0. CSCE441: Computer Graphics 3D Transformations
Jinxiang Chai

1. 3D Transformation A 3D point (x,y,z) – x,y, and z coordinates
We will still use column vectors to represent points.
Homogeneous coordinates of a 3D point
(x,y,z,1)
Transformation will be performed using 4x4 matrix

2. Right-handed Coordinate System
Left hand coordinate system
Not used in this class and
Not in OpenGL

3. Homogenous coordinates
3D Transformation
Very similar to 2D transformation
Translation transformation
Homogenous coordinates

4. Homogenous coordinates
3D Transformation
Very similar to 2D transformation
Scaling transformation
Homogenous coordinates

5. 3D Transformation 3D rotation is done around a rotation axis
Fundamental rotations – rotate about x, y, or z axes
Counter-clockwise rotation is referred to as positive rotation (when you look down negative axis) y + x z

6. 3D Transformation Rotation about z – similar to 2D rotation
Keep z constant! y + x z

7. 3D Transformation Rotation about y: z -> y, y -> x, x->z z y

8. 3D Transformation Rotation about x (z -> x, y -> z, x->y) x z
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters, y x z

9. Inverse of 3D Transformations
Invert the transformation
In general, X= AX’-->X’=A-1X
T(-tx,-ty,-tz)
T(tx,ty,tz)

10. 3D Rotation about Arbitrary Axes
Rotate p about the by the angle
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters, 10

11. 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
The general rotation matrix is a combination of coordinate-axis rotations and translations!

12. 3D Rotation about Arbitrary Axes
Rotate p about the by the angle

13. 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
Standard approach: express general rotation as composition of canonical rotations
Rotations about x, y, z

14. Composing Canonical Rotations
Goal: rotate about arbitrary vector r by θ
Idea: we know how to rotate about x,y,z
Set up a transformation that superimposes rotation axis onto one coordinate axis
Rotate about the coordinate axis
Translate and rotate object back via inverse of the transformation matrix

15. Composing Canonical Rotations
Goal: rotate about arbitrary vector r by θ
Idea: we know how to rotate about x,y,z
So, rotate about z by - until r lies in the xz plane
Then rotate about y by -β until r coincides with +z
Then rotate about z by θ
Then reverse the rotation about y (by β )
Then reverse the rotation about z (by ) 15

16. 3D Rotation about Arbitrary Axes
Rotate p about the by the angle
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

17. 3D Rotation about Arbitrary Axes
Translate so that rotation axis passes through the origin
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

18. 3D Rotation about Arbitrary Axes
Rotation by about z-axis to place the rotation vector on xoz plane
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

19. 3D Rotation about Arbitrary Axes
Rotation by about y-axis to align the rotation vector with z axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

20. 3D Rotation about Arbitrary Axes
Rotation by about z-axis (rotation vector)
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

21. 3D Rotation about Arbitrary Axes
Rotation by about y-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

22. 3D Rotation about Arbitrary Axes
Rotation by about z-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

23. 3D Rotation about Arbitrary Axes
Translate the object back to original point

24. 3D Rotation about Arbitrary Axes
Final transformation matrix for rotating about an arbitrary axis

25. 3D Rotation about Arbitrary Axes
Final transformation matrix for rotating about an arbitrary axis 25

26. 3D Rotation about Arbitrary Axes
Final transformation matrix for rotating about an arbitrary axis

27. 3D Rotation about Arbitrary Axes
Final transformation matrix for rotating about an arbitrary axis
A 3 by 3 Rotation matrix—orthogonal matrix

28. Rotation Matrices Orthonormal matrix:
orthogonal (columns/rows linearly independent)
normalized (columns/rows length of 1)

29. Rotation Matrices Orthonormal matrix:
orthogonal (columns/rows linearly independent)
normalized (columns/rows length of 1)
The inverse of an orthogonal matrix is just its transpose:

30. Rotation Matrices Orthonormal matrix:
orthogonal (columns/rows linearly independent)
normalized (columns/rows length of 1)
The inverse of an orthogonal matrix is just its transpose:

31. Rotation Matrices Orthonormal matrix:
orthogonal (columns/rows linearly independent)
normalized (columns/rows length of 1)
The inverse of an orthogonal matrix is just its transpose:

32. Rotation Matrices
Why?
is a 3-by-3 identity matrix

33. Rotation Matrices
Why?
is a 3-by-3 identity matrix

34. Rotation Matrices
Why?
is a 3-by-3 identity matrix

35. Rotation Matrices
Why?
is a 3-by-3 identity matrix

36. Rotation Matrices Orthonormal matrix:
orthogonal (columns/rows linearly independent)
normalized (columns/rows length of 1)
The inverse of an orthogonal matrix is just its transpose: e.g.,

37. Next Lecture 2D coordinate transformations
Lots of vector and matrix operations! 37