5064

4761

كارشناسي ارشد

هندسه

اصفهان:دانشگاه صنعتي اصفهان،دانشكده علوم رياضي

1388

[هشت]،102ص.:مصور

ص.ع.به: فارسي و انگليسي

قهرمان طاهريان

اعظم اعتماد دهكردي

88/12/9

محمدمهدي ابراهيمي،امير هاشمي

رياضي

ID4761

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

Legender Like Theorems in a General Absolute Geometry Mahfouz Rostamzadeh m rostamzadeh@math iut ac ir October 2009 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr S Ghahraman Taherian taherian@cc iut ac ir2000 MSC 51F05 51F20 Key words Absolute Plane Lambert Saccheri Quadrangle Angle Sum of Angles of a Triangle Measureof Segment Abstract In this thesis we present an expanded account of Legender Like Theorems in a GeneralAbsolute Geometry based on an article by Helmut Karzel Mario Marchi and Silvia Pianta 2007 The axiomatic basis will be a general absolute plane A P L in the senseof 10 where P and L denote respectively the set of points and the set of lines the orderstructure and the congruence and where furthermore the word general means that noclaim is made on any kind of continuity assumptions We show that an absolute geometry has either a singular a hyperbolic or an ellipticcongruence i e for any quadruple a b c d of distinct points contained in a plane witha d a b b c c d a b c d is then called Lambert Saccheri quadrangle if a a c d c d then either a d singular case or a c d hyperbolic case or d c a elliptic case For an absolute plane A ful lling the h parallel axiom H we show that the congruence ofa general hyperbolic plane is a hyperbolic congruence by means of the notion of congruence singular or hyperbolic or elliptic we get now a complete characterization of the di erentpossibilities which can occur in a general absolute plane studying the value of the angle de ned in any Lambert Saccheri quadrangle or equivalently the sum of the angles ofany triangle This yields in particular a Archimedes free proof of a statement generalizingthe classical rst Legendre theorem for absolute planes Then the absolute planes arecharacterized by defect of an arbitrary triangle In the following we get two classi cations for general absolute planes with midlines ofany right triangle and measure of the segment joining two midpoints of sides of any trianglerespect to the measure of its third side Finally we characterize the general absolute geome tries with Lambert Saccheri quadrangles rst case with interior points of the angle andthe other two cases with diameters of any Lambert Saccheri quadrangle by using the metricfunction de ned in 12 1

قهرمان طاهريان

اعظم اعتماد دهكردي

محمدمهدي ابراهيمي،امير هاشمي