mbhaub@mpsaz.org

Geometry

7.5 Glide Reflections and Compositions

Tue, Dec 8, 11:56 AM 7.5 Glide Reflections and Compositions 2

Goals

Identify glide reflections in the plane.

Represent transformations as compositions of simpler transformations.

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Glide Reflection

A glide reflection is a transformation where a translation (the glide) is followed by a reflection.

Line of Reflection

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Glide Reflection

1. A translation maps P onto P’.2. A reflection in a line k parallel to

the direction of the translation maps P’ to P’’.

Line of Reflection3

1 2

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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

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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

(-4, 2)

(-2, 5)

(1, 3)

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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

(-4, 2)

(-2, 5)

(1, 3)

(5, 5)

(3, 2)(8, 3)

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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

(-4, 2)

(-2, 5)

(1, 3)

(5, 5)

(3, 2)(8, 3)

(5, -5)

(3, -2)(8, -3)

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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

(-4, 2)

(-2, 5)

(1, 3)

(5, 5)

(3, 2)(8, 3)

(5, -5)

(3, -2)(8, -3)

Glide

Reflection

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

M

NO

P

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You do it. Translate by 0, 7.

M

NO

PM

NO

P

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You do it. Translate by 0, 7.

M

NO

P

M’

N’

O’

P’

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You do it. Reflect over y-axis.

M

NO

P

M’

N’

O’

P’

M’’

N’’O’’

P’’

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Compositions

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

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Composition Example

A

B

1.Reflect AB in the y-axis.

2.Reflect A’B’ in the x-axis.

A’

B’

A’’

B’’

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Try it in a different order.

A

B

1.Reflect AB in the x-axis.

2.Reflect A’B’ in the y-axis.

A’

B’

A’’

B’’

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The order doesn’t matter.

A

B

A’

B’

A’’

B’’

A’

B’

This composition is commutative.

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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.

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Are all compositions commutative?

Rotate RS 90 CW.

Reflect R’S’ in x-axis.

R

S

R’

S’

R’’

S’’

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Reverse the order.

Reflect RS in the x-axis.

Rotate R’S’ 90 CW.

R

S

R’

S’

R’’

S’’

All compositions are NOT commutative. Order matters!

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Compositions & Isometries

If each transformation in a composition is an isometry, then the composition is an isometry.

A Glide Reflection is an isometry.

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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.

MN

Both compositions are isometries, but the composition is not commutative.

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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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Homework