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ps13_solutions - Malt H02 Pwflfiew get $3 Solution A M...

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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 a line to us) then the inversion in 0 looks to us as the reflection in a line. Mathematically, prove that iff—>0then :1 —> 1: it E: m .1; z a “L, 5-90 Y x P’X [HM PE * :r9 to0 * P il‘M/ersiom 0,!» P , Problem 2. Show that a dilation is a composition of two circle inversions. So given 0 and A you have to exhibit a pair of circles (centers and radéi) such that the composition of the inversions in the two circles is the dilation. gay we Jo iwgvamug gab. mam g Chi vacli's Y: 3?»; MM: the, (swarm: 5&3?sz Then a Pale—t P Wt 30 to Piaiiev Me Liz‘s}; mega/ma mist «méh «if; E” all» its gé‘isfxw.’ sue J wtwe 1 1 6?“: 12/ ¢\) 5 6;” 2 OP' 0?: '7. 7. it k r, 3 or» We,» = .3 a? at if.» go «Una {owjacsfti‘eu cal M534 lit/«2 iuvzrsf'oas is a Wall” mm ,\= ti“ Cm x we I . ‘1 Com choose r g r r1 l 1 YO 400% r _ PE.” X (luau), chokes). Heme a dilakw it as sampflgktfi“ a; 41W Swevaéawl {it we“? ways} 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: (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 plug-in 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 ““{kéflgg Lee 210’?) Problem 2. We use Cartesian coordinates on the 1/2-plane so that the boundary line is the zit-axis. 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“? n l A, I \w I I C\\\_ a :1!“ ‘ R: 1 P F :OR ~— — ) X 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/} For the. (allot/m5 questions we need la {mot 3&6, céwiei XE aw! KC gxphci'llvlg; («as TLQ 930(le *9; AC is 5‘; X‘s—L 0‘ its Vast: 5 ‘C ,i 1 I W , 1 m» A x (*(i) ,5 13:? 1' AAC I; L E) l x z i ,2 l!- \ D’ Z X x E ~J~ ~ # w A—‘v M ”l ~45. —- > {31 o )1 4 \R‘Ei X 2 I 2 To Land the: Ceulev oi, the. cit/file l: Wfl “W e‘él‘é" gym“, gag ifLiceLiov 0’} SC 9‘, Salv¢ “1g 33145:};sz «(by 4&5 garish; (X’O) 8 eqwolkslom; geek» Eaud C fix Media“ 2 me“; x . , -W\C 1 «Line Cate-#3:; i i x X1+H : I~QX+XH at? __\i '13 I& X = “1 =3 - 4 a f _ 5 4k tenet, o; E: is (4’0) H ; 4? 51 6" Wu: m W cer w WW ac we mm £2 (F2?! 0?) 0&1 (Aflé‘)? Hugak: {h1g1 04g 0: (E. €13: TREE“ 5W WJ:+\9+ $4 I i M pfiwfi) ‘W J92“ ' K V V a “V: ,‘ © AhgieS Lereeu Céwiéfs :1 au5§€§ éeéweeu {aufigmi hum {#Q‘Cugwg L49 Paola) (A A ,~ 4 l , 0 xx Amtrak 7;;3kafl22163 ;‘ i: \X \\ , £6; \ «fl 4 FLA K.” a i K n _, 5 e Lanai: miwaz I! I ’2, V\\'\ M “ , ‘7 a“ -. W ~i O defeci “Swag/3;: 180°»{é30fe‘5 24%» 0.39% :> @ Avea {AAEC‘} : Véd‘mS/L' HQPKEC}; (‘4)1‘030" {[19 Maia: garage; 9; iLe Emholm A MAJ/L. 9.1M (0_§‘3’O_63é’[email protected]€);9 936.? ”01%) fa Avea (AAgC) ls veawmaéfa‘ . 0) 53 rag @ AU Chm/{Q8 @ch Mew/m agmaQa—iécfi wiiia mm ave. 7% —sm¥e:( MGM? a‘ ‘ 53/43:: > Hey/g £33352? ff aAg . , . . « l . {fiwgggawims} await, amauwevvah “a ...
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