1. CAP4730: Computational Structures in Computer Graphics
2D Transformations

2. 2D Transformations World Coordinates Translate Rotate Scale
Viewport Transforms
Putting it all together

3. Transformations
Rigid Body Transformations - transformations that do not change the object.
Translate
If you translate a rectangle, it is still a rectangle
Scale
If you scale a rectangle, it is still a rectangle
Rotate
If you rotate a rectangle, it is still a rectangle

4. Vertices We have always represented vertices as (x,y)
An alternate method is:
Example:

5. Matrix * Vector

6. Matrix * Matrix
Does A*B = B*A?
What does the identity do?

7. Practice

8. Translation
Translation - repositioning an object along a straight-line path (the translation distances) from one coordinate location to another.
(x’,y’)
(tx,ty)
(x,y)

9. Translation
Given:
We want:
Matrix form:

10. Translation Examples P=(2,4), T=(-1,14), P’=(?,?)

11. Which one is it?
(x’,y’)
(tx,ty)
(tx,ty)
(x,y)
(x,y)

12. Recall
A point is a position specified with coordinate values in some reference frame.
We usually label a point in this reference point as the origin.
All points in the reference frame are given with respect to the origin.

13. Applying to Triangles
(tx,ty)

14. What do we have here?
You know how to:

15. Scale Scale - Alters the size of an object. Scales about a fixed point
(x’,y’)
(x,y)

16. Scale
Given:
We want:
Matrix form:

17. Non-Uniform/Differential Scalin’
(x’,y’)
(x,y)
S=(1,2)

18. Rotation Rotation - repositions an object along a circular path.
Rotation requires an  and a pivot point

19. Rotation

20. Example
P=(4,4)
=45 degrees

21. What is the difference? Revisited
V(-0.6,0) V(0,-0.6) V(0.6,0.6)
Translate (1.2,0.3)
V(0,0.6) V(0.3,0.9) V(0,1.2)
Translate (1.2,0.3)
V(0.6,0.3) V(1.2,-0.3) V(1.8,0.9)
V(0,0.6) V(0.3,0.9) V(0,1.2)

22. Rotations V(-0.6,0) V(0,-0.6) V(0.6,0.6) Rotate -30 degrees

23. Combining Transformations
Q: How do we specify each transformation?

24. Specifying 2D Transformations
Translation
T(tx, ty)
Translation distances
Scale
S(sx,sy)
Scale factors
Rotation
R()
Rotation angle

25. Combining Transformations
Using translate, rotation, and scale, how do we get:

26. Combining Transformations
Note there are two ways to combine rotation and translation. Why?

27. Let’s look at the equations

28. Combining them
We must do each step in turn. First we rotate the points, then we translate, etc.
Since we can represent the transformations by matrices, why don’t we just combine them?

29. 2x2 -> 3x3 Matrices
We can combine transformations by expanding from 2x2 to 3x3 matrices.

30. Homogenous Coordinates
We need to do something to the vertices
By increasing the dimensionality of the problem we can transform the addition component of Translation into multiplication.

31. Homogenous Coordinates
Homogenous Coordinates - term used in mathematics to refer to the effect of this representation on Cartesian equations. Converting a pt(x,y) and f(x,y)=0 -> (xh,yh,h) then in homogenous equations mean (v*xh,v*yh,v*h) can be factored out.
What you should get: By expressing the transformations with homogenous equations and coordinates, all transformations can be expressed as matrix multiplications.

32. Final Transformations - Compare Equations

33. Combining Transformations

34. How would we get:

35. How would we get:

36. Coordinate Systems Object Coordinates World Coordinates
Eye Coordinates

37. Object Coordinates

38. World Coordinates

39. Screen Coordinates

40. Coordinate Hierarchy

41. Let’s reexamine assignment 2b

42. Transformation Hierarchies
For example:

43. Transformation Hierarchies
Let’s examine:

44. Transformation Hierarchies
What is a better way?

45. Transformation Hierarchies
What is a better way?

46. Transformation Hierarchies
We can have transformations be in relation to each other

47. More Complex Models