1. Transformations Aaron Bloomfield CS 445: Introduction to Graphics
Fall 2006
(Slide set originally by David Luebke)

2. Graphics coordinate systems
X is red
Y is green
Z is blue

3. Graphics coordinate systems
If you are on the +z axis, and +y is up, then +x is to the right
Math fields have +x going to the left

4. Outline Scaling Rotations Composing Rotations Homogeneous Coordinates
Translations
Projections

5. Scaling
Scaling a coordinate means multiplying each of its components by a scalar
Uniform scaling means this scalar is the same for all components:
 2

6. Scaling Non-uniform scaling: different scalars per component:
How can we represent this in matrix form?
X  2, Y  0.5

7. Scaling
Scaling operation:
Or, in matrix form:
scaling matrix

8. Outline Scaling Rotations Composing Rotations Homogeneous Coordinates
Translations
Projections

9.  2-D Rotation (x’, y’) (x, y) x’ = x cos() - y sin()
y’ = x sin() + y cos()

10. 2-D Rotation This is easy to capture in matrix form:
3-D is more complicated
Need to specify an axis of rotation
Simple cases: rotation about X, Y, Z axes

11. Rotation example: airplane

12. 3-D Rotation
What does the 3-D rotation matrix look like for a rotation about the Z-axis?
Build it coordinate-by-coordinate
2-D rotation from last slide:

13. 3-D Rotation
What does the 3-D rotation matrix look like for a rotation about the Y-axis?
Build it coordinate-by-coordinate

14. 3-D Rotation
What does the 3-D rotation matrix look like for a rotation about the X-axis?
Build it coordinate-by-coordinate

15. Rotations about the axes

16. 3-D Rotation
General rotations in 3-D require rotating about an arbitrary axis of rotation
Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra
Another approach: express general rotation as composition of canonical rotations
Rotations about X, Y, Z

17. Outline Scaling Rotations Composing Rotations Homogeneous Coordinates
Translations
Projections

18. Composing Canonical Rotations
Goal: rotate about arbitrary vector A by 
Idea: we know how to rotate about X,Y,Z
So, rotate about Y by  until A lies in the YZ plane
Then rotate about X by  until A coincides with +Z
Then rotate about Z by 
Then reverse the rotation about X (by -)
Then reverse the rotation about Y (by -)
Show video…

19. Composing Canonical Rotations
First: rotating about Y by  until A lies in YZ
Draw it…
How exactly do we calculate ?
Project A onto XZ plane
Find angle  to X:
 = -(90° - ) =  - 90 °
Second: rotating about X by  until A lies on Z
How do we calculate ?

20. 3-D Rotation Matrices
So an arbitrary rotation about A composites several canonical rotations together
We can express each rotation as a matrix
Compositing transforms == multiplying matrices
Thus we can express the final rotation as the product of canonical rotation matrices
Thus we can express the final rotation with a single matrix!

21. Compositing Matrices So we have the following matrices:
p: The point to be rotated about A by 
Ry : Rotate about Y by 
Rx  : Rotate about X by 
Rz : Rotate about Z by 
Rx  -1: Undo rotation about X by 
Ry-1 : Undo rotation about Y by 
In what order should we multiply them?

22. Compositing Matrices
Remember: the transformations, in order, are written from right to left
In other words, the first matrix to affect the vector goes next to the vector, the second next to the first, etc.
This is the rule with column vectors (OpenGL); row vectors would be the opposite
So in our case:
p’ = Ry-1 Rx  -1 Rz Rx  Ry p

23. Rotation Matrices Notice these two matrices:
Rx  : Rotate about X by 
Rx  -1: Undo rotation about X by 
How can we calculate Rx  -1?
Obvious answer: calculate Rx (-)
Clever answer: exploit fact that rotation matrices are orthonormal
What is an orthonormal matrix?
What property are we talking about?

