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In this paper we are presenting few aspects of how GeoGebra software can be used in hyperbolic geometry such as the construction of hyperbolic triangle. Also, GeoGebra offers so many virtual tools that can be used for the construction of other hyperbolic figures. The new version of GeoGebra 3D is another advantage. The main topic of our paper is the use of GeoGebra software to investigate what properties of Euclidean geometry are preserved in hyperbolic geometry and what properties are not,
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GeoGebra Konferencia
Budapest januaacuter 2014
GEOGEBRA FOR SIMILARITIES AND DIFFERENCES BETWEEN DIFFERENT GEOMETRIES
GeoGebra Konferencia Budapest januaacuter 2014
bull GeoGebra software can be used for constructing figures and studying and illustrating different properties Using GeoGebra software we can - construct hyperbolic figures
- perform measurements of lengths and of angles by using the respective GeoGebra virtual tools to investigate if the properties satisfied by some particular figures of Euclidean geometry are satisfied by the homologues figures in hyperbolic geometry For example consider the case of the hyperbolic triangle constructed with GeoGebra software as displayed in following figure
GeoGebra Konferencia Budapest januaacuter 2014bull For the Euclidean triangle the sum of the angles is 180deg As to
hyperbolic triangle (triangle BCE in the following figure with blue wider arches) the angles are defined by the tangents to the arches passing through the vertices The measurements are done using the angle virtual tool and for clarity reasons in figure are shown the values of the obtuse angles defined by the two respective tangents The measurements show that their sum isbull (180deg - 16387deg) + (180deg - 17611deg) + (180deg - 17859deg) = 3143degbull The result the sum of its angular measures is less than 180deg
Sum of the interior angles of the hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull In the next figure is shown the hyperbolic triangle with sides DG DF
and GF that are arches with centre C E and F respectively The tangents are not shown for reasons of clarity of the figure In figure are shown the exterior angles (denoted by ) of the angles between two tangents passing through the verticesbull The sum of the interior angles of the hyperbolic triangle can be
computed using the expression The address for the applet created for purpose of demonstration of the property with respect to the sum of the angles is bull httpwwwgeogebratubeorgmaterialshowid61143
Applet of observing the sum of the angles Using the mouse or the touch pad displace each centre of the arch to
observe the property
GeoGebra Konferencia Budapest januaacuter 2014bull Many other properties are not preserved such as those related to the
distance Also in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry Play with the above applet and observe the sum bull The definitions and perceptions of vertical angles and supplementary
angles in hyperbolic geometry are similar to those of Euclidean geometry There are properties of Euclidean geometry that are preserved in hyperbolic geometry Look at the next figure it is obvious that the vertical angles are equal and the sum of supplementary angles is 180deg This derives from the fact that the angle between two intersecting arches is defined by the angle formed by the two tangents passing through the intersecting point
The property of vertical and supplementary angles
GeoGebra Konferencia Budapest januaacuter 2014bull Is Pythagorean Theorem true in hyperbolic geometry The hyperbolic
triangle with vertices B C E (light brown colour in the next figure) is selected in a special way in order to make easy the calculation of the lengths of its arches The hyperbolic triangle BCE (the one consisted of and bounded by the three arches) is got by the construction of three arches of angular measures 90deg each one of them in such way that the angle with vertex B of the hyperbolic triangle be 90deg The radii of the arches are 2 4 and 6 units respectivelybull Looking at the notes in figure can be easy checked that the
Pythagorean Theorem does not hold true for the hyperbolic triangle The lengths of the arches are respectively but
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
GeoGebra Konferencia Budapest januaacuter 2014
bull GeoGebra software can be used for constructing figures and studying and illustrating different properties Using GeoGebra software we can - construct hyperbolic figures
- perform measurements of lengths and of angles by using the respective GeoGebra virtual tools to investigate if the properties satisfied by some particular figures of Euclidean geometry are satisfied by the homologues figures in hyperbolic geometry For example consider the case of the hyperbolic triangle constructed with GeoGebra software as displayed in following figure
GeoGebra Konferencia Budapest januaacuter 2014bull For the Euclidean triangle the sum of the angles is 180deg As to
hyperbolic triangle (triangle BCE in the following figure with blue wider arches) the angles are defined by the tangents to the arches passing through the vertices The measurements are done using the angle virtual tool and for clarity reasons in figure are shown the values of the obtuse angles defined by the two respective tangents The measurements show that their sum isbull (180deg - 16387deg) + (180deg - 17611deg) + (180deg - 17859deg) = 3143degbull The result the sum of its angular measures is less than 180deg
Sum of the interior angles of the hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull In the next figure is shown the hyperbolic triangle with sides DG DF
