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OBJECTIVE

OBJECTIVE. The student will learn the basic concepts of translations, rotations and glide reflections. IMPORTANT. Reflections are the building blocks of other transformations. We will use the material from the previous lesson on line reflections to create new transformations. 2. DEFINITION.

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OBJECTIVE

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  1. OBJECTIVE The student will learn the basic concepts of translations, rotations and glide reflections.

  2. IMPORTANT Reflections are the building blocks of other transformations. We will use the material from the previous lesson on line reflections to create new transformations. 2

  3. DEFINITION A translation, is the product of two reflections R (l) and R (m) where l and m are parallel lines.

  4. That is R (l) • R (m) d l m P P P 2d

  5. Distance Direction Translation

  6. If you draw a figure on a piece of paper and then slide the paper on your desk along a straight path, your slide motion models a translation. In a translation, points in the original figure move an identical distance along parallel paths to the image. In a translation, the distance between a point and its image is always the same. A distance and a direction together define a translation.

  7. Theorems A translation is an isometry, and is a direct transformation, and has no fixed points. Proof: Use the components of a translation.

  8. Given two parallel lines you should be able to construct the translation of any set of points and describe that translation.

  9. Play Time Consider the y-axis and the line x = 2 as lines of reflection. Find the image of the ABC if A (- 3, -1), B (- 3, 3) and C (0, 3) reflected in the y-axis and then x = 2.

  10. C C’ C” B B’ B” A A’ A” x = 2 Play Time Consider the y-axis and the line x = 2 as lines of reflection. Find the image of the ABC if A (- 3, -1), B (- 3, 3) and C (0, 3) reflected in the x-axis and then x = 2. y x Write this transformation with algebraic notation. i.e. x’ = f (x, y) y’ = f (x, y)

  11. Given a translation you should be able to construct the two lines whose reflections produce the necessary transformation. They are not unique.

  12. Play Time Consider the point P and its image P”, find two lines l and m so that the reflection in l followed by the reflection in m moves P to P”. P P”

  13. l is any arbitrary line perpendicular to PP’. l m P’ m is a line parallel to l and the distance from l to m is ½ the distance from P to P”. Play Time Consider the point P and its image P”, find two lines l and m so that the reflection in l followed by the reflection in m moves P to P’. P P” H

  14. You should be able to do the previous construction using a straight edge and a compass.

  15. Rotations P θ P O

  16. Definition A rotation is the product of two line reflections R (l) • R (m), where l and m are not parallel. The center of the rotation is O = lm . The direction of the rotation is about O from l toward m, and the angular distance of rotation is twice the angle from l to m.

  17. That is R (O, θ ) = R (l) • R (m) m P θ P l θ /2 P

  18. Theorems A rotation is an isometry, and is a direct transformation, and has one fixed points. Proof: Use the components of a rotation.

  19. Given two lines that are not parallel you should be able to reflect a set of points in the first line and then again in the second line.

  20. θ Play Time Reflect ABC in l and then in m. m l

  21. Given a rotation you should be able to construct the two lines whose reflections produce the necessary transformation. They are not unique.

  22. l θ/2 m H

  23. You should be able to do the previous construction using a straight edge and a compass.

  24. Lemma The only isometry that has three noncollinear fixed points is the identity mapping e, that fixes all points. What about three collinear points?

  25. Lemma: The only isometry with three fixed points is the identity mapping e. B = B’ P C = C’ A = A’ What is given? Three fixed points; A, B, C. Prove: T = e. What will we prove? Let P be any point in the plane. We will show P’ = P (1) T(A)=A’, T(B)=B’, T(C)=C’ Why? Given. (2) AP = AP’, BP = BP’, CP = CP’, Why? Def of isometry Note: You need all three of these distances. Why? (3) P = P’ Why? Congruent triangles. (4) T is the identity map e. Why? Def of identity map. QED 25

  26. Glide Reflections P P P P P P 26

  27. Definition A glide reflection, G (l, AB) is the product of a line reflection R (l)and a translation T (AB) in a direction parallel to the axis of reflection. That is, AB‖l. P P P 27

  28. This combination of reflection and translation can be repeated over and over: reflect then glide, reflect then glide, reflect then glide, etc. An example of this is the pattern made by someone walking in the sand. This calls for a field trip to Ocean City. The same line of reflection is used to reflect each figure to a new position followed by a glide of a uniform distance. 28

  29. 29

  30. Theorems A glide reflection is an isometry, and is an opposite transformation, and there are no invariant points under a glide reflection. Proof: Use the components of a glide reflection. 30

  31. * Transformations in General * C P Q A B R Theorem. Given any two congruent triangles, ΔABC and ΔPQR, there exist a unique isometry that maps one triangle onto the other.

  32. Theorem 1 Given three points A, B, and C and their images P, Q, and R, there exist a unique isometry that maps these points onto their images. Proof: It will suffice to show you how to find this isometry. It will be the product of line reflections which are all isometries.

  33. Focus – You will need to do this for homework and for the test.

  34. Theorem 1 If it is a direct transformation then it is a translation or a rotation. We can use the methods shown previously in “Rotation” or “Translation”. If it is an indirect transformation the following method will always work.

  35. Theorem 1 Q R C P A B B’ C’ A’ Prove that given three points A, B, and C and their images P, Q, and R, there exist a unique isometry that maps these points onto their images. Do you see that it is an indirect transformation?

  36. Theorem 2: Fundamental Theorem of Isometries. The previous proof shows that every isometry on the plane is a product of at most three line reflections; exactly two if the isometry is direct and not the identity.

  37. Corollary A nontrivial direct isometry is either a translation or a rotation.

  38. Corollary: A nontrivial direct isometry is either a translation or a rotation. What do we know? It is a direct isometry. It is a product of two reflections. If the two lines of reflection meet then it is a rotation! If the two lines of reflection do not meet (they are parallel) it is a translation!

  39. Corollary A nontrivial indirect isometry is either a reflection or a glide reflection.

  40. Corollary: A nontrivial indirect isometry is either a reflection or a glide reflection. What do we know? It is an indirect isometry. It is a product of one or three reflections. If one it is a reflection! If the three line reflection then it could be either a reflection or a glide reflection!

  41. Assignment Bring graph paper for the next three classes.

  42. Assignments T3 & T4 & T5

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