1. January 19, 20161

2. y X Z

3. Translations Objects are usually defined relative to their own coordinate system. We can translate points in space to new positions by adding offsets to their coordinates. The following figure shows the effect of translating a teapot. January 19, 20163

4. 4 Properties of Translation = = = = January 19, 2016

5. 5 Translation Revisited = = T(t x, t y, t z ) January 19, 2016

6. 6 Scaling Uniform scaling iff January 19, 2016

7. 3D Rotations January 19, 20167

9. 9

10. 10

11. 11 An Alternative View We can view the rotation around an arbitrary axis as a set of simpler steps We know how to rotate and translate around the world coordinate system Can we use this knowledge to perform the rotation? January 19, 2016

12. 12 Rotations about an arbitrary axis Rotate by around a unit axis January 19, 2016

13. 13 Rotation about an arbitrary axis A rotation matrix for any axis that does not coincide with a coordinate axis can be set up as a composite transformation involving combination of translations and the coordinate-axes rotations. 1.Translate the object so that the rotation axis passes through the coordinate origin 2. Rotate the object so that the axis rotation coincides with one of the coordinate axes 3. Perform the specified rotation about that coordinate axis 4. Apply inverse rotation axis back to its original orientation 5. Apply the inverse translation to bring the rotation axis back to its original position January 19, 2016

14. T = T -1 = Translate origin to rotation axis January 19, 201614

15. A, B, C V (0, B, C) θ Sin θ = B / V Cos θ = C / V January 19, 201615

16. Rx = Sin θ = B / V Cos θ = C / V R x -1 = Rotation about X-axis January 19, 201616

17. L ß A V January 19, 201617

18. Rotation about Y-axis R y = sin = V / L cos = A / L R y = R y -1 = January 19, 201618

19. Finally Rotate about Z-axis with angle R z = Rotation about an arbitrary axis with a angle R = T R x R y R z R y -1 R x -1 T -1 January 19, 201619

20. Shear (K xy, K xz, K yz ) = Shear January 19, 201620