and GF that are arches with centre C E and F respectively The tangents are not shown for reasons of clarity of the figure In figure are shown the exterior angles (denoted by ) of the angles between two tangents passing through the verticesbull The sum of the interior angles of the hyperbolic triangle can be
computed using the expression The address for the applet created for purpose of demonstration of the property with respect to the sum of the angles is bull httpwwwgeogebratubeorgmaterialshowid61143
Applet of observing the sum of the angles Using the mouse or the touch pad displace each centre of the arch to
observe the property
GeoGebra Konferencia Budapest januaacuter 2014bull Many other properties are not preserved such as those related to the
distance Also in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry Play with the above applet and observe the sum bull The definitions and perceptions of vertical angles and supplementary
angles in hyperbolic geometry are similar to those of Euclidean geometry There are properties of Euclidean geometry that are preserved in hyperbolic geometry Look at the next figure it is obvious that the vertical angles are equal and the sum of supplementary angles is 180deg This derives from the fact that the angle between two intersecting arches is defined by the angle formed by the two tangents passing through the intersecting point
The property of vertical and supplementary angles
GeoGebra Konferencia Budapest januaacuter 2014bull Is Pythagorean Theorem true in hyperbolic geometry The hyperbolic
triangle with vertices B C E (light brown colour in the next figure) is selected in a special way in order to make easy the calculation of the lengths of its arches The hyperbolic triangle BCE (the one consisted of and bounded by the three arches) is got by the construction of three arches of angular measures 90deg each one of them in such way that the angle with vertex B of the hyperbolic triangle be 90deg The radii of the arches are 2 4 and 6 units respectivelybull Looking at the notes in figure can be easy checked that the
Pythagorean Theorem does not hold true for the hyperbolic triangle The lengths of the arches are respectively but
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
GeoGebra Konferencia Budapest januaacuter 2014bull For the Euclidean triangle the sum of the angles is 180deg As to
hyperbolic triangle (triangle BCE in the following figure with blue wider arches) the angles are defined by the tangents to the arches passing through the vertices The measurements are done using the angle virtual tool and for clarity reasons in figure are shown the values of the obtuse angles defined by the two respective tangents The measurements show that their sum isbull (180deg - 16387deg) + (180deg - 17611deg) + (180deg - 17859deg) = 3143degbull The result the sum of its angular measures is less than 180deg
Sum of the interior angles of the hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull In the next figure is shown the hyperbolic triangle with sides DG DF
and GF that are arches with centre C E and F respectively The tangents are not shown for reasons of clarity of the figure In figure are shown the exterior angles (denoted by ) of the angles between two tangents passing through the verticesbull The sum of the interior angles of the hyperbolic triangle can be
computed using the expression The address for the applet created for purpose of demonstration of the property with respect to the sum of the angles is bull httpwwwgeogebratubeorgmaterialshowid61143
Applet of observing the sum of the angles Using the mouse or the touch pad displace each centre of the arch to
observe the property
GeoGebra Konferencia Budapest januaacuter 2014bull Many other properties are not preserved such as those related to the
distance Also in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry Play with the above applet and observe the sum bull The definitions and perceptions of vertical angles and supplementary
angles in hyperbolic geometry are similar to those of Euclidean geometry There are properties of Euclidean geometry that are preserved in hyperbolic geometry Look at the next figure it is obvious that the vertical angles are equal and the sum of supplementary angles is 180deg This derives from the fact that the angle between two intersecting arches is defined by the angle formed by the two tangents passing through the intersecting point
The property of vertical and supplementary angles
GeoGebra Konferencia Budapest januaacuter 2014bull Is Pythagorean Theorem true in hyperbolic geometry The hyperbolic
triangle with vertices B C E (light brown colour in the next figure) is selected in a special way in order to make easy the calculation of the lengths of its arches The hyperbolic triangle BCE (the one consisted of and bounded by the three arches) is got by the construction of three arches of angular measures 90deg each one of them in such way that the angle with vertex B of the hyperbolic triangle be 90deg The radii of the arches are 2 4 and 6 units respectivelybull Looking at the notes in figure can be easy checked that the
Pythagorean Theorem does not hold true for the hyperbolic triangle The lengths of the arches are respectively but
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
Sum of the interior angles of the hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull In the next figure is shown the hyperbolic triangle with sides DG DF
and GF that are arches with centre C E and F respectively The tangents are not shown for reasons of clarity of the figure In figure are shown the exterior angles (denoted by ) of the angles between two tangents passing through the verticesbull The sum of the interior angles of the hyperbolic triangle can be
computed using the expression The address for the applet created for purpose of demonstration of the property with respect to the sum of the angles is bull httpwwwgeogebratubeorgmaterialshowid61143
