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Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations...

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Transformational Transformational Geometry Geometry Math 314 Math 314
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Page 1: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Transformational Transformational GeometryGeometry

Math 314Math 314

Page 2: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Game PlanGame Plan

DistortionsDistortions OrientationsOrientations Parallel PathParallel Path TranslationTranslation RotationRotation ReflectionReflection

Page 3: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Game Plan Con’tGame Plan Con’t

Combination – Glide ReflectionCombination – Glide Reflection CombinationsCombinations Single Isometry Single Isometry Similtudes Dilutation Similtudes Dilutation Series of TranformationSeries of Tranformation

Page 4: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

TransformationTransformation Any time a figure is moved in the plane we call this a Any time a figure is moved in the plane we call this a

transformation.transformation. As mathematicians we like to categorize these As mathematicians we like to categorize these

transformations. transformations. The first category we look at are the ugly ones or The first category we look at are the ugly ones or

distortions distortions Transformation Formula Transformation Formula Format (x, y) Format (x, y) (a, b) The (a, b) The

old x becomes aold x becomes a old y becomes bold y becomes b

Page 5: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

ExamplesExamples

Eg (x,y) Eg (x,y) (x + y, x – y) (x + y, x – y) A (2, -5) A (2, -5) K (-4, 6) K (-4, 6) Eg #2 (x,y) Eg #2 (x,y) (3x – 7y, 2x + 5) (3x – 7y, 2x + 5) B (-1, 8) B (-1, 8)

(-3, 7) A’

(2, -10) K’

(-59, 3) B’

Page 6: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Using a GraphUsing a Graph

Let’s try one on graph paperLet’s try one on graph paper Consider A (1,4) B (7,2) C (3, –1)Consider A (1,4) B (7,2) C (3, –1) (x,y) (x,y) (x + y, x – y) (x + y, x – y) Step 1: Calculate the new pointsStep 1: Calculate the new points Step 2: Plot the points i.e A A’ B B’ etc. Step 2: Plot the points i.e A A’ B B’ etc. A (1,4) A (1,4) (5, -3) A’ (5, -3) A’ B (7,2) B (7,2) (9,5) B’ (9,5) B’ C (3 – 1) C (3 – 1) (2,4) C’ (2,4) C’

Page 7: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

1 2 3 4 5-1-2-3-4-5

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Ex#1: Put on Graph PaperEx#1: Put on Graph Paper

A C’

B

A’

B’

CNotice, this graph is off the page… make sure yours does not

(x,y) (x,y) (x+y, x-y) (x+y, x-y)A (1,4) A (1,4) (5,-3) A’ (5,-3) A’B (7,2) B (7,2) (9,5) B’ (9,5) B’C (3,–1) C (3,–1) (2,4) C’ (2,4) C’

Formula BoxFormula Box

Page 8: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

OrientationOrientation

To examine figures, we need to To examine figures, we need to know how they line up.know how they line up.

We are concerned with We are concerned with

Clockwise (CW)

Counterclockwise (CCW)

Page 9: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Orientation Orientation

Consistency is KeyConsistency is Key Start with A go ccwStart with A go ccw EgEg A

C’B’CBOrientation ABC and A’ B’ C’ Orientation is the same

A’

Page 10: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Orientation Con’tOrientation Con’t

A

B’C’CB

What happened to the orientation?

Orientation has changed

A’

Page 11: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Orientation VocabularyOrientation Vocabulary

Orientation the Orientation the same… orsame… or

preservedpreserved unchangedunchanged constantconstant

Orientation Orientation changed orchanged or

not preservednot preserved changedchanged not constantnot constant

Page 12: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Parallel PathsParallel Paths

When we move or transform an When we move or transform an object, we are interested in the object, we are interested in the path the object takes. To look at path the object takes. To look at that we focus on paths taken by that we focus on paths taken by the vertices the vertices

Page 13: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Parallel PathParallel Path

These are a parallel path

A

C’

C

B’

B

A’

We say line AA’ is a path

We say a transformation where all the vertices’ paths are parallel, the object has experienced a parallel path

Page 14: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Parallel PathParallel Path

A

BC

B’

A’

C’

These are not parallel paths

It is called Intersecting Paths

Page 15: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Parallel PathParallel PathA

B’

CA’

C’B

Which two letters form a parallel path? If you choose A, must go with A’; B with B’ etc.

