Proximity and Deformation

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Proximity and Deformation

• Leonidas Guibas
• Stanford University

Tutto cambia perchè nulla cambi T. di
Lampedusa, Il Gattopardo (1860)
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Proximity Maintenance in Physical Simulation

• Most forces in nature are short range

• Collision detection

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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 MundNicht manch Geheimnis
würde kundDaß ich nicht mehr mit saurem
SchweißZu sagen brauche, was ich nicht weißDaß
ich erkenne, was die WeltIm Innersten
zusammenhält,Schau alle Wirkenskraft und
Samen,Und tu nicht mehr in Worten kramen.
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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
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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
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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

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Maintaining the Sphere Hierarchy under Deformation
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How well does it work?

• Very well, except when necklace gets really
  folded
• The power diagram (Delaunay) is better in packed
  situations

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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 a
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Spanners for Continuous Objects
Add a sparse set of shortcuts, sufficient to
guarantee the spanning property
A protein example with a 5
3HVT
with Agarwal, Gao, Nguyen, Zhang
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Spanners are Useful for Proximity/Collision
Detection

• To find all points at distance d from p, find all
  points within distance ad 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

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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

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Maintaining the Shortcuts under Deformation
Many open algorithmic issues
a 3
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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