s12_mthsc481_hw03

# s12_mthsc481_hw03 - Homework 3 | Due February 7 (Tuesday) 1...

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Homework 3 | Due February 7 (Tuesday) 1 Read : Stahl, Chapters 4.1, 4.2, 4.3. 1. In this problem, we will prove Proposition 3.1.3 in Stahl in a more rigorous and elegant fashion, using only the analytic methods we developed in class. (a) Let f be a Euclidean isometry, let p 0 be a circle, and let f ( p 0 ) = p 1 . Prove that if I C,k maps p 0 to a circle, then f I C,k f - 1 maps p 1 to a circle. (b) Let f be a Euclidean isometry, let 0 be a line, and let f ( 0 ) = 1 . Prove the if I C,k maps 0 to a line, then f I C,k f - 1 maps 1 to a line. (c) Prove that if p is a circle and q is either a circle or a line, then there exists g Isom( E 2 ) such that g ( p ) is centered at the origin, and g ( q ) is orthogonal to the x -axis. (d) Prove that if I O,k maps circles and lines orthogonal to the x -axis to circles and lines, then for any C E 2 , I C,k maps circles and lines to circles and lines. Draw the appropriate commutative diagram illustrating your argument. (e) Let

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## This note was uploaded on 03/11/2012 for the course MTHSC 481 taught by Professor Staff during the Spring '12 term at Clemson.

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s12_mthsc481_hw03 - Homework 3 | Due February 7 (Tuesday) 1...

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