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Chapter 5 - Chapter 5 Isometries II Topics 1 Even and Odd...

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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

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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 orientation-reversing isometries. We shall refer to the property of an isometry of being even or odd as the parity . But is this concept “well-defined” ? 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 . a50 Note : Not only are there lines such that the given product of four reflections is equal to σ M σ L , but our proof even tells us how to find such lines.

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Chapter 5 - Chapter 5 Isometries II Topics 1 Even and Odd...

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