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Unformatted text preview: Chapter 2 Translations and Halfturns Topics : 1. Translations 2. Halfturns Copyright c circlecopyrt Claudiu C. Remsing, 2006. All rights reserved. 24 C.C. Remsing 25 2.1 Translations Let E 2 be the Euclidean plane. 2.1.1 Definition. A translation (or parallel displacement ) is a map ping τ : E 2 → E 2 , ( x, y ) mapsto→ ( x + h, y + k ) . We use to say that such a translation τ has equations x prime = x + h y prime = y + k. Given any two of ( x, y ) , ( x prime , y prime ) , and ( h, k ), the third is then uniquely deter mined by this last set of equations. Hence, a translation is a transformation. Note : We shall use the Greek letter tau only for translations. 2.1.2 Proposition. Given points P and Q , there is a unique translation τ P,Q taking P to Q . Proof : Let P = ( x P , y P ) and Q = ( x Q , y Q ). Then there are unique numbers h and k such that x Q = x P + h and y Q = y P + k. So the unique translation τ P,Q that takes P to Q has equations x prime = x + x Q x P y prime = y + y Q y P . a50 By the proposition above, if τ P,Q ( R ) = S , then τ P,Q = τ R,S for points P, Q, R, S . 26 M2.1  Transformation Geometry Note : The identity is a special case of a translation as ι = τ P,P for each point P. 2.1.3 Corollary. If τ P,Q ( R ) = R for point R , then P = Q . 2.1.4 Proposition. Suppose A, B, C are noncollinear points. Then τ A,B = τ C,D if and only if a50 CABD is a parallelogram. Proof : The translation τ A,B has equations x prime = x + x B x A y prime = y + y B y A . Then the following are equivalent : (1) τ A,B = τ C,D . (2) D = τ A,B ( C ). (3) D = ( x D , y D ) = ( x C + x B x A , y C + y B y A ). (4) 1 2 ( A + D ) = 1 2 ( B + C ) · (5) a50 CABD is a parallelogram. a50 Exercise 26 Prove the equivalence (3) ⇐⇒ (4). Exercise 27 What happens (in Proposition 2.1.4 ) if we drop the requirement that the points A, B, C are noncollinear ? It follows that a translation moves each point the same distance in the same direction . For nonidentity translation τ A,B , the distance is given by AB and the direction by (the directed line segment)→ AB . C.C. Remsing 27 Note : The translation τ A,B can be identified with the (geometric) vector v = bracketleftBigg x B x A y B y A bracketrightBigg where A = ( x A , y A ) and B = ( x B , y B ). A vector is really the same thing as a translation, although one uses different phraseologies for vectors and translations. It may be helpful to make this idea more precise. What is a vector ? The school textbooks usually define a vector as a “quantity having magnitude and direction”, such as the velocity vector of an object moving through space (in our case, the Eu clidean plane). It is helpful to represent a vector as an “arrow” attached to a point of the space. But one is not supposed to think of the vector as being firmly rooted just at one point. For instance, one wants to add vectors, and the recipe for doing this...
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 Spring '11
 ytalu
 Geometry, NonEuclidean Geometry, Euclidean geometry, Euclidean space, Claudiu C. Remsing

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