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ps13b - Math402 Problem Set 13 Second Part This problem set...

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Unformatted text preview: Math402 Problem Set 13 Second Part This problem set is about the Hyperbolic Plane of radius —-1 and its 1/2-plane model/map. I would like numerical answers in Problem 2 — use your calculator. Problem 1. In class we defined the hyperbolic distance dH(P, Q) between two points P and Q: Q r , r~ 3‘. \ / \ (*) dH(PQ)— _ llnm— PBQA 9.. \, A It: __6:_.__.-..._. *0“-.. This definition does not work if P and Q are on the same vertical line since then there is only one intersection point A of the hyperbolic line through P and Q with the boundary line. The idea in this case is to take the second intersection point B to be 00. Of course we cannot use 00 in PB and QB, so we take B to be a point on the vertical line (P-Q “very high up”. Show that this trick works, 216. show that if you plug—in points P, Q, A, and B, into the formula * and take the limit l, —-> 00, the limit exists. The answer (limit) should be a nice formula for dH(P, Q) in terms of x and y. Problem 2. We use Cartesian coordinates on the 1/2-plane so that the boundary line is the x—axis. Let A = (0,1), B = (0,2), C = (1,1), P = (3.0, 0.1), Q = (30,02), R = (3.1,0.1): we; a: A f o C <1. PuR Don’t reuse the picture, redraw it for each question below. a. Draw the hyperbolic triangle AABC. b. Find the hyperbolic lengths of the three sides of the triangle AABC, i.€. dH(A, B), dH(B, C), and d H(C’ A) Check that they satisfy the triangle inequality. c. Find the three hyperbolic angles of the triangle AABC, 2'. e the angles between the hyperbolic lines AB, BC, and CA Check that they add to less than 180°. d. Find the hyperbolic area of the triangle AABC. e. Find the hyperbolic lengths of the three sides of the triangle APQR. ...
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