Proximity and Deformation Leonidas Guibas Stanford University Tutto

Proximity and Deformation Leonidas Guibas Stanford University “Tutto cambia perchè nulla cambi” T. di Lampedusa, Il Gattopardo (1860+)

Proximity Maintenance in Physical Simulation Most forces in nature are short range Collision detection

Large-Scale Deformation Most deformable models represent an object as a collection of many small elements At each time step of a simulation, most elements move We want to capture and maintain, under element motion, information that is useful for proximity detection, but is relatively stable at the same time (the KDS “Faustian dilemma”) Drum hab ich mich der Magie ergeben, Ob mir durch Geistes Kraft und Mund Nicht manch Geheimnis würde kund; Daß ich nicht mehr mit saurem Schweiß Zu sagen brauche, was ich nicht weiß; Daß ich erkenne, was die Welt Im Innersten zusammenhält, Schau alle Wirkenskraft und Samen, Und tu nicht mehr in Worten kramen.

Bounding Volume Hierarchies for Deformable Objects Bounding volume hierarchies (BVH), using spheres, bounding boxes, etc. , have been very successfully used for collision checking of rigid objects Deformation brings up the issue of hierarchy recomputation or update Tight Hierarchy Loose Hierarchy Frequent updates Faster collision checking More stable More wasted intersection tests

Implicit Hierarchies, Defined by Object Features Exploit what stays the same: object topology Example: a smallest enclosing sphere hierarchy for a deforming `necklace’, based on a fixed balanced binary tree Each sphere is implicitly defined by four elements Note that children spheres can stick out of parent spheres [with Agarwal, Nguyen, Russel, Zhang]

Combinatorial Descriptions are Stable As the necklace deforms, bounding spheres evolve following the motions of their defining elements We need to verify that each sphere continues to enclose its assigned geometry When this condition fails, the repair is a simple basis element swap, like pivoting in LP

Maintaining the Sphere Hierarchy under Deformation

How well does it work? Very well, except when necklace gets really folded The power diagram (Delaunay) is better in packed situations Separating pairs Sphere packing

Graph and Geometric Spanners Graph setting: Replace a dense graph with a sparse subgraph (the spanner), while approximately preserving shortest paths Geometry setting: Approximate all distances between points using shortest paths on a sparse set of edges (the spanner) Widely used in communication networks expansion ratio = α

Spanners for Continuous Objects Add a sparse set of shortcuts, sufficient to guarantee the spanning property 3 HVT [with Agarwal, Gao, Nguyen, Zhang] A protein example with α = 5

Spanners are Useful for Proximity/Collision Detection To find all points at distance d from p, find all points within distance αd along the object and its shortcuts Before two points p and q on a deformable object collide, there has to be a shortcut between them Spanners can have sublinear complexity

Sampling from the Delaunay Triangulation Discretize object into elements Compute the Delaunay triangulation Cluster the Delaunay edges into groups (à la n-body or well-separated pair decompositions). Clusterheads form the shortcuts (spanner). Converges to a limit as element size decreases

Maintaining the Shortcuts under Deformation Many open algorithmic issues … α=3

Conclusions Stable structures exist that encode proximity for deformable objects Many open issues: Improved theoretical analysis Further experimental validation Extension from space curves to surfaces Everything rests by changing