1. Geometrical Transformations 2 Adapted from Fundamentals of Interactive Computer Graphics, Foley and van Dam, pp. 245-315, by Geb Thomas

2. Learning Objectives Learn how 2D transformations are represented in 3D. Recognize the inverse matrix for a homogeneous transformation. Understand how the transformations also represent coordinate frame transformations. Understand the concept of a matrix stack.

3. 2D Translation and 3D Translation

4. 2D Rotation

5. 3D Rotation – Z Axis About the Z axis x y z

6. 3D Rotation – X Axis About the Z axis x y z

7. 3D Rotation – Y Axis About the Z axis x y z

8. Inverse Matrices The 3x3 rotation submatrix is orthogonal. The inverse of the 3x3 matrix is the transform of the original matrix The inverse of the translation component is just the reverse translation.

9. Composition of Transforms Various motions can be tacked, one after the other, in a long sequence of matrices: T 1 T 2 T 3 …T 4 These combinations will maintain the relative shape of the vectors processed, but will shift them around the original coordinate frame.

10. Thinking of Reference Frames Another way to think of this mathematics, is to imagine transforming the coordinate frame to a new place y0y0 z0z0 x1x1 y1y1 z1z1 T (5,0,2) Transformation, T, moves things 5 to the right and 2 up. The whole coordinate frame moves to a new position x0x0

11. Coordinate Rotation y0y0 z0z0 x1x1 y1y1 R (y=-35) z1z1 x0x0

12. The Matrix Stack While drawing a world, you often want to draw with respect to a convenient coordinate frame The graphics card need only keep track of the current position (current transformation) If you want to shift to the left, multiply the current frame by a left translation When done, shift back to the right

13. The Matrix Stack, FILO T(15,20,35) R(z=12) T(2,1,1) R(y=15) R(x=35) T(3,2,5) Current position of Drawing reference frame Programmer’s Next desired Shift of reference frames

14. Learning Objectives Learn how 2D transformations are represented in 3D. Recognize the inverse matrix for a homogeneous transformation. Understand how the transformations also represent coordinate frame transformations. Understand the concept of a matrix stack.