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PDN Matematica Arquitectura

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6ROXFLyQ FP $7  ==

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6ROXFLyQ&RQYHUWLPRVFPDPHWURV\ P $ % =×=

PD 3  $ %DVH / =×=×= ⇒ P $ $ $ / %DVH7  =+=+=

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Page 49: PDN Matematica Arquitectura

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3ROLHGURV,UUHJXODUHV

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35,60$

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• $ULVWDVODWHUDOHVVRQORVVHJPHQWRVGHUHFWDGHWHUPLQDGRVSRUODVLQWHUVHFFLRQHVGHODVFDUDVODWHUDOHV7RGDVODVDULVWDVODWHUDOHVGHXQSULVPDVRQSDUDOHODV\FRQJUXHQWHV

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 SULVPDÈUHDODWHUDO(VODVXPDGHODViUHDVGHODVFDUDVODWHUDOHV

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3DUDXQSULVPDGHQODGRV

ÈUHDWRWDO(VODVXPDGHOiUHDODWHUDO\HOiUHDGHODVEDVHVHVGHFLU9ROXPHQ6HOODPDYROXPHQGHXQSULVPDDODPHGLGDGHOHVSDFLRTXHRFXSD

K $9  %$6(  ×= (MHUFLFLR&DOFXOHHOiUHDWRWDOGHOSULVPDSHQWDJRQDOGHODILJXUDDGMXQWD6ROXFLyQ

( )

FP

QO DS $ % =

××=

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FPKO Q $ / =×=×= (QWRQFHV

FP $ $ $ % /7  =+=+=

(MHUFLFLR /DV EDVHV GH XQ SULVPD UHFWR VRQ SHQWiJRQRV UHJXODUHV GH FP GHDSRWHPD\GHiUHDFP6LODDOWXUDGHOSULVPDHVGHFPGHWHUPLQHHOiUHDWRWDO\YROXPHQGHOSULVPD6ROXFLyQ

FPO FPO  $ %

=⇒=×= FP $ / =×=

⇒ FP $ $ $ / %7  =+=+= FPK $9  % =×=×=

 %$6(  /7  $ $ $ +=

KFP

OFP

FP

KFP

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K

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K 3 KDQ $ %DVH/$7(5$/ ×=××=

Page 51: PDN Matematica Arquitectura

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3,5È0,'(8QD SLUiPLGH HV XQ SROLHGUROLPLWDGRSRUXQSROtJRQREDVHGHOD SLUiPLGH \ SRU WULiQJXORV TXH

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• $ULVWDVODWHUDOHV9$9%9&«VRQODVLQWHUVHFFLRQHVGHODVFDUDVODWHUDOHVHVWRHVGHORVWULiQJXORVTXHGHOLPLWDQODVFDUDV(TXLYDOHQWHPHQWHVHVHxDODTXHVRQORVVHJPHQWRVTXHXQHQHOYpUWLFHGHODSLUiPLGHFRQORVYpUWLFHVGHOSROtJRQRTXHKDFHODVYHFHVGHEDVH

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EDVH

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 SLUiPLGHWULDQJXODUVRQWULiQJXORVHTXLOiWHURVVHOODPDWHWUDHGURUHJXODU• 3RUODVFDUDFWHUtVWLFDVGHODEDVH

D5HJXODUFXDQGRODEDVHHVXQSROtJRQRUHJXODU\HOSLHGHODDOWXUDFRLQFLGHFRQHOFHQWURGHODEDVH(QXQDSLUiPLGHUHJXODUWRGDVODVDULVWDVODWHUDOHVWLHQHQODPLVPDPHGLGD\ODVFDUDVODWHUDOHVVRQWULiQJXORVLVyVFHOHVFRQJUXHQWHV

E,UUHJXODUFXDQGRODEDVHHVXQSROtJRQRLUUHJXODU

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Page 52: PDN Matematica Arquitectura

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• 7RGR SODQR TXH LQWHUVHFD D XQD SLUiPLGH \ HV SDUDOHOR D OD EDVH GHWHUPLQD SROtJRQRVVHPHMDQWHVHQWUHODEDVH\ODVHFFLyQWUDQVYHUVDO

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K

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• 'DGDVGRVSLUiPLGHVODViUHDVGHODVEDVHVVRQSURSRUFLRQDOHVDORVFXDGUDGRVGH

