Non Euclidian Geometry Poincaré model (Perpendiculars and inversion of circles) To solve the exercise it is necessary to use definitions, theorems and lemmas in addition to the inversion properties of circles. Draw each of the exercises in Geogebra and then solve: Given the following theorems: Theorem 1: Given a circle C with center at O , let A, B, P, Q be four distinct points of C and denote by A´, B´, P´, Q their inverse points in C respectively. So (AB,PQ) = (A'B',P'Q') . Theorem 2: Let C be a circle with center at O and radius r and D a circle with center at C and radius s. Suppose that O is exterior to D, that p is the power of O with respect to D, and that k=r^2/p. Then D is transformed under the inversion of the circle C into the circle D' of radius 'ks' with center in the image C* of C under the dilation of O with radius k. Also, if P is any point at D and P´ its inverse with respect to D, then the tangent line t´ to D' at P´ is the reflection through the bisector of the segment PP´ of the line tangent D to P . Let L be a Poincaré line that is not a diameter of C, then L is an arc of a circle D orthogonal to C. Using Theorems 1 and 2 show that the reflections through L are represented in the Poincaré model by inversions with respect to D Please be as clear as possible. Show and explain all the steps.Thank you a lot.
Non Euclidian Geometry
Poincaré model (Perpendiculars and inversion of
To solve the exercise it is necessary to use definitions, theorems and lemmas in addition to the inversion properties of circles. Draw each of the exercises in Geogebra and then solve:
Given the following theorems:
Theorem 1: Given a circle C with center at O , let A, B, P, Q be four distinct points of C and denote by A´, B´, P´, Q their inverse points in C respectively. So (AB,PQ) = (A'B',P'Q') .
Theorem 2: Let C be a circle with center at O and radius r and D a circle with center at C and radius s. Suppose that O is exterior to D, that p is the power of O with respect to D, and that k=r^2/p. Then D is transformed under the inversion of the circle C into the circle D' of radius 'ks' with center in the image C* of C under the dilation of O with radius k. Also, if P is any point at D and P´ its inverse with respect to D, then the tangent line t´ to D' at P´ is the reflection through the bisector of the segment PP´ of the line tangent D to P .
Step by step
Solved in 3 steps
