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Unformatted text preview: S UMMARY OF 10/20 L ECTURE We began by reviewing what a rigid motion (or isometry ) of the plane is. We will denote the usual cartesian plane by R 2 . A map f : R 2 R 2 is an isometry if it preserves distances, i.e. if for any two points x and y in the plane, d ( f ( x ) ,f ( y ) ) = d ( x , y ) . Here d ( a , b ) denotes the usual distance between the two points a and b . To be more specific, if a = ( a 1 ,a 2 ) and b = ( b 1 ,b 2 ) , then d ( a , b ) = p ( b 1- a 1 ) 2 + ( b 2- a 2 ) 2 . The composition of any two isometries is an isometry, and this motivates proving that the set of all isometries is a group under composition. We came up with some examples of isometries: 1. The trivial or identity map, which sends every point to itself. 2. Rotations (around any point in the plane) 3. Reflections (about any fixed line in the plane) 4. Translations (along any fixed line in the plane) 5. Glide reflections (wrt any fixed line in the plane: first reflect over the line, then translate along the same line)...
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- Fall '10