Homework 4|Due February 21 (Tuesday)1Read: 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) andradius 3.2. Prove that the hyperbolic circumference of a circle with hyperbolic radiusRis 2πsinh(R).3. Prove Proposition 5.1.4 in Stahl, that in hyperbolic geometry, every angle is congruent toan angle in standard position. Recall that in class we showed this for right angles, whichwas the hyperbolic analog of Euclid’s Postulate 4.4. LetPbe a point inH2, and let‘be a Euclidean line segment throughP. Prove that thereexists a unique hyperbolic geodesicgthat passes throughPand is tangent to‘. (In otherwords, prove that, given a pointPand a unit vectorv, there is a unique geodesic throughPthat leavesPin the direction ofv.) Also, give an explicit construction/description ofg, given‘, using either ruler-and-compass or analytic methods.
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