼½o~��j>�~j�npjbo>�Yµ | �M�ÌsM��j>m&m]�?lS�5j>m¾sct � jbySj>��l�±.j>yAqr ySj�n�sct | m | o�wcsMyQsc�Kjby�� � �ì»í]�o r �]sM�Bzr ��q| ���±.jby~qr ySj�n�jb�M�,��sM�~sM�?¶&zscwcsMyä��sMkFk | ��sct � q| oAo~t>jbo�³ | m]n7¶&zsc��sMms�N�s r o~o½�?�'q| �A�fqr ySj�nazj>��n�zjbkMj Ø

� ��s,sct � jbySj>��l1±.jbyAqr y�jbn�wcsMySsM��j>y��«u � �?�5j>y�l;o>zÑ kFm�sMk¦jb�M�¾��sc�psM�?¶~s���sMkFk | ��sAN�s r o~o½�?�'q| �A�fqr ySj�nzjb�An�zjbkMjIsMt�q| oAo~t>jbo�³ | m]np��scmÌí��°�t_sMy{zsM�&�&lBkFlжAj>y�j>m]n�zjboAn�s � � �?��i§zjbnpjby�j>�M�?l�k,sMySkVsMy��asct�zsMo&zsMm�sMkÅySjbµ&jbn�n~j>kFlSm]npjbm&l3�

3^ [JVWbN?JMHV`ÖXJ è � � + -�/�1 -�2 3 N8� 3 Nºu � -HR T6M = 5 - ; N = P K- D 2 -�T<-0/�1 ; R�/ 3 D KR /BAC > E Y -�M�AR Me- P - � aì í]� G� N � ; N Y K- D 3 E K3 D > -�M#1 -�Me- >h= Y -�M AR M�- P - P T K- � - N C 5�3 D 3 E ��K3 P 7H- ; ; N AR T � 3 N P -�/�1 ; K� T E -�M = P 3 D�P C > K3�2 1 D 32 - > M�-�X - P = N C > - P�D = T R ; 3�2 M�-�T K- � - N 2 = GG10�\?LBTUT 2&é63^ Z 4 3LYOSJVW �7JDOSTUH�Xg è TUHçJMZ>P è WbPfP è W�JMé?Z�X\�O h�\

ô�j>�M�Mj>m sM� | n~nÅsMt � jbySjb�Kl ±.j>yAqr y�jbnÂzj>oÅsMmm�sMkÂj>�M� G8Î 7 Ï u � ³�sM�~sM��zj�npjb�~j>o�j>y~�| zsMySy3zÑ n'zscops�ô�jb�M�Djbm´s�îËjb�M� | yÓ�]sMm U | �A��sMm?�?�Òzj>�Aµ&jbnM�| t�zsM��nvndsM��n | ��zsMm]�¦u � �?�5j>m���sM��j>yÓ�À�?zjbo~tbµ�sMyS�KsMt>s,s 7³&sM�psM�Òzjbn~j>��ndsM��n | ��zsMm]�?m�sMk'�

I �

Page 48: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

� ��� � � J� è Xg ê PVX^ � ° N�s r o~o½�?�'q| �A�fqr ySj�npjbm&j>k�sMt G � î��Yk | �K³�sckFn�±.jby~qr ySj�np��sc�psM� | m�wMjbn~n�±.j>y�o~tMzÑ m�o~tbj>�~l�m]npll�m]npj>�M��zsMy ¶&zsMm«s� � G � î��~���]\ \�� � ��?�¢p  �� ¹ ÈFâ>ãDJ �@?�¢~  ��È ? È  

oAt�zsM� | n,zj>��n7¶Vqr k'�� JMéçhUJMé��'H1XJcT � ��q| mmF�Mj>mÅj>y�ySjbmf�| �~l�t>µ&j�nD�| �çµ | �M�Is%N�s r o~o½�?�'q| �A�fqr ySj�n�j>�M�IsM� | n~n;±.jbyAqr y�jbnp�&sM�psM� | m�wMjbn~n±.jbySoAtDzÑ m«oAt>j>�AlSm]npl�l�mFn~j>�D�&zsMyж~sIm&j>�Ù±dqr �M�Ks�±.jby~qr ySj�n�³�sc�psM��zjbn~j>�~jbt5zjbo?zj>mj>k«�Kjb�Mw#zsMyQsco~tbn�zsMo�zsVn'z| yä�

