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OBJECTIVE

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

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

IMPORTANT

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

DEFINITION

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That is

R (l)

l

d

2d

m

• R (m)

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Direction

Translation

Distance

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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.

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an isometry, and

is a direct transformation, and

has no fixed points.

A translation is

Theorems

Proof: Use the components of a translation.

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

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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.

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Play TimeConsider 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.

x = 2

A

B C

A’

B’C’

A”

B” C”

Write this transformation with algebraic notation. i.e.x’ = f (x, y) y’ = f (x, y)

x

y

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

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Play TimeConsider 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”

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Play TimeConsider 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”

l l is any arbitrary line perpendicular to PP’.

m

m is a line parallel to l and the distance from l to m is ½ the distance from P to P”.

P’

H

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

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Rotations

θ

O

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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.

Definition

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That is

R (O, θ ) = R (l) • R (m)

l

m

θ

θ /2

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an isometry, and

is a direct transformation, and

has one fixed points.

A rotation is

Theorems

Proof: Use the components of a rotation.

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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.

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Play TimeReflect ABC in l and then in m.

l

m

θ

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produce the necessary transformation. They are not unique.

Given a rotation you should be able to construct the two lines whose reflections

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m

θ/2

l

H

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

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Lemma

The only isometry that has three noncollinear fixed points is the identity mapping e, that fixes all points.

What about three collinear points?

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Lemma: The only isometry with three fixed points is the identity mapping e.

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Three fixed points; A, B, C. Prove: T = e.

(1) T(A)=A’, T(B)=B’, T(C)=C’ Given.(2) AP = AP’, BP = BP’, CP = CP’, Def of isometry

(3) P = P’ Congruent triangles.

(4) T is the identity map e. Def of identity map.

What is given? What will we prove?

Why?

Why?

Why?

Why?

QED

A = A’

P

C = C’

B = B’

Let P be any point in the plane. We will show P’ = P

Note: You need all three of these distances. Why?

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Glide Reflections

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DefinitionA 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.

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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.

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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.

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* Transformations in General *

Theorem. Given any two congruent triangles, ΔABC and ΔPQR, there exist a unique isometry that maps one triangle onto the other.

A B

CP Q

R

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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.

Theorem 1

Proof: It will suffice to show you how to find this isometry. It will be the product of line reflections which are all isometries.

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

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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”.

Theorem 1

If it is an indirect transformation the following method will always work.

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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.

A B

C P

Q

R

B’

C’

A’

Theorem 1

Do you see that it is an indirect transformation?

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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.

Theorem 2: Fundamental Theorem of Isometries.

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

Corollary

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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!

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

Corollary

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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!

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Assignment

Bring graph paper for the next three classes.

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Assignments T3 & T4 & T5