Animação e Visualização Tridimensional. Collision Detection Corpo docente de AVT / CG&M / DEI / IST / UTL

Transcription

1 Animação e Visualização Triimensional Collision Detection

2 Collision Hanling Collision Detection Collision Determination Collision Response

3 Collision Hanling Collision Detection Collision Determination Collision Response

4 Collision Hanling Collision Detection Collision Determination Collision Response

5 Collision Hanling Collision Detection Collision Determination Collision Response

6 Collision Hanling Collision Detection Collision Determination Collision Response

7 Collision Hanling Collision Detection Collision Determination Collision Response

8 Collision Hanling Collision Detection Collision Determination Collision Response

9 What you nee to know Basic geometry vectors, points, homogenous coorinates, affine transformations, ot prouct, cross prouct, vector projections, normals, planes Math helps Linear algebra, calculus, ifferential equations

10 Plane Equation A 3D Plane is efine by a normal an a istance along that normal Plane Equation: (N, Ny, Nz) (, y, z) = Fin : (N, Ny, Nz) (P, Py, Pz) = For test point (,y,z), if plane equation > : point on front sie (in irection of normal), < : on back sie = : irectly on plane

11 Cross an Dot Proucts point point::operator^(point p) { // epens on the choice of orientation point res; res. = y*p.z - z*p.y; res.y = z*p. - *p.z; res.z = *p.y - y*p.; return res; } ouble point::operator*(point p) { } return (p.* p.y*y p.z*z);

12 So where o you start.? First you have to etect collisions With iscrete timesteps, every frame you check to see if objects are intersecting (overlapping) Testing if your moel s actual volume overlaps another s is too slow Use bouning volumes (BV s) to approimate each object s real volume

13 BouningVolumes? Conve-ness is important spheres, cyliners, boes, polyhera, etc. Spheres are mostly use for fast culling For boes an polyhera, most intersection tests start with point insie-outsie tests That s why conveity matters! There is no general insie-outsie test for a 3D concave polyheron.

14 D Point Insie-OutsieTests Conve Polygon Test Test point has to be on same sie of all eges Concave Polygon Tests 36 egree angle summation Compute angles between test point an each verte Insie if they sum to 36 Slow, ot prouct an acos for each angle! Several other methos eists Eplore them!

15 Closest point on a line Hany for all sorts of things

16 Spheres as BouningVolumes Simplest 3D Bouning Volume Center point an raius Point in/out test: Calculate istance between test point an center point If istance <= raius, point is insie You can save a square root by calculating the square istance an comparing with the square raius!!! (this makes things a lot faster) It is ALWAYS worth it to o a sphere test before any more complicate test.

17 Ais-Aligne Bouning Boes Specifie as two points: Normals are easy to calculate Simple point-insie test:

18 ProblemsWithAABB s Not very efficient Rotation can be complicate Must rotate moel an rebuil AABB but this is not efficient

19 Oriente Bouning Boes Define by: Center point, 3 normalize ais, 3 ege half-lengths Can be store as 8 points sometimes more efficient Can become not-a-bo after transformations Ais are the 3 face normals Better at bouning than spheres an AABB s

20 Simple Collision Detection Only shoot rays to fin collisions, i.e., approimate an object with a set of rays Cheaper, but less accurate Test for point in plane or point in sphere

21 Simple Collision Detection Only shoot rays to fin collisions, i.e., approimate an object with a set of rays Cheaper, but less accurate Test: point insie sphere

22 Simple Collision Detection Only shoot rays to fin collisions, i.e., approimate an object with a set of rays Cheaper, but less accurate Test: intersection with sphere r r r R( t) = R t< behin o object center t= center t=1 borer t>1 ousie object R t

23 Intersection Ray Sphere Ray Sphere Replacing ray s, y, z in sphere equation, we have: (,y,z ) ( 1,y 1,z 1 ) ( c,y c,z c ) = C B t A t ( ) ( ) ( ) t z z t z z z z t y y t y y y y t t = = = = = = ( ) ( ) ( ) = r z z y y c c c ( ) ( ) ( ) ( ) ( ) ( ) r z z y y C z z z y y y B z y A c c c c c c = = =

24 Intersection Ray Sphere Normalizing ray vector Simplifying the equation: y z = 1 A = 1 t = B ± B C B - C Conclusão < Ray oes not intersect sphere = Ray is tangent to sphere > Ray intersects sphere

25 Intersection Ray Sphere In principle we want the lower t: But... t = B B C (a) t<, not intersecting (b) origin insie sphere (C>): two solutions, we want the higher value, not the lower! (c) normal case (a) (b) (c)

26 Intersection Ray Sphere Não queremos calcular tuo antes e saber se há intersecção ou não... Solução: calcular Se <, não há intersecção possível t min = R D OC (outra forma: se t min oc r e B<, não intersecta) t min R R C C C C

27 Intersection Ray-Plane Plane Equation A B y C z D = Replacing in the parametric e equation an solving in orer to t, we have: t i ( A B y C z D) = A B y C z t i N R D = N R Ray is parallell to plane if: N R =

28 Collision Detection Packages Bullet Physics Library - library for performing rigi-boy collision etection an response. Open source an free for commercial use, an is integrate with Blener an COLLADA. V-clip - a low level object collision library. ODE - a free rigi boy ynamics package which inclues collision etection. ColDet - a free collision etection library for generic polyhera. Havok - the most popular commercial library for games is free for non-commercial use.

29 Conclusion cannot test every pair of triangles: O(n ) use BVs because these are cheap to test better: use a hierarchical scene graph