°�t G ³�sM�~sM��zjbnpjb�~jbt5zj>oAµ&j>t�ndsc�An | t�z| � Î 7 Ï u � m | �~��zsMy�lSo5j>�M�?o?zjb�MwDjbk]n | �A�Kjbt�| �Aj�s �Õj>lSm�]sM�An~j>m?�y�j>kfzj>³5j>t5zj>o��&j!&�m�zÑ �>l{z| ¶&zsM��z| y�sM��z| ��z| sMm,npjbyжAjboMqr y< ¡ �Ë� Ê F ¡½¡ � < ¡ G Ê F � ¡ � < � G ¢< � �Ë� Ê F ¡ � � < ¡ G Ê F � � � < � G £

��m&mj>k«kBq| wDj�npkMjbtbn�zj>�5j>mÀnpj�npoAt�| y�j>�Djbo �@?�¢p  � Ç 7 µ&jby��Mj>m«±.j>m&m�zscySy�s< ¡ � ��?�¢p  �#á < � � ��?�¢p  �f� � �@?�¢~  � � < ¡ G �@?�¢~  �Äá < � G �@?�¢p  � � � � i �

q| o~o~tbjb±dqr �D�1zj>o>�f��t�w?lSoAt | m]n1sMtbn1� r npscn7¶~s��Vµ | �M�_sM��j>m&m]�?lS�5j>mIsMt�� jbySjb�KlD±.jby~qr ySj�npm&jbk�m&l�m �>o�³�sM�~sM� | Þy�lSk r o}³ | m]n7¶~s�'sck?k | �ºsMt �àÎ 7 Ï u � �l �Bj>�~jbm �>l{zsMySµ&scn'z| y�j>k�zj>³5j>t5zj>oº�~j>� r y�zsM�AlSob�i�j>kFlSm]npocqr k,sMt�u � �?�5j>ySlYj>�M�?o?zjb�MwDjbk]n | � | k»zsMy�npsMy�sMy�k | n | n~n�� � �æð � Ç u � ñ5· �}·�� Ø óÉo~t�±�zj>��zsVn{��fw?lS�j>m&ob�Yµ | �c�«µ�sKsct �¸Î 7 Ï u � wDjbkFn | �A±dqr �M�Mw�zj>m]��j>ySjb�Djbnºnpjbo~tIs i � Ø ��.�j/&�m�zÑ �>l�z| ��sMm«o~tbj>�~jb³&y~�|±.jby�n�zjbnpjbySjbk?mj>k'�5sMkFk | �ºsMt � zsMyÓndsMyBySjMzÑ �An}jbySjb�KlY±.jbyAqr y�jbn��~sç¶�nds�wcsMmÌsMtX� � ��q| ���±.jby~qr ySj�npjbm��°�t � ±.j>yAqr ySj�npjbnKsMkFk | �a� | m&�V¶ r k jbySy�lS³n~lSk r o~m&sMk � l�ySy�jbn~wMj«µ&lS³5j>�A� | y�lSk r oAm�sMk����ĵ�s´sMt´q| o~oAt>j>o³ | m]n7¶~sÅj>y�ySl�³npl�k r o � l�ySy�jbn~wMj�µ&l�³Bjb�~� | ySl�k r od���

°�t«scy�zsM�&�l,zsMy�y zÑ n�zsMo_o~t>jb�~l�mFn�sON�s r oAo �?�'q| �~�;qr y�jbnpmj>kÕsÀk | ��³�sMk]n�±.j>yAqr ySj�np��sM�~sM� | mÕwMjbn~nű.jbySo~tMzÑ moAt>jb�~lSm]n~l�lSm]npjb�D��zscyж&zscnÄ�Kjb��ySjbµ&jbn�µ�scn'zsc� | tbm&l | yÓ���©z| � | m§lSob�µ | �M�akFl�o~t�zsc�¦zÑ n7¶ r k�jb�M�§�'q| ���&l'±.j>yAqr y�jbn�Þ�&sM�psM�«±.j>y�o~tMzÑ mÄzj�n{�� ����� XG�OQO Xg W�X\?T � + -�/�1 -�2<3 N � -�/�1 C M(1�3�2 -�M�- > =�Y -�M�AR M�- P � 3�X C M 3 N G � 3 D 3 > K- P - D - N K- ; 2 -�T > -�/ Y -�M�-�M C�àÎ 7 Ï u � 2 C'D > K3�M =V; -�/�1 ; K-�/�7H-�T P C'D > - N C -�/�1 -�M�- > =�Y -�M�AR Me- P - P K� D M�-HG�� -0/�1 AR�2�T 3 7 � 3 D 3 > K- P - D �P 3 DFP C > K3�2 1 E 3�2 -�/�1 î � C'D 5�3�2 � > K- D X - P C N K3 D�P P 3 DFP C > K3�2 1 P G

