Compositions of Rigid Transformations
Objectives: Find and classify a composition of reflections; identify glide reflections.
Theorem 8-1
The composition of two or more rigid transformations is a rigid
transformation.
Vocabulary
A glide reflection is the composition of a glide (translation) and a
reflection across a line parallel to the direction of translation.
A computer can translate an image and then
reflect it, or vice versa. The two rabbit images are
glide reflection images of each other.
Compositions of Rigid Transformations
Objectives: Find and classify a composition of reflections; identify glide reflections.
Compositions of Rigid Transformations
Objectives: Find and classify a composition of reflections; identify glide reflections.
A translation or rotation is a composition of two reflections.
Theorem 8-2
A composition of reflections across two parallel lines is a translation.
Compositions of Rigid Transformations
Theorem 8-3
A composition of reflections across two intersecting lines is a rotation.
The center of rotation is the point where the lines intersect,
and the angle is twice the angle formed by the intersecting lines.
So the letter D is rotated 86° clockwise, or 274° counterclockwise,
with the center of rotation at point A.
Compositions of Rigid Transformations
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a.
Draw J between parallel lines / and m. What is the image of 𝑅𝑚 ◦ 𝑅𝑙
What is the distance of the resulting translation?
𝐽 ?
The image is a translation. The distance is twice the distance between l and m.
b. Use the results of part (a) and Problem 1. Make a conjecture about the distance of any
translation that is the result of a composition of reflections across two parallel lines.
Conjecture: The distance between the preimage and the image is twice the
distance between the lines of reflection.
Compositions of Rigid Transformations
The center of rotation is C (point of intersection).
The angle of rotation is 90 clockwise.
Compositions of Reflections
Objectives: Use a composition of reflections; identify glide reflections.
ABC has vertices A(4, 5), B(6, 2), and C(0, 0). Find the image of ABC for a glide
reflection where the translation is (x, y) →(x, y + 2) and the reflection line is x = 1.
Compositions of Reflections
Objectives: Use a composition of reflections; identify glide reflections.
a. Reflect the letter R across a and then b. Describe
the resulting rotation.
R rotates clockwise through the angle shown by the green
arrow. The center of rotation is C and the measure of the
angle is twice m1.
Compositions of Reflections
Objectives: Use a composition of reflections; identify glide reflections.
b. Find the image of R for a reflection across line l and then
across line m. Describe the resulting translation.
R is translated the distance and direction shown by the green
arrow. The arrow is perpendicular to lines l and m with length
equal to twice the distance from l to m.
Compositions of Reflections
Objectives: Use a composition of reflections; identify glide reflections.
a. Find the image of TEX under a glide reflection where
the translation is (x, y) →(x + 1, y) and the reflection
line is y = 2. Draw the translation first, then the
reflection.
b. Would the result of part (a) be the same if you
reflected TEX first, and then translated it? Explain.
Yes; if you reflected it and then
moved it right, the result would
be the same.