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Chapter 3 - Chapter 3 Reections and Rotations Topics 1...

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Chapter 3 Reflections and Rotations Topics : 1. Equations for a Reflection 2. Properties of a Reflection 3. Rotations a1 a1 a1 a1 a1 a1 a65 a65 a65 a65 a65 a65 Copyright c circlecopyrt Claudiu C. Remsing, 2006. All rights reserved. 38
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C.C. Remsing 39 3.1 Equations for a Reflection A reflection will be defined as a transformation leaving invariant every point of a fixed line L and no other points. (An optical reflection along L in a mirror having both sides silvered, would yield the same result.) We make the following definition. 3.1.1 Definition. Reflection σ L in line L is the mapping σ L : E 2 E 2 , P mapsto→ P , if point P is on L Q , if point P is off L and L is the perpendicular bisector of PQ. The line L is usually referred to as the mirror of the reflection. Note : We do not use the word reflection to denote the image of a point or of a set of points. A reflection is a transformation and never a set of points. Point σ L ( P ) is the image of point P under the reflection σ L . 3.1.2 Proposition. Reflection σ L is an involutory transformation that interchanges the halfplanes of L . Reflection σ L fixes point P if and only if P is on L . Reflection σ L fixes line M pointwise if and only if M = L . Reflection σ L fixes line M if and only if M = L or M ⊥ L . Proof : It follows immediately from the definition that σ L negationslash = ι but σ 2 L = ι as the perpendicular bisector of PQ is the perpendicular bisector of QP . Hence, σ L is onto as σ L ( P ) is the point mapped onto the given point P since σ L ( σ L ( P )) = P for any point P . Also, σ L is one-to-one as σ L ( A ) = σ L ( B ) implies A = σ L ( σ L ( A )) = σ L ( σ L ( B )) = B .
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40 M2.1 - Transformation Geometry Therefore, σ L is an involutory transformation. Then, from the definition of σ L , it follows that σ L interchanges the halfplanes of L . Note : In fact, any involutory mapping (on E 2 ) is a transformation (and hence an involution). Clearly, σ L fixes point P if and only if P is on L . Not only does σ L fix line L , but σ L fixes every point on L . Note : In general, transformation α is said to fix pointwise set S of points if α ( P ) = P for every point P in S ; that is, if α leaves invariant (unchanged) every point in S . Observe the difference between fixing a set and fixing a set pointwise. Every line perpendicular to L is fixed by σ L , but none of these lines is fixed pointwise as each contains only one fixed point. Suppose line M is distinct from L and is fixed by σ L . Let Q = σ L ( P ) for some point P that is on M but off L . Then P and Q are both on M since M is fixed, and L is the perpendicular bisector of PQ . Hence, L and M are perpendicular. a50 Exercise 41 Show that if the nonidentity mapping α : A A is involutory (i.e. α 2 is the identity mapping), then it is invertible.
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