Plane Isometries and the Complex Numbers
نویسنده
چکیده
The length of a vector v = ( x y ) in R2 is || ( x y ) || = √ x2 + y2, and the distance between two vectors v and w is the length of their difference: ||v − w||. A function h : R2 → R2 is called an isometry when it preserves distances: ||h(v)− h(w)|| = ||v −w|| for all v and w in R2. Examples of isometries include translation by ( 1 −1/2 ) , a rotation by 40 degrees counterclockwise around the origin, and reflection across the line y = 1− 2x. The effect of these isometries on some line segments is illustrated in the figures below.
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