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Unformatted text preview: Chapter 5 Isometries II Topics : 1. Even and Odd Isometries 2. Classification of Isometries 3. Equations for Isometries a1 a1 a1 a1 a1 a65 a65 a65 a65 a65 Copyright c circlecopyrt Claudiu C. Remsing, 2006. All rights reserved. 65 66 M2.1  Transformation Geometry 5.1 Even and Odd Isometries A product of two reflections is a translation or a rotation. By considering the fixed points of each, we see that neither a translation nor a rotation can be equal to a reflection. Thus, for lines L , M , N σ N σ M negationslash = σ L . When a given isometry is expressed as a product of reflections, the number of reflections is not invariant . Although the product of two reflections cannot be a reflection, we know that in some cases a product of three reflections is a reflection. (We shall see this is possible only because both 3 and 1 are odd integers.) We make the following definitions. 5.1.1 Definition. An isometry that is a product of an even number of reflections is said to be even . 5.1.2 Definition. An isometry that is a product of an odd number of reflections is said to be odd . Note : It is intuitively clear that the product of an even number of reflections preserves the sense of a clockwise oriented circle in the plane, whereas the product of an odd number of reflections reverses it. We say that even isometries are orientation preserving and that odd isometries are orientationreversing isometries. We shall refer to the property of an isometry of being even or odd as the parity . But is this concept “welldefined” ? Observe that, since an isometry is a product of reflections, an isometry is even or odd . Of course, no integer can be both even and odd, but is it not conceivable some product of ten reflections could equal to some product of seven reflections ? We shall show this is impossible. Exercise 71 Show that if P is a point and A and B are lines, then there are lines C and D with C passing through P such that σ B σ A = σ D σ C . C.C. Remsing 67 Based on this simple fact, we can now prove the following 5.1.3 Proposition. A product of four reflections is a product of two re flections. Proof : Suppose product σ S σ R σ Q σ P is given. We want to show this product is equal to a product of two reflections. Let P a point on line P . There are lines Q prime and R prime such that σ R σ Q = σ R prime σ Q prime with P on Q prime . Also, there are lines R primeprime and M such that σ S σ R prime = σ M σ R primeprime with P on R primeprime . Since P , Q prime , R primeprime are concurent at P , then there is a line L such that σ R primeprime σ Q prime σ P = σ L . Therefore, σ S σ R σ Q σ P = σ S σ R prime σ Q prime σ P = σ M σ R primeprime σ Q prime σ P = σ M σ L ....
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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.
 Spring '11
 ytalu
 Geometry, Equations

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