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Unformatted text preview: Malt H02. Pwﬂﬁew, get $3 Solution: / A M swears Problem 1. Show that if we stand near the circumference of a very large circle c (so 0 looks like
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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: (a dH(PQ)=’1nPBQA , 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 PQ “very high up”. Show that this trick works, 218. show that if you plugin points P, Q, A, and B, into the formula *
and take the limit I —> 00, the limit exists. The answer (limit) should be a nice formula for
dH(P, Q) in terms of m and y. (K121 = Let, n
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zitaxis. Let A = (0,1), B = (0,2), C = (1,1), P = (3.0,0.1), Q = (3.0, 0.2), R = (3.1,0.1): A“?
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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, zle. dH(A, B), dH(B, C),
and d H(C', A). Check that they satisfy the triangle inequality. 0. Fiﬁthﬂhree hyperbolic angles of the triangle AABC, Lee. 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. @ See above 6/}
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 Spring '09
 malkin

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