Commute : if 2 transformations S,T happen to have the property ST=TS Commutative : a collection of transformations in which every pair commute
Reflections: Rm The fundamental type of motion. m Q A A^m AQ = QA^m m : line of reflection for a A^m : it own image. A : is the reflection of A^m with respect to Q Q: point of reflection
Reflections: Rm … R R R m m Rm = (Rm) -1 RmRm = I A reflection: ( or flip) is an isometry in which a figure and its images have opposite orientations. Isometry : ( or motion) A transformation T of the entire plane onto itself, it length is invariant under T.
Reflections preserve : collinearity, betweeness of points m S X T Y U Z
Reflections preserve : Angle measure and distance measure y x A B’ B C C’ A’ ABC ≡ A’B’C’ ≡ Proposition 9.5
Isometries As Products of Reflections <ul><li>The four Euclidean isometries: </li></ul><ul><li>Reflection </li></ul><ul><li>Translation </li></ul><ul><li>Rotation </li></ul><ul><li>Glide reflection </li></ul>
Translation and Reflection X units Translate 2X units to the right m n <ul><li>Translation is equivalent to the composition of 2 reflections, one across m and the other across n </li></ul><ul><li>- A composition of reflection in 2 parallel lines is a translation </li></ul>Proposition 9.12. Given a line t, the set of translation along t is a commutative group
Proposition 9.7 A motion T = I is a rotation if and only if T has exactly one fixed point
Rotation and Reflection m n C A B < ACB = 2 < mCn - Rotation is then a composition of the 2 reflections over m and n - A composition of reflections in 2 intersecting lines is a rotation Proposition 9.8
Proposition 9.9: Given a point A, the set of rotations about A is a commutative group.
Glide and Reflection A glide reflection: is the composition of translation and a reflection in a line parallel to glide vector X units Translate 2X units to the right m n