1. Transformational Geometry
Math 314

2. Game Plan Distortions Orientations Parallel Path Translation Rotation
Reflection

3. Game Plan Con’t Combination – Glide Reflection Combinations
Single Isometry
Similtudes Dilutation
Series of Tranformation

4. Transformation Transformation Formula Format (x, y)  (a, b) The
Any time a figure is moved in the plane we call this a transformation.
As mathematicians we like to categorize these transformations.
The first category we look at are the ugly ones or distortions 
Transformation Formula
Format (x, y)  (a, b) The
old x becomes a
old y becomes b

5. Examples Eg (x,y)  (x + y, x – y) A (2, -5)  K (-4, 6)  (-3, 7) A’
B (-1, 8) 
(-3, 7) A’
(2, -10) K’
(-59, 3) B’

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

7. Ex#1: Put on Graph Paper Formula Box (x,y)  (x+y, x-y)C’
Formula Box
(x,y)  (x+y, x-y)
A (1,4)  (5,-3) A’
B (7,2)  (9,5) B’
C (3,–1)  (2,4) C’ B
Notice, this graph is off the page… make sure yours does not  C A’

8. Orientation To examine figures, we need to know how they line up.
We are concerned with
Clockwise (CW)
Counterclockwise (CCW)

9. Orientation Consistency is Key Start with A go ccw Eg A’ A B C B’ C’
Orientation ABC and A’ B’ C’
Orientation is the same

10. Orientation Con’t A A’ B C B’ C’ What happened to the orientation?
Orientation has changed

11. Orientation Vocabulary
Orientation the same… or
preserved
unchanged
constant
Orientation changed or
not preserved
changed
not constant

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

13. Parallel Path
We say a transformation where all the vertices’ paths are parallel, the object has experienced a parallel path A’ A B’ C’ B C
We say line AA’ is a path
These are a parallel path

14. It is called Intersecting Paths
Parallel Path A’ A C’ B’ B C
These are not parallel paths
It is called Intersecting Paths

15. Parallel Path Solution: A + CB
Which two letters form a parallel path? If you choose A, must go with A’; B with B’ etc. B’ A’ C
Solution: A + C
Do stencil #1-3 

16. Isometry
It is a transformation where a starting figure and the final figure are congruent.
Congruent: equal in every aspect (side and angle)

17. Isometry Example Are these figures congruent? Since 16 = 24 = 32K A 16 12 6 24 B C 32 T 9 P
Since 16 = 24 = 32
8/3 = 8/3 = 8/3
Are these figures congruent?

18. Translation Sometimes called a slide or glide Formula t (a,b)
Means (x,y)  (x + a, y + b)
Eg t (-3,4) 
Eg Given A (7,1) B (3,5) C(4,-1)
Draw t (-3,4) Include formula box and type box on graph
Type box means label and answer orientation (same / changed)
Parallel Path (yes / no)

19. Type Box Orientation – same Parallel Path - yes
Given A (7,1) B (3,5) C(4,-1) Draw t (-3,4) A’ B’ B
Formula Box
(x,y)  (x-3, y+4)
A (7,1)  (4,5) A’
B (3,5)  (0,9) B’
C (4,–1)  (1,3) C’ C’ A
Type Box C
Orientation – same
Parallel Path - yes

20. Rotation In theory we need a rotation point An angle A direction
In practice – we use the origin as the rotation point
Angles of 90° and 180°
Direction cw and ccw
Note in math counterclockwise is positive

21. Rotation Formula r (0, v ) Rotation Origin Angle & Direction
r (0, -90°) means a rotation about the origin 90° clockwise
(x,y)  (y, -x)
When x becomes -x it changes sign. Thus – becomes +; + becomes –
Notice the new position of x and y.

22. Rotation
r (0, 90) means rotation about the origin 90° counterclockwise
(x,y)  (-y, x)
r (0, 180) means rotation about the origin (direction does not matter)
(x,y)  (-x, -y)

23. Rotation Practice Given A (-4,2) B (-2,4) C (-5,5)
Draw r (0,90); include formula box on graph
You try it on a graph!

24. r (0, 90) A (-4,2) B (-2,5) C(-5,-5) Orientation – same
(x,y)  (-y, x)
A (-4,2)  (-2,-4) A’
B (-2,5)  (-5,-2) B’
C (-5,-5)  (5,-5) C’ A
Orientation – same
Parallel Path - no B’ C C’ A’

25. Reflections In theory we need a reflection line
Sx = reflection over x axis
(x,y)  (x, -y)
Sy = reflection over y axis
(x,y)  (-x,y)
S reflection over y = x
(x,y)  (y,x)
S reflection over y = -x
(x,y)  (-y,-x)

26. Memory Aid It is very important to put all these formulas on one page.
P 160 #7 Put on separate sheet
P161 #9
You should be able to do all these transformation and understand how they work.

27. Combination Notation
When we perform two or more transformations we use the symbol °
It means after
A ° B
Means A after B
t (-3,2) ° Sy means
A translation after a reflection (you must start backwards!)

28. Combination Glide Reflection
Draw t (-3,2) ° Sy
(x,y)  (-x,y)  (x-3, y+2)
A (4,3) 
C (-1,2)  (1,2) C’  (-2,4) C’’
Orientation changed, Parallel Path no
What kind of isometry is this? It is a GLIDE REFLECTION
Let us look at the four types of isometries
(-4,3) A’  (-7,5) A’’
B (1,-3)  (-1,-3)  B’ (-4,-1) B’’

29. Single Isometry
Orientation Same? Parallel Path?
Any transformation in the plane that preserves the congruency can be defined by a single isometry.
TRANSLATION
YES
YES No
ROTATION
YES
REFLECTION No
GLIDE REFLECTION No

30. Table Representation Orientation Same (maintained)
Orientation Different (changed)
With Parallel Path
Translation
Reflection
Without Parallel Path
Rotation
Glide Reflection

31. Similtudes & Dilitations
When a transformation changes the size of an object but not its shape, we say it is a similtude or a dilitation.
Note – we observe size by side length and shape by angles
The similar shape we will create will have the same angle measurement and the sides will be proportional.
The 1st part we need is this proportionality constant or scale factor.

32. Similtudes & Dilitations
The 2nd part we need is a point from which this increase or decrease in size will occur.
Note – this is an exercise in measuring so there can be some variation
Consider transform ABC by a factor of 2 about point 0 (1,5).
The scale factor is sometimes called k

33. Similtudes & Dilitations
Sign of the scale factor
Positive – both figures (original & new) are on the same side of point
Negative – both figures (original and new) are on the opposite sides of point
The point is sometimes called the hole point

34. h ((1,5),2) A B A’ C B’ C’
mOA=2
moA’=2x2=4

35. Other Examples P23 Example #8 P24 Spider Web Discuss scale factor
Beam or light beam method