� 3 3 N� Y -�M�AR�M�- P -�M?M = � P = T R ; � 3�T6T C'D Y -�2�2[K3:MVM � � G � î��A��� ( � � � î��A� £� 3 � -�5 = /93 N� Y -�M AR M�- P X = � - DHE C M = T R ; � 3�T6T C'D � � G � î��A� � Ê ( � � � î��~� P -�M ��- ; AR�MSG

GIHKJcé �'TUHçJMZ �L8H;X\FZbWIé63^ Z 4 JKWbL 4 LBT �7JDO 3LBOSJcW�Jô�j>�M�Mj>m�sM� | n~nIj>�M� | yÓ�]sMm�zÑ w?µ | oAo~t§oAt>j>�AlSm]n�³&sM�psM�Òzjbn~j>�Aj>t>j�n~n;�$Î �Ðí ¢ $ � Ï u � jb�M�?o~t>jb�F�r t�zsM�An��q| �~�5jM�5sM��j>yÓ�åzsMyÓndsMy{zsMm | o#n{zÑ ³ r oYzr � ���Ky�zj>kMjbt>t�qr m&k,��zs��µ | �M� � ��? ��� · � � � �@? �ç· ��?�Ç �Ðí ¢ $ � �#n~j>yжAjbocqr yä�n~j>µ�zsVn�s �x�'q| �~�;qr y�jbnpj�oAj>µ | y oAj>� n?�r m&lSkÌj>y3��° � Î��Ðí ¢ $ � Ï u$�'q| �A�fqr ySj�npl;±dqr �D�cw�zjbm]�¾�asçÛl�� r �Ksy�j>�M�Mj>m � " � .¾zsMyQsco~o~t r m&k§jb�M� | yÓ�]sMmAa³ | tbl�n{zÑ waoAt�zsM� | n>��sM��j>y��?�Aj�±.j>m&m�zscySy aò ¡

� ��

i�j>kFlSm]npocqr kÂs ���Kjbm]n�zj>m�s � ¡ � � � ¢ � �Å� Ø· � � � · � � � ¢ � � � � ¡ á � ��wDjbk]n | �A�Kjbt�| kcjbn{�

.�jb�M��qr kasct�u � �F�BjbySl 7 � � í ¢ $ � áb�Ðí ¢ ��� � n�zjb�DyQscyQsM³ | mKsMtbn�sct�G�Î 7 Ï u � wDjbk]n | ��±Uqr �D�Mw�zj>m]�Fn{�'sMµ | y±.jbm&m�zsMy�y G �@?�¢~  �#�*� ��? ��×�aoAlSm   � � ��? ��×' � | o   � � ��? � n~jbn~o~t�| ySj>�Mj>o �@?�¢~  � Ç 7 j>oAjbn�zjbm��v°�t G³&sM� �bl�zsMy�lSo��&jb�~lÓw�zscy�n7¶~sclSm�sMk�wMj>k]n | �Al�zsMy�lSo}oAt | �At{scn�zsM�psÅs

< ¡ G �@?�¢~  �Äá < � G �@?�¢p  �f� � Ø Ê � ��? ��o~l�m   � � B o~lSm   � � �@? �Y× � | o   � � ��? � DI �

