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Course: PMATH 360, Fall 2009
School: W. Alabama
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360 Set PMath 9 9 9.1 More reection and inversion Poincar reection e Let be the boundary circle of a Poincar model of the hyperbolic plane. e Let O be the center of . Let A and B be two P-points in this hyperbolic plane. Suppose A, B, and O are not collinear. 9.1.1 P-line (A, B) Find the construction steps of the P-line on A and B following these steps: Find the circle on A and B that is orthogonal to . Call...

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360 Set PMath 9 9 9.1 More reection and inversion Poincar reection e Let be the boundary circle of a Poincar model of the hyperbolic plane. e Let O be the center of . Let A and B be two P-points in this hyperbolic plane. Suppose A, B, and O are not collinear. 9.1.1 P-line (A, B) Find the construction steps of the P-line on A and B following these steps: Find the circle on A and B that is orthogonal to . Call it . Find the arc on that is interior to . (*) Hints: (i) The P-line lies on a circle that is on A and B and is orthogonal to . (ii) The segment connecting the centers of and meets in a point that is on the P-line. (iii) Show the end points of the arc as open circles. (iv) Before dening the macro, hide all the objects other than the three input objects (, A and B) and your three output objects, the arc and its end-points. Get experience by experimenting with your macro by creating several lines. Construct some parallel lines. Construct several non-intersecting line pairs. 9.1.2 P-pbis (A, B) Find the construction for a second P-line so that the P-reection with respect to it maps A to B and maps B to A. (*) Hints: Since the P-reection maps A to B and B to A, we see that P-line(A,B) P-line(B,A) = P-line(A,B). Any circle that inverts A to B must have its center on (the Euclidean) line(A,B). There is another pair of points, U and V, on P-line(A,B) that, like A and B, must also change places. They give another lien on the center. Find a circle with such a center that is orthogonal to and hence also to . When you nd your construction, use Cabri tools to verify/conrm to your satisfaction that your work is right. answer Your will include the construction of the arc on that is inside . 1 of 3 PMath 360 Set 9 9.2 9.2.1 Inversions in the complex plane H1 , f1 Let be the circle in the complex plane with center 0 and radius 10. 9.2.1.1 Find H1 , the Hermitian matrix of . (*) 9.2.1.2 Use the numbers in the matrix H1 to nd a function f1 so that the the formula w = f1 (z) represents inversion with respect to . (*) 9.2.1.3 Verify that the xed points (**) of f1 are exactly the points that satisfy the equation determined by the matrix H1 . That is, z = f1 (z) (z, 1)H1 (, 1)t = 0. (*) z 9.2.2 H2 , f2 Let be circle with center (-4, -3) and radius 5. Let H2 be the matrix, (and f2 be the function) representing . Do the same three tasks for H2 and f2 that you did for H1 and f1 in 9.2.1. (*) 9.2.3 The product 9.2.3.1 Convince yourself that the product of the three inversions, rst with respect to , then with respect to , and then with respect to again, might be expected to be a reection with respect to a line. 9.2.4 9.2.4.1 (*) The composition f1 f2 f1 Find (and simplify) the composition f3 (z) = f1 (f2 (f1 (z))). 2 of 3 PMath 360 Set 9 9.2.4.2 (*) Use the function f3 to write the corresponding matrix H3 . 9.2.4.3 Fixed points points of a line. (*) Verify that the xed points (**) of f3 are the 2008-07-04 (*) Items marked with an asterisk should be submitted for marking. (**) We say z is a xed point of f if and only if z = f (z). The reason is that we sometimes think of a function f as causing or representing motion. Something get moved from one place to another. If something does not move, it has a xed location. 3 of 3
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W. Alabama - PMATH - 360
PMath 360Set 777.17.1.1Inversion Constructions, experienceThe inverse of a pointConstructionLet be a circle, let O be its centre, and let P be any point distinct from O. The problem is to Let L1 be the line OP. Let M be a point on . Let
W. Alabama - PMATH - 360
PMath 360Solutions to Set 999.1More reection and inversionPoincar reection eLet be the boundary circle of a Poincar model of the hyperbolic plane. e Let O be the center of . Let A and B be two P-points in this hyperbolic plane. Suppose A,
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PMath 360Sample Final1999Instructions:1. There are 8 questions. 2. Show your work in the space provided. 3. If you need more space, continue your work on the back of the previous page or on the blank page at the end, but indicate in the space
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W. Alabama - PMATH - 360
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W. Alabama - PMATH - 360
PMath 360Sample Final20001. Homogeneous Coordinates Let P, Q, R, and S be points given by P : p = (p1 , p2 , p3 ), Q : q = (q1 , q2 , q3 ), R : r = (r1 , r2 , r3 ) and S : s = (s1 , s2 , s3 ). For each of the following geometric statements abou
W. Alabama - PMATH - 360
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W. Alabama - PMATH - 360
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W. Alabama - PMATH - 360
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W. Alabama - PMATH - 360
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W. Alabama - PMATH - 360
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W. Alabama - PMATH - 360
PMath 360Set 555.1Conics, poles and polarsA particular conicLet be the conic whose Cartesian equation is 2x2 + 3y 2 + 2x 4y = 0. 1. Find the 3 by 3 symmetric matrix M that represents . 2. Show that the point A : (8, 4, 5.5) is on . 3. Fin
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PMath 360Set 101010.110.1.1Train, Necklace and PorismTrain of circlesFoundation lineLet L be any line. A horizontal line will do. Hide any point(s) used to dene L. Let P1 and P2 be any two, new, distinct points on L. Let and L1 and L2 be
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The Radical Centre of Three CirclesThe Radical Centre of three circles is the unique point in the plane for which the power of that point with respect to all three circles is the same. The radical centre is the intersection of the three radical axes
W. Alabama - MATH - 348
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