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Geometry

Geometry. Glide Reflections and Compositions. Goals. Identify glide reflections in the plane. Represent transformations as compositions of simpler transformations. Glide Reflection. A glide reflection is a transformation where a translation (the glide) is followed by a reflection.

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Geometry

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  1. Geometry Glide Reflections and Compositions

  2. Goals • Identify glide reflections in the plane. • Represent transformations as compositions of simpler transformations.

  3. Glide Reflection • A glide reflection is a transformation where a translation (the glide) is followed by a reflection. Line of Reflection

  4. Glide Reflection • A translation maps P onto P’. • A reflection in a line k parallel to the direction of the translation maps P’ to P’’. 1 2 Line of Reflection 3

  5. Example Find the image of ABC after a glide reflection. A(-4, 2), B(-2, 5), C(1, 3) Translation: (x, y)  (x + 7, y) Reflection: in the x-axis

  6. Find the image of ABC after a glide reflection. A(-4, 2), B(-2, 5), C(1, 3) Translation: (x, y)  (x + 7, y) Reflection: in the x-axis (-2, 5) (1, 3) (-4, 2)

  7. Find the image of ABC after a glide reflection. A(-4, 2), B(-2, 5), C(1, 3) Translation: (x, y)  (x + 7, y) Reflection: in the x-axis (5, 5) (-2, 5) (1, 3) (8, 3) (-4, 2) (3, 2)

  8. Find the image of ABC after a glide reflection. A(-4, 2), B(-2, 5), C(1, 3) Translation: (x, y)  (x + 7, y) Reflection: in the x-axis (5, 5) (-2, 5) (1, 3) (8, 3) (-4, 2) (3, 2) (3, -2) (8, -3) (5, -5)

  9. Find the image of ABC after a glide reflection. A(-4, 2), B(-2, 5), C(1, 3) Translation: (x, y)  (x + 7, y) Reflection: in the x-axis (5, 5) (-2, 5) Glide (1, 3) (8, 3) Reflection (-4, 2) (3, 2) (3, -2) (8, -3) (5, -5)

  10. You do it. • Locate these four points: • M(-6, -6) • N(-5, -2) • O(-2, -1) • P(-3, -5) • Draw MNOP O N P M

  11. O O N N P P M M You do it. • Translate by 0, 7.

  12. O N P M You do it. • Translate by 0, 7. O’ N’ P’ M’

  13. O N P M You do it. • Reflect over y-axis. O’ O’’ N’ N’’ P’ P’’ M’’ M’

  14. Compositions • A composition is a transformation that consists of two or more transformations performed one after the other.

  15. Composition Example • Reflect AB in the y-axis. • Reflect A’B’ in the x-axis. A A’ B’ B B’’ A’’

  16. Try it in a different order. • Reflect AB in the x-axis. • Reflect A’B’ in the y-axis. A B B’ B’’ A’’ A’

  17. The order doesn’t matter. A A’ B’ B B’ B’’ A’’ A’ This composition is commutative.

  18. Commutative Property • a + b = b + a • 25 + 5 = 5 + 25 • ab = ba • 4  25 = 25  4 • Reflect in y, reflect in x is equivalent to reflect in x, reflect in y.

  19. Are all compositions commutative? R’ R Rotate RS 90 CW. Reflect R’S’ in x-axis. S S’ S’’ R’’

  20. Reverse the order. R R’’ Reflect RS in the x-axis. Rotate R’S’ 90 CW. S S’’ S’ R’ All compositions are NOT commutative. Order matters!

  21. Compositions & Isometries • If each transformation in a composition is an isometry, then the composition is an isometry. • A Glide Reflection is an isometry.

  22. Example Reflect MN in the line y = 1. Translate using vector 3, -2. Now reverse the order: Translate MN using 3, -2. Reflect in the line y = 1. N M Both compositions are isometries, but the composition is not commutative.

  23. Summary • A Glide-Reflection is a composition of a translation followed by a reflection. • Some compositions are commutative, but not all.

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