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ÈUHD WRWDO (V LJXDO DO VHPLSHUtPHWUR GH OD EDVH SRU OD VXPD GHO DSRWHPD GH OD

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Page 53: PDN Matematica Arquitectura

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7521&2'(3,5È0,'((V OD SDUWH GH OD SLUiPLGH UHJXODUFRPSUHQGLGDHQWUHODEDVH\XQSODQRSDUDOHORDODEDVHTXHFRUWHDWRGDVODVDULVWDVODWHUDOHV

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( )

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+=

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6 6  $ $ /$7(5$/727$/ ++=

9ROXPHQ(VLJXDODXQWHUFLRGHODDOWXUDGHOWURQFRPXOWLSOLFDGRSRUODVXPDGHODViUHDVGHODVEDVHVPiVODUDt]FXDGUDGDGHOSURGXFWRGHODVPLVPDV

( )

6 6 6 6 K

9 ++

=

(MHUFLFLR+DOOHHOiUHDWRWDOGHXQDSLUiPLGHKH[DJRQDOYHUILJXUDFRQDULVWDVODWHUDOHVGHFP\DULVWDVGHODEDVHGHFP6ROXFLyQ• &iOFXORGHODDOWXUDGHODFDUDODWHUDO

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FPK =−=

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FPD =−=

FP $ % =

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FP $ / =

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$SRWHPD$OWXUD

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(MHUFLFLR /DV EDVHV GH XQ WURQFR GH SLUiPLGH UHJXODU VRQFXDGUDGRV GHFP\ FP GHODGR UHVSHFWLYDPHQWH /DV DULVWDVODWHUDOHVVRQGHFPKDOOHVXiUHDWRWDO

6ROXFLyQ&iOFXORGHODDOWXUDGHXQDFDUDODWHUDO

FPK =−= FP $ %$6(6  =+=

FP $ / = 

  

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&LOLQGUR&LOLQGURFLUFXODUUHFWRRFLOLQGURGHUHYROXFLyQHVHOVyOLGRJHQHUDGRSRUODURWDFLyQFRPSOHWDGHXQUHFWiQJXORDOUHGHGRUGHXQRGHVXVODGRV

(OHPHQWRV• %DVHV VRQ ORV FtUFXORV GH FHQWURV 2 \ 2¶

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2

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8QLYHUVLGDG3HUXDQDGH&LHQFLDV$SOLFDGDV&XDGHUQR$XWRLQVWUXFWLYRSDUDOD3UXHEDGH'HILQLFLyQGH1LYHOHVGH0DWHPiWLFD

• *HQHUDWUL] HV HO VHJPHQWR $% TXH DO JLUDU DOUHGHGRU GHO HMH 22¶ JHQHUD OD

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KU $ /$7(5$/ π = ÈUHDWRWDO(VLJXDODODVXPDGHOiUHDODWHUDOFRQHOiUHDGHODVEDVHV

U $ $ /$7(5$/727$/ π +=

9ROXPHQ(VLJXDODOSURGXFWRGHOiUHDGHODEDVHSRUODDOWXUDJHQHUDWUL]

KU9  π = (MHUFLFLR'HWHUPLQHHOYROXPHQGHXQFLOLQGURGHGLiPHWURGHODEDVHFP\FRQDOWXUDFP6ROXFLyQ FPKU9  === π π 

(MHUFLFLR&DOFXOHODFDQWLGDGGHFDUWyQTXHVHQHFHVLWDUiSDUDKDFHUERWHVGHIRUPDFLOtQGULFDVLQWDSDGHFPGHGLiPHWUR\FPGHDOWXUD6ROXFLyQ UKU $727$/ π π  +=

π π  +=727$/ $ FP $727$/ =

FPFDUWyQGH&DQWLGDG  ==

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&RQR(VHOVyOLGRJHQHUDGRSRUODURWDFLyQFRPSOHWDGHXQWULiQJXORUHFWiQJXORDOUHGHGRUGHXQRGHVXVFDWHWRV

(OHPHQWRV• %DVHHVHOFtUFXORGHFHQWUR³2´• $OWXUD K HV HO VHJPHQWR

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• *HQHUDWUL]HVHOVHJPHQWRTXHXQH

