CS361

1. Week 13 - Friday CS361

2. Last time • What did we talk about last time? • Ray/sphere intersection • Ray/box intersection • Slabs method • Line segment/box overlap test • Ray/triangle intersection • Ray/polygon intersection

3. Questions?

4. Assignment 5

5. Project 4

6. Plane/box intersection • Simplest idea: Plug all the vertices of the box into the plane equation n • x + d = 0 • If you get positive and negative values, then the box is above and below the plane, intersection! • There are more efficient ways that can be done by projecting the box onto the plane

7. BV/BV intersection • Seeing if bounding volumes collide is a fundamental part of most collision detection algorithms • Does this ship hit that asteroid? • Bounding volumes are often arranged in hierarchies of bounding volumes • Bounding volumes allow for easy reject cases

8. Sphere/sphere intersection • Sphere/sphere is the easiest • Is the distance between their centers bigger than the sum of their radii? • If yes, they are disjoint • If no, they overlap r2 r1 c2 c1

9. Sphere/box intersection • Remember that an AABB is defined by two points, amin and amax • We go through each axis (x, y, and z) and first check to see if we can reject based on distance on that axis being too large • If not, we add the squared axis's distance to the total • If you want to do a sphere/OBB intersection, transform the sphere's center into the axes of the OBB and then the OBB will be an AABB

10. Sphere/box pseudocode intersect( c, r, A ) { d = 0 for i in x,y,z if( (e = ci – aimin) < 0 ) if( e < -r ) return DISJOINT d = d + e2 else if ( (e = ci – aimax) > 0 ) if( e > r ) return DISJOINT d = d + e2 if ( d > r2 ) return DISJOINT return OVERLAP }

11. AABB/AABB intersection • We test each dimension to see if the min of one box is greater than the max of the other or vice versa • If that's ever true, they're disjoint • If it's never true, they overlap intersect(A, B ) { for i in x,y,z • if(aimin > bimax or bimin > aimax ) return DISJOINT return OVERLAP }

12. k-DOP/k-DOP intersection • Note that an AABB is a special case of a 6-DOP • We can apply the same test as for an AABB, looking at the mins and maxes of each slab • Because the axes of a k-DOP are not necessarily orthogonal, this test is inexact (unlike for the AABB) • It is conservative: Some disjoint k-DOPs will report that they overlap, but overlapping k-DOPs will never report disjoint intersect(A, B ) { for i in 1 … k/2 • if(diA,min > diB,max or diB,min > diA,max ) return DISJOINT return OVERLAP }

13. OBB/OBB intersection • Again, an OBB is surprisingly complex • The fastest way found involves the separating axis test • There are 15 different axes you've got to test for overlap before you can be sure that the boxes overlap • When all the math is worked out, the test is quite fast

14. View frustum intersection • Because anything visible on the screen will be in the view frustum, we can save time by ignoring objects that are not • Remember that the frustum is defined by 6 planes: near, far, left, right, top and bottom • When test the frustum against bounding volumes, we will want three answers: outside, inside, and intersect

15. Frustum planes • To test frustum intersection, it is necessary to know the plane equations for each of the six frustum planes • If the view matrix is V and the projection matrix is P, the final transform is M = PV • If mi means the ith row of M, the equations for each plane are as follows: • -(m3 + m0) • (x, y, z, 1) = 0 (left) • -(m3 – m0) • (x, y, z, 1) = 0 (right) • -(m3 + m1) • (x, y, z, 1) = 0 (bottom) • -(m3 – m1) • (x, y, z, 1) = 0 (top) • -(m3 + m2) • (x, y, z, 1) = 0 (near) • -(m3 – m2) • (x, y, z, 1) = 0 (far)

16. Frustum/sphere intersection • We take the center of the sphere p and plug it into each of the plane equations, getting signed distance values • Note that the normals of the planes point outwards • If the distance to any given plane is greater than radius r, the sphere is outside the frustum • If the distances to all six planes are less than –r, the sphere is inside • Otherwise, the sphere intersects • This test is conservative: Reports some outside spheres as intersections

17. Frustum/box intersection • We skipped over the section that says how to test a plane for intersection with a box, but it's a simple calculation • To do frustum/AABB intersection, we test the AABB against every plane of the frustum • If the box is outside any plane, we return outside • If the box is not outside any plane but intersects some plane, we return intersects • Otherwise, we return inside

18. Line/line intersection • We will only look at the 2D problem, but the book has discussion of 3D lines as well • For a 2D vector (x, y), we define its perp dot product (x, y) = (-y, x) • Thus, we can work through the equations of a line (only the path to the value of s is shown) • r1(s) = r2(t) • o1 + sd1 = o2 + td2 • sd1 • d2 = (o2 – o1) • d2 • s = ((o2 – o1) • d2) / d1 • d2

19. Upcoming

20. Next time… • Collision detection

21. Reminders • Keep working on Project 4 • Start Assignment 5 • Read Chapter 17