s12_mthsc481_hw04

# s12_mthsc481_hw04 - Homework 4 | Due February 21(Tuesday 1...

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Homework 4 | Due February 21 (Tuesday) 1 Read : Stahl, Chapters 4.4., 5.1, 5.2, 5.3, 5.4. 1. Find the Euclidean center and radius of the circle that has hyperbolic center (5 , 4) and radius 3. 2. Prove that the hyperbolic circumference of a circle with hyperbolic radius R is 2 π sinh( R ). 3. Prove Proposition 5.1.4 in Stahl, that in hyperbolic geometry, every angle is congruent to an angle in standard position. Recall that in class we showed this for right angles, which was the hyperbolic analog of Euclid’s Postulate 4. 4. Let P be a point in H 2 , and let be a Euclidean line segment through P . Prove that there exists a unique hyperbolic geodesic g that passes through P and is tangent to . (In other words, prove that, given a point P and a unit vector v , there is a unique geodesic through P that leaves P in the direction of v .) Also, give an explicit construction/description of g , given , using either ruler-and-compass or analytic methods.
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