This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ) ....
View
Full
Document
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.
 Spring '08
 Staff
 Math, Geometry

Click to edit the document details