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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 the exercise in Geogebra and then solve: Lemma 1: If d(OB) = d, then OB = (ed-1), where e is the base of the natural logarithms +1 and r is the radius of C. Let c be a point in the interior of C. Prove that a circle centered at point c in the hyperbolic sense is represented in the Poincaré model by a Euclidean circle whose center is other than C to unless C=O, where O is the center of C. Suggestion: Make the case C-O first and use Lemma 1. Then the set of all circles centered at O is transformed into the set of all circles with hyperbolic center at C through the reflections in the Poincaré line perpendicular to the bisector of the Poincaré segment OC.

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 the exercise in Geogebra and then solve: Lemma 1: If d(OB) = d, then OB = (ed-1), where e is the base of the natural logarithms +1 and r is the radius of C. Let c be a point in the interior of C. Prove that a circle centered at point c in the hyperbolic sense is represented in the Poincaré model by a Euclidean circle whose center is other than C to unless C=O, where O is the center of C. Suggestion: Make the case C-O first and use Lemma 1. Then the set of all circles centered at O is transformed into the set of all circles with hyperbolic center at C through the reflections in the Poincaré line perpendicular to the bisector of the Poincaré segment OC.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.2: Ellipses
Problem 67E
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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 the exercise in Geogebra and then solve: Lemma 1: If d(OB) = d, then OB and r is the radius of C. = r(ed-1) ed +1 2 where e is the base of the natural logarithms Let c be a point in the interior of C. Prove that a circle centered at point c in the hyperbolic sense is represented in the Poincaré model by a Euclidean circle whose center is other than C to unless C=O, where O is the center of C. Suggestion: Make the case C=O first and use Lemma 1. Then the set of all circles centered at O is transformed into the set of all circles with hyperbolic center at C through the reflections in the Poincaré line perpendicular to the bisector of the Poincaré segment OC. Please be as clear as possible. Show and explain all the steps of the proof. Thank you a lot.
Transcribed Image Text: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 the exercise in Geogebra and then solve: Lemma 1: If d(OB) = d, then OB and r is the radius of C. = r(ed-1) ed +1 2 where e is the base of the natural logarithms Let c be a point in the interior of C. Prove that a circle centered at point c in the hyperbolic sense is represented in the Poincaré model by a Euclidean circle whose center is other than C to unless C=O, where O is the center of C. Suggestion: Make the case C=O first and use Lemma 1. Then the set of all circles centered at O is transformed into the set of all circles with hyperbolic center at C through the reflections in the Poincaré line perpendicular to the bisector of the Poincaré segment OC. Please be as clear as possible. Show and explain all the steps of the proof. Thank you a lot.
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