1. Collisions using separating-axis tests Christer Ericson Sony Computer Entertainment Slides @ http://realtimecollisiondetection.net/pubs/

2. Problem statement Determine if two (convex) objects are intersecting. Possibly also obtain contact information. ?!

3. Underlying theory Set C is convex if and only if the line segment between any two points in C lies in C.

4. Underlying theory Separating Hyperplane Theorem States: two disjoint convex sets are separable by a hyperplane.

5. Underlying theory Nonintersecting concave sets not generally separable by hyperplane (only by hypersurfaces). Concave objects not covered here.

6. Underlying theory Separation w.r.t a plane P separation of the orthogonal projections onto any line L parallel to plane normal.

7. Underlying theory A line for which the projection intervals do not overlap we call a separating axis.

8. Separated if or Testing separation Compare absolute intervals

9. Testing separation For centrally symmetric objects: compare using projected radii Separated if

10. Code fragment void GetInterval(Object o, Vector axis, float &min, float &max) { min = max = Dot(axis, o.getVertex(0)); for (int i = 1, n = o.NumVertices(); i < n; i++) { float value = Dot(axis, o.getVertex(i)); min = Min(min, value); max = Max(max, value); }

11. Axes to test But which axes to test? Simplification: Deal only with polytopes Convex hulls of finite point sets Planar faces

12. Axes to test Handwavingly: Look at the ways A and B can come into contact. In 3D, reduces to vertex-face and edge- edge contacts. Vertex-face: a face normal from either polytope will serve as a separating axis. Edge-edge: the cross product of an edge from each will suffice.

13. Axes to test Theoretically: Consider the Minkowski difference C of A and B. When A and B disjoint, origin outside C, specifically outside some face F. Faces of C come from A, from B, or from sweeping the faces of either along the edges of the other. Therefore the face normal of F is either from A, from B, or the cross product of an edge from either.

14. Axes to test Four axes for two 2D OBBs:

15. 3D Objects Face dirs (A) Face dirs (B) Edge dirs (AxB) Total Segment–Tri011x34 Segment–OBB031x36 AABB–AABB30(3)0(3x0)3 OBB–OBB333x315 Tri–Tri113x311 Tri–OBB133x313 Axes to test

16. Code fragment bool TestIntersection(Object o1, Object o2) { float min1, max1, min2, max2; for (int i = 0, n = o1.NumFaceDirs(), i < n; i++) { GetInterval(o1, o1.GetFaceDir(i), min1, max1); GetInterval(o2, o1.GetFaceDir(i), min2, max2); if (max1 < min2 || max2 < min1) return false; } for (int i = 0, n = o2.NumFaceDirs(), i < n; i++) { GetInterval(o1, o2.GetFaceDir(i), min1, max1); GetInterval(o2, o2.GetFaceDir(i), min2, max2); if (max1 < min2 || max2 < min1) return false; } for (int i = 0, m = o1.NumEdgeDirs(), i < m; i++) for (int j = 0, n = o2.NumEdgeDirs(), j < n; j++) { Vector axis = Cross(o1.GetEdgeDir(i), o2.GetEdgeDir(j)); GetInterval(o1, axis, min1, max1); GetInterval(o2, axis, min2, max2); if (max1 < min2 || max2 < min1) return false; } return true; }

17. Moving objects When objects move, projected intervals move:

18. Moving objects Objects intersect when projections overlap on all axes. Objects are in contact over the interval [max i { t i first }, min i { t i last }], where t i first and t i last are time of first and last contact on axis i. No contact if max i { t i first } > min i { t i last }

19. Moving objects Optimization 1: Consider relative movement only. Shrink interval A to point, growing interval B by original width of A. Becomes moving point vs. stationary interval. Optimization 2: Exit as soon as max i { t i first } > min i { t i last }

20. Nonpolyhedral objects Spheres, capsules, cylinders, cones, etc. E.g. sphere vs. OBB: Closest point on OBB to sphere center

21. Nonpolyhedral objects Capsule tests: Split into three features Test axes that can separate features of capsule and features of second object.

22. Robustness warning Cross product of edges can result in zero-vector. if (Dot(foo, axis) > Dot(bar, axis)) return false; if (epsilon1 > epsilon2) return false; Typical test: Becomes, due to floating-point errors: Results in: Chaos!

23. Contact determination Covered by Erin Catto (later)

24. References Ericson, Christer. Real-Time Collision Detection. Morgan Kaufmann 2005. http://realtimecollisiondetection.net/ Levine, Ron. Collisions of moving objects. gdalgorithms-list mailing list article, November 14, 2000. http://realtimecollisiondetection.net/files/levine_swept_sat.txt Boyd, Stephen. Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. http://www.stanford.edu/~boyd/cvxbook/ Rockafellar, R. Tyrrell. Convex Analysis. Princeton University Press, 1996.