24. Rotation Matrices Rotation matrix is orthogonal
Columns/rows linearly independent
Columns/rows sum to 1
The inverse of an orthogonal matrix is just its transpose:

25. Rotation Matrix for Any Axis
Given glRotated (angle, x, y, z)
Let c = cos(angle)
Let s = sin(angle)
And normalize the vector so that ||(x,y,z|| == 1
The produced matrix to rotate something by angle degrees around the axis (x,y,z) is:

26. Outline Scaling Rotations Composing Rotations Homogeneous Coordinates
Translations
Projections

27. Translations
For convenience we usually describe objects in relation to their own coordinate system
We can translate or move points to a new position by adding offsets to their coordinates:
Note that this translates all points uniformly

28. Translation Matrices?
We can composite scale matrices just as we did rotation matrices
But how to represent translation as a matrix?
Answer: with homogeneous coordinates

29. Homogeneous Coordinates
Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector
[x, y, z, 0]T represents a point at infinity (use for vectors)
[0, 0, 0]T is not allowed
Note that typically w = 1 in object coordinates

30. Homogeneous Coordinates
Homogeneous coordinates seem unintuitive, but they make graphics operations much easier
Our transformation matrices are now 4x4:

31. Homogeneous Coordinates
Homogeneous coordinates seem unintuitive, but they make graphics operations much easier
Our transformation matrices are now 4x4:

32. Homogeneous Coordinates
Homogeneous coordinates seem unintuitive, but they make graphics operations much easier
Our transformation matrices are now 4x4:

33. Homogeneous Coordinates
Homogeneous coordinates seem unintuitive, but they make graphics operations much easier
Our transformation matrices are now 4x4:
Performing a scale:

34. More On Homogeneous Coords
What effect does the following matrix have?
Conceptually, the fourth coordinate w is a bit like a scale factor

35. More On Homogeneous Coords
Intuitively:
The w coordinate of a homogeneous point is typically 1
Decreasing w makes the point “bigger”, meaning further from the origin
Homogeneous points with w = 0 are thus “points at infinity”, meaning infinitely far away in some direction. (What direction?)
To help illustrate this, imagine subtracting two homogeneous points

36. Outline Scaling Rotations Composing Rotations Homogeneous Coordinates
Translations
Projections

37. Homogeneous Coordinates
How can we represent translation as a 4x4 matrix?
A: Using the rightmost column:
Performing a translation:

38. Translation Matrices
Now that we can represent translation as a matrix, we can composite it with other transformations
Ex: rotate 90° about X, then 10 units down Z:

39. Translation Matrices
Now that we can represent translation as a matrix, we can composite it with other transformations
Ex: rotate 90° about X, then 10 units down Z:

40. Translation Matrices
Now that we can represent translation as a matrix, we can composite it with other transformations
Ex: rotate 90° about X, then 10 units down Z:

41. Translation Matrices
Now that we can represent translation as a matrix, we can composite it with other transformations
Ex: rotate 90° about X, then 10 units down Z:

42. Transformation Commutativity
Is matrix multiplication, in general, commutative? Does AB = BA?
What about rotation, scaling, and translation matrices?
Does RxRy = RyRx?
Does RAS = SRA ?
Does RAT = TRA ?

43. Outline Scaling Rotations Composing Rotations Homogeneous Coordinates
Translations
Projections

44. Perspective Projection
In the real world, objects exhibit perspective foreshortening: distant objects appear smaller
The basic situation:

45. Perspective Projection
When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world
The view plane
How tall should this bunny be?

46. Perspective Projection
The geometry of the situation is that of similar triangles. View from above:
What is x?
P (x, y, z) X Z
View plane d
(0,0,0)
x’ = ?

47. Perspective Projection
Desired result for a point [x, y, z, 1]T projected onto the view plane:
What could a matrix look like to do this?

48. A Perspective Projection Matrix
Answer:

49. A Perspective Projection Matrix
Example:
Or, in 3-D coordinates:

50. A Perspective Projection Matrix
OpenGL’s gluPerspective() command generates a slightly more complicated matrix:
Can you figure out what this matrix does?

51. Projection Matrices
Now that we can express perspective foreshortening as a matrix, we can composite it onto our other matrices with the usual matrix multiplication
End result: can create a single matrix encapsulating modeling, viewing, and projection transforms
Though you will recall that in practice OpenGL separates the modelview from projection matrix (why?)