LectureNotes2U - Math 497C Sep 9, 2004 1 Curves and...

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Unformatted text preview: Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 1.5 Isometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics. We say that a mapping f : M 1 M 2 is an isometry provided that d 1 ( p, q ) = d 2 ( f ( p ) , f ( q ) ) , for all pairs of points in p , q M 1 . An orthogonal transformation A : R n R n is a linear map which preserves the inner product, i.e.,- A ( p ) , A ( q ) = h p, q i for all p , q R n . One may immediately check that an orthogonal transfor- mation is an isometry. Conversely, we have: Theorem 1. If f : R n R n is an isometry, then f ( p ) = f ( o ) + A ( p ) , where o is the origin of R n and A is an orthogonal transformation. Proof. Let f ( p ) := f ( p ) f ( o ) . We need to show that f is a linear and h f ( p ) , f ( q ) i = h p, q i . To see the latter note that h x y, x y i = k x k 2 + k y k 2 2 h x, y i . Thus, using the definition of f , and the assumption that f is an isometry, we obtain 2- f ( p ) , f ( q ) = k f ( p ) k 2 + k f ( q ) k 2 k f ( p ) f ( q ) k 2 = k f ( p ) f ( o ) k 2 + k f ( q ) f ( o ) k 2 k f ( p ) f ( q ) k 2 = k p k 2 + k q k 2 k p q k 2 = 2 h p, q i . 1 Last revised: September 17, 2004 1 Next note that, since f preserves the inner product, if e i , i = 1 . . . n , is an orthonormal basis for R n , then so is f ( e i ). Further,- f ( p + q ) , f ( e i ) = h p + q, e i i = h p, e i i + h q, e i i =- f ( p ) , f ( e i ) +- f ( q ) , f ( e i ) =- f ( p ) + f ( q ) , f ( e i ) . Thus if follows that f ( p + q ) = f ( p ) + f ( q ) . Similarly, for any constant c ,- f ( cp ) , f ( e i ) = h cp, e i i =- c f ( p ) , f ( e i ) , which in turn yields that f ( cp ) = f ( p ), and completes the proof f is linear. If f : R n R n is an isometry with f ( o ) = o we say that it is a rotation , and if A = f f ( o ) is identity we say that f is a translation . Thus another way to state the above theorem is that an isometry of the Euclidean space is the composition of a rotation and a translation. Any mapping f : R n R m given by f ( p ) = q + A ( p ), where q R n , and A is any linear transformation, is called an ane map with translation part q and linear part A . Thus yet another way to state the above theorem is that any isometry f : R n R n is an ane map whose linear part is orthogonal An isometry of Euclidean space is also referred to as a rigid motion . Recall that if A T denotes the transpose of matrix A , then- A T ( p ) , q =- p, A ( q ) ....
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This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

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LectureNotes2U - Math 497C Sep 9, 2004 1 Curves and...

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