Use the following argument to show that in the Poincaré model, two points P and Q inside the unit circle with center O determine a unique (hyperbolic) line. Consider the case in the picture below, where P and Q are not collinear with O. (See picture.) Let P' be on ray OP so that OP'=1/OP. Then P, Q and P' are not collinear and the there exists a unique circle C passing through them. You need to show C is perpendicular to the unit circle. To do this, show that the tangent from O to C has length 1, so that point of tangency is also on the unit circle. So at the intersection of the circles, the radius OR is tangent to C. Use that fact to explain why the circles are perpendicular. Unit circle

Use the following argument to show that in the Poincaré model, two points P and Q inside the unit circle with center O determine a unique (hyperbolic) line. Consider the case in the picture below, where P and Q are not collinear with O. (See picture.) Let P' be on ray OP so that OP'=1/OP. Then P, Q and P' are not collinear and the there exists a unique circle C passing through them. You need to show C is perpendicular to the unit circle. To do this, show that the tangent from O to C has length 1, so that point of tangency is also on the unit circle. So at the intersection of the circles, the radius OR is tangent to C. Use that fact to explain why the circles are perpendicular. Unit circle

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.4: Plane Curves And Parametric Equations

Problem 61E

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