Solution: A + C

Do stencil #1-3

Page 16: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

IsometryIsometry

It is a transformation where a It is a transformation where a starting figure and the final figure starting figure and the final figure are congruent.are congruent.

Congruent: equal in every aspect Congruent: equal in every aspect (side and angle)(side and angle)

Page 17: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Isometry ExampleIsometry Example

16 12

CB

A

9

24 6

32PT

K

Since 16 = 24 = 32

6 9 12

8/3 = 8/3 = 8/3

Are these figures congruent?

Page 18: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

TranslationTranslation Sometimes called a slide or glideSometimes called a slide or glide Formula Formula tt (a,b)(a,b)

Means (x,y) Means (x,y) (x + a, y + b) (x + a, y + b) Eg Eg tt (-3,4) (-3,4) Eg Given A (7,1) B (3,5) C(4,-1)Eg Given A (7,1) B (3,5) C(4,-1) Draw t Draw t (-3,4) (-3,4) Include formula box and Include formula box and

type box on graphtype box on graph Type box means label and answer Type box means label and answer

orientation (same / changed) orientation (same / changed) Parallel Path (yes / no)Parallel Path (yes / no)

Page 19: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

1 2 3 4 5-1-2-3-4-5

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Given A (7,1) B (3,5) C(4,-1) Draw Given A (7,1) B (3,5) C(4,-1) Draw tt (-3,4) (-3,4) Given A (7,1) B (3,5) C(4,-1) Draw Given A (7,1) B (3,5) C(4,-1) Draw tt (-3,4) (-3,4)

A

C’

BA’B’

C

(x,y) (x,y) (x-3, y+4) (x-3, y+4)A (7,1) A (7,1) (4,5) A’ (4,5) A’B (3,5) B (3,5) (0,9) B’ (0,9) B’C (4,–1) C (4,–1) (1,3) C’ (1,3) C’

Formula BoxFormula Box

Type Box

Orientation – same

Parallel Path - yes

Page 20: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

RotationRotation

In theory we need a rotation pointIn theory we need a rotation point An angleAn angle A directionA direction In practice – we use the origin as the In practice – we use the origin as the

rotation pointrotation point Angles of 90Angles of 90°° and 180 and 180°° Direction cw and ccwDirection cw and ccw Note in math counterclockwise is positiveNote in math counterclockwise is positive

Page 21: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

RotationRotation FormulaFormula r (0, v ) r (0, v )

Rotation Origin Angle & DirectionRotation Origin Angle & Direction r (0, -90r (0, -90°°) means a rotation about the ) means a rotation about the

origin 90origin 90°° clockwise clockwise (x,y) (x,y) (y, -x) (y, -x) When x becomes -x it changes sign. When x becomes -x it changes sign.

Thus – becomes +; + becomes –Thus – becomes +; + becomes – Notice the new position of x and y.Notice the new position of x and y.

Page 22: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

RotationRotation

r (0, 90) means rotation about the r (0, 90) means rotation about the origin 90origin 90°° counterclockwise counterclockwise

(x,y) (x,y) (-y, x) (-y, x) r (0, 180) means rotation about r (0, 180) means rotation about

the origin (direction does not the origin (direction does not matter)matter)

(x,y) (x,y) (-x, -y) (-x, -y)

Page 23: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Rotation PracticeRotation Practice

Given A (-4,2) B (-2,4) C (-5,5) Given A (-4,2) B (-2,4) C (-5,5) Draw r (0,90); include formula box Draw r (0,90); include formula box

on graphon graph You try it on a graph!You try it on a graph!

Page 24: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

1 2 3 4 5-1-2-3-4-5

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

r (0, 90) A (-4,2) B (-2,5) C(-5,-5)r (0, 90) A (-4,2) B (-2,5) C(-5,-5)

A

C’

B

A’

B’

C

(x,y) (x,y) (-y, x) (-y, x) A (-4,2) A (-4,2) (-2,-4) A’ (-2,-4) A’B (-2,5) B (-2,5) (-5,-2) B’ (-5,-2) B’C (-5,-5) C (-5,-5) (5,-5) C’ (5,-5) C’

Orientation – same

Parallel Path - no

Page 25: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

ReflectionsReflections

In theory we need a reflection lineIn theory we need a reflection line SSxx = reflection over x axis = reflection over x axis (x,y) (x,y) (x, -y) (x, -y) SSyy = reflection over y axis = reflection over y axis (x,y) (x,y) (-x,y) (-x,y) S reflection over y = xS reflection over y = x (x,y) (x,y) (y,x) (y,x) S reflection over y = -x S reflection over y = -x (x,y) (x,y) (-y,-x) (-y,-x)

Page 26: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Memory AidMemory Aid

It is very important to put all these It is very important to put all these formulas on one page. formulas on one page.

