s12_mthsc481_hw01

# s12_mthsc481_hw01 - 6 Prove that any pair of lines that...

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Homework 1 | Due January 24 (Tuesday) 1 Read : Stahl, Chapters 1.1, 1.2, 1.3, 1.4, 2.1. 1. Let X be a geometry, and let f and g be isometries of X . (a) Prove that f is invertible and f - 1 is an isometry of X . (b) Prove that f g is an isometry of X . For problems 2 6 , you are only allowed to use our analytic deﬁnition of Euclidean geom- etry and the results that we have derived from it. 2. Prove that Euclidean length of a path is invariant under orientation-reversing change of parameter (recall that we did the orientation-preserving case in class). 3. Prove that the Euclidean angle between two paths at a given point of intersection is well-deﬁned, that is, it is invariant under orientation-preserving change of parameter. 4. Prove that the translation τ v is a Euclidean isometry that preserves lines. 5. Prove that any two points in E 2 can be moved to two points on the x -axis using a sequence of rotations around the origin and translations.
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Unformatted text preview: 6. Prove that any pair of lines that meet at an angle θ can be moved by a Euclidean isometry onto any other pair of lines that meet at an angle θ . For problems 7 – 8 , you may use both analytic and ruler-and-compass methods. 7. Prove that if two rigid motions agree on two distinct points, then they agree everwhere on the line joining those two points. 8. Let τ v be a translation, and let R C,α be a rotation around a point C ∈ E 2 . (a) Describe, geometrically, the isometry f := τ v ◦ R C,α ◦ τ-1 v . This is the unique map that makes the following diagram commute: E 2 R C,α / τ v ± E 2 τ v ± E 2 f / _ _ _ _ _ _ E 2 What do R C,α and τ v ◦ R C,α ◦ τ-1 v have in common? (b) Express R C,α as a composition of translations and rotations around the origin. MthSc 481 | Topics in Geometry and Topology | Spring 2012 | M. Macauley...
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