Page 49: Eötvös Loránd Universityweb.cs.elte.hu/geometry/vl/dgjegy11.pdf · !#"$ %& ' ( ) *,+.-&/103254'- +768-&9;:&=?+A@ 2B+DC =?E>=F9 GIHKJMLBN'OQPSRBJMTUHVP;WYX JMZ []\?OVX

kFlÓ±.j ¶Aj>t5zj>o�n�m]�Djb�.¶Vqr k'�5sM��l�sMt�n�� r ndscn.¶~s�µ | �M�«sMt�GI�&l �Bj>�Aj>m �bl�zsMy�µ�scn'z| y�j>kfzj>³5j>t5zj>oº�~jb� r y�zsc�~lSob�°���j>mmF�?l��BjbmÌsct8K³ | t>lÓn�zÑ wKo~t�zsM� | n}j>y�j>�Djbm&�f�| jbmÌkFl �>o~l�m&j>k�w#zsMyQsMoAtbn.¶ r k'�'sMkFk | �}jby�zj>�Aµ&jbnM�| �&µ | �M�sct � � G � 7§��sMyQsMkFt>scn�j>�M�§o~lS�Ks�±.j>yAqr ySj�n�ySjb�M�Djbm�u � �?��scm��� �� � � J� è Xg ê PVX^ � °$±.j>m]npl�j>k¾o | ��zsMmæzj>�An~j>y��Kjbt>jbnAn � ±.j>yAqr ySjbn~jbn_s=�²ÎB�Ðí ¢ $ � Ï u � j>�c��oAt>jb�?�r t�zsM�An��q| �~�Äzjbµ&j>t�ndsc�An | t�z| ao r �]sM�Bzr n r � r o�±.j>yAqr y�jbn~m&j>k§� | m&�ç¶ r k'�� JMéçhUJMé��'H1XJcT � °�t�u � �?�BjbySl�jb�M�?o~t>jb�F�r t�zsM�AnÄ�'q| �A�Bj�n r � r oA±.jby~qr ySj�npj�µ | ��j | � | ��±�s�n'z| � r o~t�±.j>yAqr ySjbnAnpjby3�

��q| tbwMjbn~ySj>m�oAt�zsM� | y�zsco~opscy�sM��z| �&lSk'��µ | �M�8sÀn r � r oA±.jby~qr ySj�n;G̳�sM�psc��zj�npj>�Aj>t5zj>o?zjbµ&j>t«ndsc�An | t�z| j>y�oM�|±U�| �Kjbm&m]��l�o?zjb�Dj>k,��zscnp�AlÓÛ�zsM�~sI±.j>m&m}zsMySyÈFâ>ãPJ ��?�¢p  �f�9 � ��Ø Ê � ��? ��o~lSm   � �

n~jbn~o~t�| ySj>�Mj>o ��?�¢p  � Ç 7þj>o~j�n�zjbm��i_q| �&��±�zj>ySjv��z| � | m�l�o}ySj�wDj>tbjbn~µ&jbnM�| �'µ | �M�«s N�s r oAo �?�'q| �~�;qr y�jbnpj�n�ySjMzÑ ��z| � Î>7 Ï uÖ±Uqr �D�Mw�zj>m]�?�~j

� ��?�¢p  ��� Ê � �@? �}o~l�m   ��Ø Ê � �@? ��oAlSm   � � � � � �

n~j>y ¶Aj>ocqr yä�;��oAt>j>�AlSm]nÄs�n r � r oA±.j>yAqr ySj�n G ��?�¢p  ��³ | m]n7¶~svj>y�ySl�³npl�k r o>�Mµ&sv±.jbm&m�zsMy�y��Âò   ò �����;°��Kjbm&m]�?lÓÞ�5j>m�lS�]sMt í�ò   ò ���Dsck?k | �fsct G ��?�¢p  �Y³ | m]nfµ&l�³Bjb�~� | ySl�k r o>� �sº³Bjb�&lS�   ��íºwcsM�M�   � �an~j>yжAjbocqr yä�sck?k | ��sMt�G ��?�¢p  �ij | m]n�³&sM�psM� | y�lSk r o>�i�j>kFlSm]npocqr kÂu � �?�5j>m�s¦î¸� �Ðí ¢ $ � á � � ¢ ��� � t�zsM�An�n�zj>�MyQsMySsM³ | n{�«°�t � � G � 7,�vn r � r oA±.jby~qr ySj�nk�qr y�oM�| ±.j>y�zjbm&j>kÅm&j�wDjbt>t�qr kKsMt G � î��fk | �K³�sckFn ±.j>yAqr ySj�np��sM�~sM� | n{��° n r � r oA±.j>yAqr ySj�nÄk�qr y�oM�| ±.j>ySj}n~j>µ�zscn#sMt