HO YpUWLFH FRQ XQ SXQWR GH ODFLUFXQIHUHQFLDEDVH\TXHDOJLUDUDOUHGHGRUGHOHMHJHQHUDODVXSHUILFLHFyQLFD

6HFXPSOHTXH KU J  += ÈUHD ODWHUDO (V LJXDO DOVHPLSURGXFWRGHODORQJLWXGGHOD FLUFXQIHUHQFLD EDVH SRU ODJHQHUDWUL]GHOFRQR

 J U $ /DWHUDO  =

ÈUHDWRWDO(VLJXDODODVXPDGHOiUHDODWHUDOFRQHOiUHDGHODEDVH

( )U J U $7RWDO  += π 

9ROXPHQ(VLJXDODXQWHUFLRGHOSURGXFWRGHOiUHDGHODEDVHSRUODDOWXUDGHOFRQR

KU9 

π =

(MHUFLFLR'HWHUPLQHHOiUHDWRWDO\HOYROXPHQGHOFRQRGHUDGLRFP\JHQHUDWUL]FP

6ROXFLyQ  % /7RWDO  $ $ $ += FP J  5 $ /DWHUDO  === π π 

FP 5 $ % === π π 

(QWRQFHV FP $7RWDO  =+= \

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K 59  ===

π π 

9

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• $OWXUDHVODGLVWDQFLDHQWUHODVEDVHV

• 5DGLRVVRQORVFRUUHVSRQGLHQWHVDODVEDVHVFLUFXODUHV5\U

(MHUFLFLR'HWHUPLQHODFDSDFLGDGGHXQYDVRGHYLGULRTXHWLHQHIRUPDGHWURQFRGHFRQRVLHOUDGLRGHVXEDVHHVFPHOUDGLRGHVXDEHUWXUDHVFP\WLHQHXQDDOWXUDGHFPDVXPDTXHHOYLGULRHVGHHVSHVRUGHVSUHFLDEOH6ROXFLyQ

3RUVHPHMDQ]D

=→=

+ [

 [ [

FP9  π π π  =−=

(VIHUD

(VHOVyOLGRTXHVHREWLHQHDOJLUDUXQVHPLFtUFXORXQDYXHOWDFRPSOHWDDOUHGHGRUGHVXGLiPHWUR

ÈUHDGHODVXSHUILFLHHVIpULFD U $ π =

9ROXPHQ

U9  π =

[

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8QLYHUVLGDG3HUXDQDGH&LHQFLDV$SOLFDGDV&XDGHUQR$XWRLQVWUXFWLYRSDUDOD3UXHEDGH'HILQLFLyQGH1LYHOHVGH0DWHPiWLFD

(MHUFLFLR'HWHUPLQHHOiUHDVXSHUILFLDOGHXQDHVIHUDGHUDGLRFP

6ROXFLyQ

U $ π =

(QWRQFHV

FP $ == π 

(MHUFLFLR&DOFXOHHOYROXPHQGHXQDHVIHUDLQVFULWDHQXQFLOLQGURGHPGHDOWXUD6ROXFLyQ

U9  π =

(QWRQFHV

P9  == π 

(MHUFLFLR8QDFDOGHUDGHYDSRUGHIRUPDFLOtQGULFDWHUPLQDHQVXVH[WUHPRVHQXQDVHPLHVIHUDGHFPGHUDGLR+DOODUHOiUHDH[WHUQDGHHVWDFDOGHUDVLHOFLOLQGURWLHQHXQUDGLRLJXDODOGHODVHVIHUDV\ODORQJLWXGHVHOGREOHGHOGLiPHWUR

6ROXFLyQ &RPR HQ OD ILJXUD VH SUHVHQWDQ GRV VHPLHVIHUDV GHPLVPRUDGLRHQWRQFHVHOiUHDTXHKD\HQWUHODVGRVVHPLHVIHUDVHV

( )  (6)(5$6 FPπ π = = ×

(QWDQWRTXHHOiUHDODWHUDOGHOFLOLQGURHV

( )   / &,/,1'526 FPπ π − = = ×

)LQDOPHQWHHOiUHDH[WHUQDGHODFDOGHUDHV

( )   ( &$/'(5$6 FPπ − = ×

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