Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations...
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Transcript of Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations...
Transformational Transformational GeometryGeometry
Math 314Math 314
Game PlanGame Plan
DistortionsDistortions OrientationsOrientations Parallel PathParallel Path TranslationTranslation RotationRotation ReflectionReflection
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
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
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’
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’
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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
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)
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’
Orientation Con’tOrientation Con’t
A
B’C’CB
What happened to the orientation?
Orientation has changed
A’
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
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
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
Parallel PathParallel Path
A
BC
B’
A’
C’
These are not parallel paths
It is called Intersecting Paths
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
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)
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?
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)
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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
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
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.
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)
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!
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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
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)
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.
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!)
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’’
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
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
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.
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
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
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h ((1,5),2)h ((1,5),2)
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mOA=2 mOA=2 moA’=2x2=4moA’=2x2=4
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