Applet of observing the sum of the angles Using the mouse or the touch pad displace each centre of the arch to
observe the property
GeoGebra Konferencia Budapest januaacuter 2014bull Many other properties are not preserved such as those related to the
distance Also in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry Play with the above applet and observe the sum bull The definitions and perceptions of vertical angles and supplementary
angles in hyperbolic geometry are similar to those of Euclidean geometry There are properties of Euclidean geometry that are preserved in hyperbolic geometry Look at the next figure it is obvious that the vertical angles are equal and the sum of supplementary angles is 180deg This derives from the fact that the angle between two intersecting arches is defined by the angle formed by the two tangents passing through the intersecting point
The property of vertical and supplementary angles
GeoGebra Konferencia Budapest januaacuter 2014bull Is Pythagorean Theorem true in hyperbolic geometry The hyperbolic
triangle with vertices B C E (light brown colour in the next figure) is selected in a special way in order to make easy the calculation of the lengths of its arches The hyperbolic triangle BCE (the one consisted of and bounded by the three arches) is got by the construction of three arches of angular measures 90deg each one of them in such way that the angle with vertex B of the hyperbolic triangle be 90deg The radii of the arches are 2 4 and 6 units respectivelybull Looking at the notes in figure can be easy checked that the
Pythagorean Theorem does not hold true for the hyperbolic triangle The lengths of the arches are respectively but
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
GeoGebra Konferencia Budapest januaacuter 2014bull In the next figure is shown the hyperbolic triangle with sides DG DF
and GF that are arches with centre C E and F respectively The tangents are not shown for reasons of clarity of the figure In figure are shown the exterior angles (denoted by ) of the angles between two tangents passing through the verticesbull The sum of the interior angles of the hyperbolic triangle can be
computed using the expression The address for the applet created for purpose of demonstration of the property with respect to the sum of the angles is bull httpwwwgeogebratubeorgmaterialshowid61143
Applet of observing the sum of the angles Using the mouse or the touch pad displace each centre of the arch to
observe the property
GeoGebra Konferencia Budapest januaacuter 2014bull Many other properties are not preserved such as those related to the
distance Also in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry Play with the above applet and observe the sum bull The definitions and perceptions of vertical angles and supplementary
angles in hyperbolic geometry are similar to those of Euclidean geometry There are properties of Euclidean geometry that are preserved in hyperbolic geometry Look at the next figure it is obvious that the vertical angles are equal and the sum of supplementary angles is 180deg This derives from the fact that the angle between two intersecting arches is defined by the angle formed by the two tangents passing through the intersecting point
The property of vertical and supplementary angles
GeoGebra Konferencia Budapest januaacuter 2014bull Is Pythagorean Theorem true in hyperbolic geometry The hyperbolic
triangle with vertices B C E (light brown colour in the next figure) is selected in a special way in order to make easy the calculation of the lengths of its arches The hyperbolic triangle BCE (the one consisted of and bounded by the three arches) is got by the construction of three arches of angular measures 90deg each one of them in such way that the angle with vertex B of the hyperbolic triangle be 90deg The radii of the arches are 2 4 and 6 units respectivelybull Looking at the notes in figure can be easy checked that the
Pythagorean Theorem does not hold true for the hyperbolic triangle The lengths of the arches are respectively but
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
Applet of observing the sum of the angles Using the mouse or the touch pad displace each centre of the arch to
observe the property
GeoGebra Konferencia Budapest januaacuter 2014bull Many other properties are not preserved such as those related to the
distance Also in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry Play with the above applet and observe the sum bull The definitions and perceptions of vertical angles and supplementary
angles in hyperbolic geometry are similar to those of Euclidean geometry There are properties of Euclidean geometry that are preserved in hyperbolic geometry Look at the next figure it is obvious that the vertical angles are equal and the sum of supplementary angles is 180deg This derives from the fact that the angle between two intersecting arches is defined by the angle formed by the two tangents passing through the intersecting point
The property of vertical and supplementary angles
GeoGebra Konferencia Budapest januaacuter 2014bull Is Pythagorean Theorem true in hyperbolic geometry The hyperbolic
triangle with vertices B C E (light brown colour in the next figure) is selected in a special way in order to make easy the calculation of the lengths of its arches The hyperbolic triangle BCE (the one consisted of and bounded by the three arches) is got by the construction of three arches of angular measures 90deg each one of them in such way that the angle with vertex B of the hyperbolic triangle be 90deg The radii of the arches are 2 4 and 6 units respectivelybull Looking at the notes in figure can be easy checked that the
Pythagorean Theorem does not hold true for the hyperbolic triangle The lengths of the arches are respectively but
