MATH 3181 - 001 FALL 1996

Homework Assignment 5
Due Wednesday, October 30, 1996

Let A and B be two points in the Klein model of the hyperbolic plane and let P and Q denote the ideal points on the circle of radius 1 that complete the Klein line. You can consider the points A and B with complex coordinates z and w, then show that P and Q have complex coordinates and , where t and u are roots of a quadratic equation expressing the fact that P and Q lie on the unit circle. Find the coefficients D, E, and F and show that
In order to express the Klein length d_k(A,B) in terms of the coordinates (a₁,a₂) of A and (b₁,b₂) of B, prove that with a suitable ordering of the ends P and Q of the Klein line through A and B you have the formula

HINT: Use Problem 1.
Let A=(0,0), B=(0,½), the circle of radius 1, and let l be the diameter of cut out by the x-axis.
1. Find the Klein length d_k(A,B)
2. Find the coordinates of the point M on the segment AB that represents its midpoint in the Klein model.
3. Find the equation of the locus of points whose perpendicular Klein distance from l equals d_k(A,B).
In the Klein model construct a pentagon in the hyperbolic plane with five right angles.
Use our formula for the Klein length to derive a proof of the Bolyai-Lobachevsky formula for the angle of parallelism:

HINT: Take the vertex of the angle of parallelism to be the center O and show that the Klein distance corresponding to is given by

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