Euler vs Hamilton

Performance and Accuracy comparison between Rotation Matrices and Quaternions for rotating a first-person camera in 3D space.

Transcript

1. Euler vs Hamilton Daniel Smith && Jan CW Kroeze University of Pretoria
2. INTRO
3. Various ways to rotate stuff
4. Various ways to rotate stuff Quaternions
5. Various ways to rotate stuff Quaternions Pitch-Yaw-Roll
6. Various ways to rotate stuff Quaternions Pitch-Yaw-Roll Matrices
7. Various ways to rotate stuff Quaternions Pitch-Yaw-Roll Matrices Angle-Axis
8. Various ways to rotate stuff Quaternions Pitch-Yaw-Roll Matrices Angle-Axis WHICH ONE??
9. Pitch-Yaw-Roll is just bad (Gimbal Lock)
10. Angle-Axis is complicated
11. Quaternions are popular
12. Rotation Matrices ... exist
13. How do we choose?
14. METRICS
15. SPEED
16. Rotating is a lot of FP math
17. sin, cos and FP multiplication
18. We measure total ns spent on calculations
19. STABILITY
20. What is stability?
21. Rounding errors! 4.0 + 3.0 = 6.99999999
22. Rotations get a bit wonky over time
23. Orthonormalisation
24. (Orthogonal + normal) + isation
25. Orthogonalisation + normalisation
26. Un-wonky rotations
27. How do you measure stability?
28. Compare limited-precision with perfect precision
29. (no such thing as perfect precision)
30. Compromise: double is “more perfect” than single
31. Discrepancy between single- and double-precision
32. EXPERIMENT
33. Implement a 3D camera
34. Rotate the camera using different methods
35. Measure stability and speed
36. RESULTS
37. Instability / Time after about a minute of input
38. Total CPU Time
39. Rotation Matrices accumulate instability at a much faster rate
40. (unless they’re orthonormalised)
41. Instability / Time
42. Quaternions are very well behaved
43. Orthonormalising quaternions is a silly idea
45. Quaternions 3x faster than Rotation Matrices
46. 3x faster!! (Order now, 50% off!)
47. (in the order of nanoseconds, so what difference does it make?)
48. Orthonormalising Quaternions make them faster?
49. ???
50. CONCLUSION
51. What does it mean?
52. Keep using quaternions!
53. (but perhaps not for the reasons you expect)
54. SPEED
55. STABILITY
56. INTERPOLATION
57. SUNROOF!