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Simulation and Animation

Simulation and Animation. Collisions. Collisions. Detection. Broad Phase. Bounding Volumes. Key idea: Surround the object with a (simpler) bounding object (the bounding volume). If something does not collide with the bounding volume, it does not collide with the object inside.

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Simulation and Animation

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  1. Simulation and Animation Collisions

  2. Collisions

  3. Detection

  4. Broad Phase

  5. Bounding Volumes • Key idea: • Surround the object with a (simpler) bounding object (the bounding volume). • If something does not collide with the bounding volume, it does not collide with the object inside. • Often, to intersect two objects, first intersect their bounding volumes

  6. Choosing a Bounding Volume • Lots of choices, each with tradeoffs • Tighter fitting is better • More likely to eliminate “false” intersections

  7. Choosing a Bounding Volume • Lots of choices, each with tradeoffs • Tighter fitting is better • Simpler shape is better • Makes it faster to compute with

  8. Choosing a Bounding Volume • Lots of choices, each with tradeoffs • Tighter fitting is better • Simpler shape is better • Rotation Invariant is better • Easier to update as object moves

  9. Choosing a Bounding Volume • Lots of choices, each with tradeoffs • Tighter fitting is better • Simpler shape is better • Rotation Invariant is better • Convex is usually better • Gives simpler shape, easier computation

  10. Bounding Volumes: Sphere • Rotationally invariant • Usually • Usually fast to compute with • Store: center point and radius • Center point: object’s centerof mass • Radius: distance of farthest point on object from centerof mass. • Often not very tight fit

  11. Axis Aligned Bounding Box (AABB) • Very fast to compute with • Store: max and min along x,y,z axes. • Look at all points and record max, min • Moderately tight fit • Must update after rotation, unlessa loose box that encompasses thebounding sphere

  12. Common Bounding Volumes: k-dops • k-discrete oriented polytopes • Same idea as AABBs, but use more axes. • Store: max and min along fixed set of axes. • Need to project points onto other axes. • Tighter fit than AABB, but also a bit more work.

  13. Choosing axes for k-dops • Common axes: consider axes coming out from center of a cube: • Through faces: 6-dop • same as AABB • Faces and vertices: 14-dop • Faces and edge centers: 18-dop • Faces, vertices, and edge centers; 26-dop • More than that not really helpful • Empirical results show 14 or 18-dop performs best.

  14. Oriented Bounding Box (OBB) • Store rectangular parallelepiped oriented to best fit the object • Store: • Center • Orthonormal set of axes • Extent along each axis • Tight fit, but takes work to get good initial fit • OBB rotates with object, therefore onlyrotation of axes is needed for update • Computation is slower than for AABBs,but not as bad as it might seem

  15. Convex Hull • Very tight fit (tightest convex bounding volume) • Slow to compute with • Store: set of polygons forming convex hull • Can rotate CH along with object. • Can be efficient for someapplications

  16. Testing for Collision • Will depend on type of objects and bounding volumes. • Specialized algorithms for each: • Sphere/sphere • AABB/AABB • OBB/OBB • Ray/sphere • Triangle/Triangle

  17. Collision Test Example: Sphere-Sphere • Find distance between centers of spheres • Compare to sum of sphere radii • If distance is less, they collide • For efficiency, check squared distance vs. square of sum of radii d r2 r1

  18. Collision Test Example: AABB vs. AABB • Project AABBs onto axes • i.e. look at extents • If overlapping on all axes, the boxes overlap. • Same idea for k-dops.

  19. Collision Test Example: OBB vs. OBB • Similar to overlap test for k-dops • How do we find axes to test for overlap?

  20. Separating Axis Theorem • Two convex shapes do not overlap if and only if there exists an axis such that the projections of the two shapes do not overlap

  21. Enumerating Separating Axes • 2D: check axis aligned with normal of each face • 3D: check axis aligned with normals of each face and cross product of each pair of edges

  22. Enumerating Separating Axes • 2D: check axis aligned with normal of each face • 3D: check axis aligned with normals of each face and cross product of each pair of edges

  23. Enumerating Separating Axes • 2D: check axis aligned with normal of each face • 3D: check axis aligned with normals of each face and cross product of each pair of edges

