1. Physics for Games Programmers Reframing the Problem Squirrel EiserlohTechnical DirectorMumboJumbo Gamessquirrel@eiserloh.netwww.algds.org

2. Physics for Games Programmers Reframing the Problema.k.a. “It’s All Relative” Squirrel EiserlohTechnical DirectorMumboJumbo Gamessquirrel@eiserloh.netwww.algds.org

3. Overview • Tunneling • Movement Bounds • Swept Shapes • Einstein Says... • Minkowski Says... • Rotation

4. Tunneling (Sucks)

5. Tunneling • Question #1: Do objects A and B overlap? • Plenty of reference material to help solve this, but... • ...this is often the wrong question to ask (begs tunneling).

6. Tunneling

7. Tunneling

8. Tunneling

9. Tunneling

10. Tunneling (cont’d) • Tunneling is very, very bad – this is not a “mundane detail” • Things falling through world • Bullets passing through people or walls • Players getting places they shouldn’t • Players missing a trigger boundary • Tunneling is a false negative • Okay, so tunneling really sucks. What can we do about it?

11. Movement Bounds

12. Movement Bounds • Disc / Sphere

13. Movement Bounds • Disc / Sphere • AABB (Axis-Aligned Bounding Box)

14. Movement Bounds • Disc / Sphere • AABB (Axis-Aligned Bounding Box) • OBB (Oriented Bounding Box)

15. Movement Bounds • Question #2: Could A and B have collided during the frame? • Better than Question #1 (solves tunneling!), but...

16. Movement Bounds • Question #2: Could A and B have collided during the frame? • Better than Question #1 (solves tunneling!), but... • ...even if the answer is “yes”, we still don’t know for sure (false positive).

17. Movement Bounds • Conclusion • Good: They prevent tunneling! (i.e. no false negatives) • Bad: They don’t actually tell us whether A and B collided (still have false positives). • Good: They can be used as a cheap, effective early rejection test.

18. Swept Shapes

19. Swept Shapes • Swept disc / sphere (n-sphere): capsule

20. Swept Shapes • Swept disc / sphere (n-sphere): capsule • Swept AABB: convexpolytope (polygon in 2d, polyhedron in 3d)

21. Swept Shapes • Swept disc / sphere (n-sphere): capsule • Swept AABB: convex polytope (polygon in 2d, polyhedron in 3d) • Swept triangle / tetrahedron (simplex): convex polytope

22. Swept Shapes • Swept disc / sphere (n-sphere): capsule • Swept AABB: convex polytope (polygon in 2d, polyhedron in 3d) • Swept triangle / tetrahedron (simplex): convex polytope • Swept polytope: convexpolytope

23. Swept Shapes (cont’d) • Like movement bounds, only with a perfect fit!

24. Swept Shapes (cont’d) • Like movement bounds, only with a perfect fit! • Still no false negatives (tunneling).

25. Swept Shapes (cont’d) • Like movement bounds, only with a perfect fit! • Still no false negatives (tunneling). • Finally, no false positives, either!

26. Swept Shapes (cont’d) • Like movement bounds, only with a perfect fit! • Still no false negatives (tunneling). • Finally, no false positives, either! • No, wait, nevermind. Still have ‘em. Rats.

27. Swept Shapes (cont’d) • Like movement bounds, only with a perfect fit! • Still no false negatives (tunneling). • Finally, no false positives, either! • No, wait, nevermind. Still have ‘em. Rats.

28. Swept Shapes (cont’d) • Like movement bounds, only with a perfect fit! • Still no false negatives (tunneling). • Finally, no false positives, either! • No, wait, nevermind. Still have ‘em. Rats.

29. Swept Shapes (cont’d) • Conclusion • Suck? • Can be used as early rejection test, but... • ...movement bounds are better for that. • If you’re not too picky... • ...they DO solve a large number of nasty problems (especially tunneling) • ...and can serve as a poor man’s continuous collision detection for a basic engine.

31. Einstein Says... • Coordinate systems are relative

32. Relative Coordinate Systems

33. Relative Coordinate Systems • World coordinates

34. Relative Coordinate Systems • World coordinates • A’s local coordinates

35. Relative Coordinate Systems • World coordinates • A’s local coordinates • B’s local coordinates

36. Relative Coordinate Systems • World coordinates • A’s local coordinates • B’s local coordinates • Many others (e.g. origin at point of impact)

37. Relative Coordinate Systems (cont’d) • Ball vs. world...

38. Relative Coordinate Systems (cont’d) • Ball vs. world... • in world coordinates

39. Relative Coordinate Systems (cont’d) • Ball vs. world... • in world coordinates • x component • y component

40. Relative Coordinate Systems (cont’d) • Ball vs. world... • in world coordinates • x component • y component • in impact coordinates

41. Relative Coordinate Systems (cont’d) • Ball vs. world... • in world coordinates • x component • y component • in impact coordinates • parallel component • perpendicular component

42. Relative Coordinate Systems (cont’d) • Ball vs. world... • in world coordinates • x component • y component • in impact coordinates • parallel component • perpendicular component • Change in motion happens along the perpendicular axis

43. Relative Coordinate Systems (cont’d) • Ball vs. ball...

44. Relative Coordinate Systems (cont’d) • Ball vs. ball... • in world coordinates

45. Relative Coordinate Systems (cont’d) • Ball vs. ball... • in world coordinates • x component • y component

46. Relative Coordinate Systems (cont’d) • Ball vs. ball... • in world coordinates • x component • y component • in impact coordinates

47. Relative Coordinate Systems (cont’d) • Ball vs. ball... • in world coordinates • x component • y component • in impact coordinates • parallel component • perpendicular component • Energy is exchanged along the perpendicular axis

48. Relative Coordinate Systems (cont’d) Also, math is often nicer at the origin. x2 - 2xh + h2 + y2 - 2yk + k2 = r2 x2 + y2 = r2

49. Einstein Says... • Coordinate systems are relative • Motion is relative

50. Relative Motion