P 160 #7 Put on separate sheetP 160 #7 Put on separate sheet P161 #9P161 #9 You should be able to do all these You should be able to do all these

transformation and understand how transformation and understand how they work. they work.

Page 27: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Combination NotationCombination Notation

When we perform two or more When we perform two or more transformations we use the symbol transformations we use the symbol °°

It means afterIt means after A ° B A ° B Means A after BMeans A after B

t t (-3,2(-3,2) ° ) ° SSyy means means

A translation after a reflection (you must A translation after a reflection (you must start backwards!)start backwards!)

Page 28: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Combination Glide ReflectionCombination Glide Reflection Draw t Draw t (-3,2) (-3,2) ° ° SSyy

(x,y) (x,y) (-x,y) (-x,y) (x-3, y+2) (x-3, y+2) A (4,3) A (4,3) A (4,3) A (4,3) C (-1,2) C (-1,2) (1,2) C’ (1,2) C’ (-2,4) C’’ (-2,4) C’’ Orientation changed, Parallel Path noOrientation changed, Parallel Path no What kind of isometry is this? It is a GLIDE What kind of isometry is this? It is a GLIDE

REFLECTIONREFLECTION Let us look at the four types of isometriesLet us look at the four types of isometries

(-4,3) A’ (-4,3) A’ (-7,5) A’’ (-7,5) A’’

B (1,-3) B (1,-3) (-1,-3) (-1,-3) B’ (-4,-1) B’’ B’ (-4,-1) B’’

Page 29: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Single IsometrySingle Isometry Any transformation in the plane that Any transformation in the plane that

preserves the congruency can be preserves the congruency can be defined by a single isometry.defined by a single isometry.

Orientation Same?Orientation Same? Parallel Path?Parallel Path?

YES

YES

YES

No

No

No

TRANSLATION

ROTATION

REFLECTION

GLIDE REFLECTION

Page 30: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Table RepresentationTable Representation

Orientation Orientation Same Same (maintained)(maintained)

Orientation Orientation Different Different (changed)(changed)

With Parallel With Parallel PathPath

TranslationTranslation ReflectionReflection

Without Without Parallel PathParallel Path

RotationRotation Glide Glide ReflectionReflection

Page 31: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Similtudes & Dilitations Similtudes & Dilitations When a transformation changes the size of When a transformation changes the size of

an object but not its shape, we say it is a an object but not its shape, we say it is a similtude or a dilitation. similtude or a dilitation.

Note – we observe size by side length and Note – we observe size by side length and shape by angles shape by angles

The similar shape we will create will have the The similar shape we will create will have the same angle measurement and the sides will same angle measurement and the sides will be proportional. be proportional.

The 1The 1stst part we need is this proportionality part we need is this proportionality constant or scale factor. constant or scale factor.

Page 32: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Similtudes & DilitationsSimiltudes & Dilitations

The 2The 2ndnd part we need is a point from part we need is a point from which this increase or decrease in size which this increase or decrease in size will occur.will occur.

Note – this is an exercise in measuring Note – this is an exercise in measuring so there can be some variationso there can be some variation

Consider transform ABC by a factor Consider transform ABC by a factor of 2 about point 0 (1,5).of 2 about point 0 (1,5).

The scale factor is sometimes called kThe scale factor is sometimes called k

Page 33: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Similtudes & DilitationsSimiltudes & Dilitations Sign of the scale factorSign of the scale factor Positive – both figures (original & new) Positive – both figures (original & new)

are on the same side of pointare on the same side of point Negative – both figures (original and Negative – both figures (original and

new) are on the opposite sides of pointnew) are on the opposite sides of point The point is sometimes called the hole The point is sometimes called the hole

pointpoint

Page 34: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

1 2 3 4 5-1-2-3-4-5

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

h ((1,5),2)h ((1,5),2)

A

C’

B

A’

B’

C

mOA=2 mOA=2 moA’=2x2=4moA’=2x2=4

Page 35: Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation.

Other ExamplesOther Examples

P23 Example #8P23 Example #8 P24 Spider Web P24 Spider Web

Discuss scale factor Discuss scale factor

Beam or light beam methodBeam or light beam method


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