  24. Bounding Volumes Hierarchies

  25. Intersecting Bounding Volume Hierarcies • For object-object collision detection • Keep a queue of potentially intersecting BVs • Initialize with main BV for each object • Repeatedly pull next potential pair off queue and test for intersection. • If that pair intersects, put pairs of children into queue. • If no child for both BVs, test triangles inside • Stop when we either run out of pairs (thus no intersection) or we find an intersecting pair of triangles

  26. BVH Collision Test example

  27. B-Volume Examples

  28. Space partitioning Whyspacepartitioning? The bestobjectistheonethatisnot goingtobeprocessed! • Processing means • Determiningthespatialrelationshipsbetweenobjects • Do objectsintersect – collisiondetection • Naive approachhascomplexity O(n2) (n polygons in thescene) • Can also be: Rendering • Are objectswithinthevisiblespace – visibilityculling • Are objectsoccluded – occlusionculling

  29. Spatial hierarchies • Hierarchiesofspacepartitions • (Regular Grid) • Quadtree, Octree • BSP-Trees (BSP = Binary Space Partitioning) • KD-Trees • Assignmentofobjectstopartitions • Collisiondetectionthenbecomes 1) determine in whichpartitiontheobjectis 2) testonlyobjects in the same partition • Goodforstaticscenes, otherwisethehierarchyhastobere-build

  30. Regular Grid • Span course grid over the domain • Find cells in which the objects reside • Test if one cells contains more than one object • Not really a hierarchy (one level)

  31. Spatial hierarchies • Octrees • Recursiveregularsubdivisionofspaceinto 8 subspaces • Onenodeissplitinto 8 childnodes Adaptive quadtree Balancedoctree - grid

  32. Examples

  33. Spatial hierarchies • Bsp-Trees • Computational representation of space • Search structure and representation of geometry • Generalization of binary search trees for dim>1 • Search complexity to find spatial relationships between n polygons within O(n log n) • Recursive space partitioning by means of arbitrarily positioned partitioning planes

  34. Spatial hierarchies • Bsp-Trees (Binary Space Partitioning) • Every cellis a convexpolyhedron P1 A B P2 P3 C B D A P2 D P3 C P1

  35. Spatial hierarchies • Bsp-Treeexample • Inter-objectpartitioning • Binary treeoflines in 2D A A B C B D C D Partitioning Tree

  36. Spatial hierarchies • Characteristics of Bsp-Trees • Transformation of object = transformation of tree • Intra-object relationships remain static for solid objects • Merging with other trees easy to do • Objects in one halfspace cannot intersect objects in the corresponding other halfspace • Accelerates intersection test between objects

  37. Spatial hierarchies • Constructing Bsp-Trees • Elementary operation is recursive subdivision • Split convex region into two convex subregions • Use hyperplanes as cutting primitives Hyperplane R- R R+

  38. Spatial hierarchies • Bsp-Treeconstruction • Different orderingsoffacestouseas hyperplanes result in different trees • Greedyapproach (decisionsforeachstepbased on whatseems optimal) not alwaysapplicable • GoodBsp-Treerepresentsthe model assequenceofapproximations • Pruningyields different resolutionsofthe model

  39. Spatial hierarchies • Bsp-Treeconstruction • Bsp-Treesonlyperformgoodifgeometricfeaturesarelocal • Is truemostofthe time • Then, a significantsubsetofspacecanbeeliminated • Low cost (shortpaths) forreachinghighprobabilityregions • SimilartoHuffmancoding • Probabilityfor ´in a region´ is proportional tothesizeoftheregion • p+ = vol(tree.posSubspace) / vol(tree) • p- = vol(tree.negSubspace) / vol(tree)

  40. Spatial hierarchies • Collisontestforparticles in BSP-tree • Test, in whichnodetheparticleislocated • Traverse treeuntilleafs • Collision, ifat least onefilledleafishit

  41. Spatial hierarchies • Collisiontestforparticles in BSP-tree • Clip pathlinebetween 2 particlelocationsatseparating planes • Test all in-between time steps • Collision will bedetectedindependentofsizeof time step t0 + t t0 + t t0 t0 Collision not detected collision detected

  42. Spatial hierarchies • Collision test for particles with extend • Offset on surfaces (level planes) • Different offsets for different moving objects • Construction of different BSP-trees Problem: Distance too large

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