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
GeoGebra Konferencia Budapest januaacuter 2014bull Many other properties are not preserved such as those related to the
distance Also in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry Play with the above applet and observe the sum bull The definitions and perceptions of vertical angles and supplementary
angles in hyperbolic geometry are similar to those of Euclidean geometry There are properties of Euclidean geometry that are preserved in hyperbolic geometry Look at the next figure it is obvious that the vertical angles are equal and the sum of supplementary angles is 180deg This derives from the fact that the angle between two intersecting arches is defined by the angle formed by the two tangents passing through the intersecting point
The property of vertical and supplementary angles
GeoGebra Konferencia Budapest januaacuter 2014bull Is Pythagorean Theorem true in hyperbolic geometry The hyperbolic
triangle with vertices B C E (light brown colour in the next figure) is selected in a special way in order to make easy the calculation of the lengths of its arches The hyperbolic triangle BCE (the one consisted of and bounded by the three arches) is got by the construction of three arches of angular measures 90deg each one of them in such way that the angle with vertex B of the hyperbolic triangle be 90deg The radii of the arches are 2 4 and 6 units respectivelybull Looking at the notes in figure can be easy checked that the
Pythagorean Theorem does not hold true for the hyperbolic triangle The lengths of the arches are respectively but
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
The property of vertical and supplementary angles
GeoGebra Konferencia Budapest januaacuter 2014bull Is Pythagorean Theorem true in hyperbolic geometry The hyperbolic
triangle with vertices B C E (light brown colour in the next figure) is selected in a special way in order to make easy the calculation of the lengths of its arches The hyperbolic triangle BCE (the one consisted of and bounded by the three arches) is got by the construction of three arches of angular measures 90deg each one of them in such way that the angle with vertex B of the hyperbolic triangle be 90deg The radii of the arches are 2 4 and 6 units respectivelybull Looking at the notes in figure can be easy checked that the
Pythagorean Theorem does not hold true for the hyperbolic triangle The lengths of the arches are respectively but
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
GeoGebra Konferencia Budapest januaacuter 2014bull Is Pythagorean Theorem true in hyperbolic geometry The hyperbolic
triangle with vertices B C E (light brown colour in the next figure) is selected in a special way in order to make easy the calculation of the lengths of its arches The hyperbolic triangle BCE (the one consisted of and bounded by the three arches) is got by the construction of three arches of angular measures 90deg each one of them in such way that the angle with vertex B of the hyperbolic triangle be 90deg The radii of the arches are 2 4 and 6 units respectivelybull Looking at the notes in figure can be easy checked that the
Pythagorean Theorem does not hold true for the hyperbolic triangle The lengths of the arches are respectively but
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
GeoGebra Konferencia Budapest januaacuter 2014bull Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry Consider the formula for the area of a triangle which in Euclidean geometry is For simplicity we consider the hyperbolic triangle BCE of the above figure and which is a right-angle triangle Its hypotenuse is represented by the arch CE whereas arch BL serves as its height The centre of the arch is point K and it is chosen in such a way that this arch be perpendicular to arch CE The difference between the length of arch BL and the respective chord BL (153) is very small or negligible Assume now that the length of arch BL is 156 Calculate the area of the hyperbolic triangle BCE in two ways
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
GeoGebra Konferencia Budapest januaacuter 2014bull Formally applying the formula a(BCE) = 12[arch(CE)] [arch(BL)] =
123π156 = 234π asymp 735bull Doing more precise calculations in Euclidean geometry
a(BCE) = 14π36 ndash 14π4 - 14π16 - 24 = 4π ndash 8 asymp 456 ˂ 735bull If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle triangle in Euclidean geometry we havebull a(BCE) = 12[arch(BE)] [arch(BC)] = 12π2π = π2 asymp 986 ne 735
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
GeoGebra Konferencia Budapest januaacuter 2014bull Conclusionbull Properties of Euclidean geometry that are preserved in hyperbolic geometry are mainly
those linked with the angles But in Euclidean geometry the sum of the angles of the triangle is constant Not so in hyperbolic geometry
bull Many other properties are not preserved such as those related to the distance Formulas used and valid in Euclidean geometry are not valid in hyperbolic geometry The Euclidean geometry satisfies the independence property wrt the calculations of the area of the triangle (the area does not depend on the option which side is taken as a base and the height perpendicular to the base constructed from the vertex in front of it) In hyperbolic geometry the products of the lengths of the arches taken as bases with the lengths of the respective arches taken as heights give different results That is the hyperbolic geometry does not satisfy the independence property wrt the calculation of the area of the triangle in different ways The same result is true in the case of calculating the area of a hyperbolic polygon
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
See you next GeoGebra Conference
Many thanks for your attention
Pellumb Kllogjeri and Qamil KllogjeriALBANIA
