Chapter 7 - Chapter 7 Similarities Topics : 1....

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Unformatted text preview: Chapter 7 Similarities Topics : 1. Classification of Similarities 2. Equations for Similarities a106 a106 a106 a1 a1 a65 a65 a65 a65 a1 a1 a106 Copyright c circlecopyrt Claudiu C. Remsing, 2006. All rights reserved. 98 C.C. Remsing 99 7.1 Classification of Similarities The image of a triangle as seen through a “magnifying glass” is similar to the original triangle. For instance, the transformation ( x, y ) mapsto→ (2 x, 2 y ) is a “magnifying glass” for the Euclidean plane, multiplying all distances by 2. (We shall call such a mapping a stretch .) Some definitions We make the following definition. 7.1.1 Definition. If C is a point and r > 0, then a stretch (or homo- thety ) of ratio r about C is the transformation that fixes C and otherwise sends point P to point P prime , where P prime is the unique point on CP → such that CP prime = rCP (or, alternatively, where P prime is the unique point on ←→ CP such that CP prime = rCP ). We say that the point C is the centre and the (positive) factor r is the magnification ratio of the stretch. A stretch is also called a homothetic transformation . Note : We allow the identity transformation to be a stretch (of ratio 1 and any centre ). Observe, however, that we allow magnification ratios r ≤ 1, which is in slight conflict with the everyday meaning of the word “magnification”. Exercise 106 Verify that the set of all stretches with a given centre C forms a commutative group. There is nothing to stop us from allowing a negative ratio in the definition of a stretch. In this case, point P is taken to a point P prime lying on ←→ CP but on the other side of C from where P is located; that is, CP prime = rCP . Thus such a transformation is the product (in either order) of a stretch about C and a halfturn about the centre. This motivates the following definition. 100 M2.1 - Transformation Geometry 7.1.2 Definition. A dilation about point C is a stretch about C or else a stretch about C followed by a halfturn about C . Other transformations can be obtained by composing a stretch with any other transformation (e.g. an isometry). Two such special combinations will be given a name. 7.1.3 Definition. A stretch reflection is a nonidentity stretch about some point C followed by the reflection in some line through C . 7.1.4 Definition. A stretch rotation is a nonidentity stretch about some point C followed by a nonidentity rotation about C . Any of the above transformations are shape-preserving : they increase or decrease all lengths in the same ratio but leave shapes unchanged . We make the following definition. 7.1.5 Definition. If r > 0, then a similarity (or similitude ) of ratio r is a transformation α such that P prime Q prime = rPQ for all points P and Q , where P prime = α ( P ) and Q prime = α ( Q ) ....
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This note was uploaded on 12/26/2011 for the course MATH 466 taught by Professor Ytalu during the Spring '11 term at Middle East Technical University.

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Chapter 7 - Chapter 7 Similarities Topics : 1....

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