� sMt | m�³ | m]n7¶~sclS��z| yvzsMySyä����j>yÓ�Dj>kF�5j>m¦s�N�s r oAo �?��q| �~�;qr y�jbn}mj>��m&j>�]sVn�zÑ w5�� ��� � XG�OQO Xg W�X\?T ��� � 3 R ; ; � /�AC'DHE AR Me- P 2 -�TO3 P R E R ; Y -�M�AR�M�- P TZAR M ; C Y -�M6K-�2b7'- P0P Y -�M ; N K� 2 ; N - D = 2 P == 2 P -�/ D K3�M ��K3 D 3 Y -�2�2[K3�M?M � � G � î��A� �»��� �

" � ��? �&È ?�£

GIH�� � 2 4 JDOQP;Jcé �'TUHçJMZ �L8H;X\FZbWIé63^ Z 4 J�WbJMO hUJMT�é63^ Z 4 3LYOSJVWYXJc[�JDO;ND\Fe ê T ^ OS\]W ^ T_WYXJcW�JMOSJMN°Òt�zsc�An�l�mFn~j>��wMscySy r � | m�zj>��npj>y��Kjbt>j�n~n��'q| �A�Bj1npj>y ¶Aj>o��'q| �~�;qr y�jbn�zjbm&j>k�± | �]scyS��zscn�sº��� Ø ��� .�j/&�m�zÑ �bl�z| �&sMmsc�n r k���j>�&�a°�tKu � n�zj>�_j>�M�?oAt>j>�F�r t�zsM��n���q| �~�ÄzjblSm&jbkÀnpjbyжAj>o_�'q| �A�fqr ySj�n�zj�wDj>y�kVsM³ �>o | yQscn | o_sMtKsMy�zsM��&lkFl ¶Aj>ySjbm]n�zj>ob�'sM��j>y��Mjbn B�j>m �dµj>y¯�]n�zjbn~j>y�j>k�zj>m]n�o~t | k#zscoºjb�Ky3zÑ n~j>m&lä�

� ��� � * XJVW�JDO � � - P ; N C M�-�/�- ; u � ��E -�M = � -�/�1 ; N - D R N K3 DFP /BAC'DFE - P -�M ��- ; /BAC'DFE AR M�- P K- D - P -�M ��- ; AR M 3� � ���#��� � -0/�1�-�2�M�C P M�-�2 ; K-0/�G

�./�1 � -�/�1 ; N - D R N K3 D�P /BAC'DFE K- D - Y -�2�2[K3�M?M.3 � � �Ä�}�¿��� AC ;0; N - Y AR:/ /&K- ; 3�T6T C'D K- ; ��; 3�T�3�T6T C'D � X�33 � -�/�1<T C 2 7'- � N K3 DFP ; K� T /BAC'DFE -HG���Ky�zj>kcj>t>t�qr m&k¦�&zs��µ | �c�¾san | ³ | y{z| �Dl{zsM��sMm¦s§kYq| �~�~jby�lSt | n'z| ³¦j>�M�?o~tbj>�F�r t�zsc�An��'q| �A�Äzj>kcjbn�np�Al�w?l�zscySlSo

�bo | ��z| kFm�sMk � lSy�ySjbnAwDj¾kFlS� | m]npµ&scn'z| �>o | ��z| kFm�sMk��«oAt | kÄzsMoÌm&jbwMj>tbm&l3� ° kBq| wMjbn~kMj>t��| j>�Aj>�&��zj>m]�Fn¾sB�zsM�A�D�ÜÌl�ySm | � �Fn�zjbnpjbySkfzj>m]n}npsM�An.¶&zsMkaoAt�zsM� | m�s�oAt{sMkFl�� | ��sMy | ����sMm��� �"! � * XJcW�JDO � � 343 Nau � � E -�M = � -�/:1 ; N - D R N K3 D�P /BAC'DFE - K3�M P 3�MJ365 C P P ��; C > KC 2 - > T = E�C 2 P X�3 P KC �3:T T C'D 3 � P -�M ��- ; /�AC'DHE AR Me- P K- D - Y -�2�2[K3�M?M � � �Ä�#ì I � £

I �