9.4 Composition of Transformations

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9.4 Composition of Transformations Then: You drew reflections, translations, and rotations. Now: 1. Draw glide reflections and other compositions of isometries in the coordinate plane. 2. Draw compositions of reflections in parallel and intersecting lines. https://plus.maths.org/issue38/features/livio/figure5.jpg

9.4 Composition of Transformations I. Definition: The resulting transformation when a transformation is applied to a figure and then another transformation is applied to its image. Artwork by Escher http://cs.brown.edu/courses/csci1950-g/results/proj5/spotaszn/images/escher.jpg

9.4 Composition of Transformations II. Types of Composition of Transformations A. Glide reflection: the composition of a translation followed by a reflection in a line parallel to the translation vector. http://rhsweb.org/assignments/Goldsmith/Geometry/Diagrams/Transformations/Glide-Reflection-2.gif

9.4 Composition of Transformations B. Graph a Glide Reflection 1. Quadrilateral BGTS has vertices B(–3, 4), G(–1, 3), T(–1 , 1), and S(–4, 2). Graph BGTS and its image after a translation along 5, 0 and a reflection in the x-axis. B’ (_____, _____) B” (_____, _____) G’ (_____, _____) G” (_____, _____) T’ (_____, _____) T” (_____, _____) S’ (_____, _____) S” (_____, _____)

9.4 Composition of Transformations B. Graph a Glide Reflection 2. Triangle XYZ has vertices X(6, 5), Y(7, –4) and Z(5, –5). Graph △XYZ and its image after a translation along 〈1, 2〉 and a reflection in the y-axis. X’ (_____, _____) X” (_____, _____) Y’ (_____, _____) Y” (_____, _____) Z’ (_____, _____) Z” (_____, _____)

9.4 Composition of Transformations 3. From Problem #2, XYZ  X’Y’Z’ and X’Y’Z  X”Y”Z”, so by Transitive Property of Congruence, then XYZ  X”Y”Z”. 4. Glide reflections, reflections, translations, and rotations are the only four rigid motions or isometries in a plane. 5. Theorem 9.1 Composition of Isometries The composition of two (or more) isometries is an isometry. http://mathbitsnotebook.com/Geometry/Transformations/comptr.jpg

9.4 Composition of Transformations C. Graph Other Compositions of Isometries 1. ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph TUV and its image after a translation along –1 , 5 and a rotation 180° about the origin. T’ (_____, _____) T” (_____, _____) U’ (_____, _____) U” (_____, _____) V’ (_____, _____) V” (_____, _____)

9.4 Composition of Transformations C. Graph Other Compositions of Isometries 2. ΔJKL has vertices J(2, 3), K(5, 2), and L(3, 0). Graph ΔJKL and its image after a translation along 3, 1 and a rotation 90° about the origin. J’ (_____, _____) J” (_____, _____) K’ (_____, _____) K” (_____, _____) L’ (_____, _____) L” (_____, _____)

9.4 Composition of Transformations D. Composition of Two Reflections 1. Composition of two reflection in parallel lines is the same as a translation.

9.4 Composition of Transformations D. Composition of Two Reflections 2. Composition of two reflections in intersecting lines is the same as a rotation.

9.4 Composition of Transformations E. Describe the transformations that are combined to create each pattern. 1. Carpet pattern: http://www.publicdomainpictures.net/pictures/120000/nahled/seamless-carpet-pattern-1425828344vll.jpg

9.4 Composition of Transformations E. Describe the transformations that are combined to create each pattern. 2. Tile pattern: http://cdn.shopify.com/s/files/1/0094/1122/products/6268_exotic_tile_patterns_wall_floor_stencils_to_paint.jpg?v=1456446006

9.4 Composition of Transformations E. Describe the transformations that are combined to create each pattern. 3. Wallpaper pattern: https://s-media-cache-ak0.pinimg.com/736x/d5/34/35/d53435de3a39ccd9154141e4af49fd36.jpg

9.4 Composition of Transformations Assignment: P. 655-657 #8-16 evens, 21-25 all, 27-29 all, 31-35 all, 38, 45-48 all Graph: 8-16 evens, 25 & 27 (2 pieces of graph paper)