Collision Detection for Subdivision Surfaces

1
Collision Detection for Subdivision Surfaces

• Jingyu Yan
• Spring 2003, Comp239

2
Outlines

• Subdivision Surfaces
• Parametric Surfaces
• Collision Detection
• Self-interference Detection for Subdivision
  Surfaces

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Subdivision

• 2D Curves

• 3D Surfaces

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Loop Subdivision

• Targeting triangular mesh
• Loop scheme

Figure Source http//www.scg.uwaterloo.ca/hqle/s
ubdivision/images/Loop/Loop1.html
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Loop Subdivision

• Extraordinary points
• concerns the smoothness of the surface
• Generating smooth surfaces. C1 near extraordinary
  point and C2 anywhere else

Figure Source http//www.scg.uwaterloo.ca/hqle/s
ubdivision/images/Loop/Loop1.html
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Subdivision

• Establish a mathematical representation for the
  process of subdivision curves

Globally, P (1) 8 x 1 S 8 x 5 P5 x 1 Locally, P
(1)3 x 1 S 3 x 3 P3 x 1 Then P (n)3 x 1 S n3
x 3 P3 x 1
Graphics Source
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Subdivision

• Establish a mathematical representation for the
  process of subdivision surfaces

Globally, P (1) 12 x 1 S 12 x 9 P9 x
1 Locally, P (1)9 x 1 S 9 x 9 P9 x 1 Then P
(n)9 x 1 S n9 x 9 P9 x 1
Graphics Source
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Subdivision

• Eigen analysis of S
• SVD(S) UDV (D is a diagonal matrix and UV I)
• P ?i xi ci (xi are the eigenvectors of S)

P (n) S nP U D nV P
P (n) S n P S n ?i xi ci
?i ?nixi ci
Convergence of subdivision requires ?i lt 1
Direct calculation of the limit positions are
possible
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Subdivision

• Further Eigen analysis of S in 2D
• ?0 1, its eigenvector is 1
• Derived from invariance under affine
  transformation
• ?i lt ?1 lt 1 , i 2,3,
• Derived from only one tangent vector exists for
  the limit control points. (refer to Chapter 2,
  ZORIN, D., AND SCHR ODER, P., Eds. Subdivision
  for Modeling and Animation. Course Notes. ACM
  SIGGRAPH, 1998)

gt
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Subdivision

• Further Eigen analysis of S in 3D
• ?0 1, its eigenvector is 1
• Derived from the invariance under affine
  transformation
• ?i lt ?1 ?2 lt 1 , i 3,4,
• ?1 , ?2 are called subdominant eigenvalues.
• Derived from only one tangent vector exists for
  the limit control points. (refer to Chapter 3,
  ZORIN, D., AND SCHRODER, P., Eds. Subdivision for
  Modeling and Animation. Course Notes. ACM
  SIGGRAPH, 1998)

gt
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Parametric Coordinates

• Euclidean coordinates (x, y) or (x, y, z)
• Parametric coordinates (? , ?, ?) with respect to
  P0 , P1 , P2
• The mapping from parametric coordinates to
  Euclidean coordinates
• (? , ?, ?) gt ?P0 ?P1 ?P2
• Parameterize parametric coordinates (?(t) , ?(t),
  ?(t)) with respect to P0 , P1 , P2
• A mapping from t to a set of points in Euclidean
  space
• P(t) ?(t)P0 ? (t)P1 ? (t)P2
• ?i Bi(t)Pi

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Parametric Curves

• Parametric curves ?i Bi(t)Pi
• If Bi(t) are Berstein polynomials(Bezier basic
  functions), ?i Bi(t)Pi represents a Bezier curve
• If Bi(t) are Spline basic functions, ?i Bi(t)Pi
  represents a Spline curve

Figure source http//www.doc.ic.ac.uk/dfg/AndysS
plineTutorial/
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Parametric Surfaces

• Parametric surfaces ?i Bi(u,v)Pi
• Bi(u,v) is a product of two Berstein
  polynomials(B(u)B(v)) for Bezier surfaces
• Bi(u,v) is a product of two Spline basic
  functions(S(u)S(v)) for Spline surfaces

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Collision Detection

• Given the geometry of objects, detect if there is
  collision between them. If there is, where it is.
• Self-interference
• Two objects
• N objects

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Hierarchical Structure for Collision Detection
Figure Source http//www.gamasutra.com/features/2
0000330/bobic_02.htm
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Hierarchical Structure for Collision Detection

• OBB trees
• AABB trees

Siggraph96, OBB-Tree A Hierarchical Structure
for Rapid Interference Detection, S. Gottschalk,
M.C. Lin, D. Manocha Computer Graphics
J. Arvo and D. Kirk. A survey of ray tracing
acceleration techniques, An Introduction to Ray
Tracing. Academic Press, 1989
Figure Source http//www.gamasutra.com/features/2
0000330/bobic_02.htm
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Subdivision Surfaces and Collision Detection

• Between subdivision surfaces
• Subdivision surface satisfies convex hull
  property
• Bounding boxes can be applied to their control
  mesh
• Hierarchical structure can be built for collision
  detection.
• Self-interference

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Self-interference Detection for Subdivision
Surfaces

• Two sufficient conditions for ruling out
  self-interference
• The Gauss map image of a patch is restricted to a
  hemisphere
• The projection of its boundary onto a separating
  plane have no self-intersections.(This condition
  can generally be ignored in most applications)

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Gauss map image of a surface

• The Gauss map maps every point on the surface to
  its unit normal vector
• The Gauss map image is the domain of the Gauss
  map of a surface.

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Algorithm Overview

• Derive the tangent function of a patch
• Calculate the normal bounds of the patch
• If the normal bounds are within a hemisphere,
  self-interference can be ruled out

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Derive the tangent function

• Loop subdivision

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Derive the tangent function

• From the coarsest level of control points p to
  the interested level of control points q1

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Derive the tangent function

• The limit surface of a regular patch
• B(u,v) is a row vector of box spline basic
  polynomials
• The limit surface of the level j regular patch

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Derive the tangent function

• The u-tangent function
• Scaling it will not change its direction
• For v, we derive the similar function

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Calculate the normal bound

• Rewrite
• The normal function

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Calculate the normal bound

• The first term of the normal function
• The remaining terms can be calculated from the
  bounds of the following terms
• The above term determined by the bounds of

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Calculate the normal bound

• Calculation of
• i 0 0,0
• i 1,2
• i gt 2

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Results
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Reference

• Eitan Grinspun and Peter Schröder, Normal Bounds
  for Subdivision-Surface Interference Detection,
  Proceedings of IEEE Scientific Visualization,
  2001.
• ZORIN, D., AND SCHR ODER, P., Eds. Subdivision
  for Modeling and Animation. Course Notes. ACM
  SIGGRAPH, 1998.
• DEROSE, T., KASS, M., AND TRUONG, T. Subdivision
  surfaces in character animation. Proceedings of
  SIGGRAPH 98 (July 1998), 8594.   
• Ming Lin, Stefan Gottschalk, Collision detection
  between geometric models a survey. Proceedings
  of IMA Conference on Mathematics of Surfaces 1998
• M. Lin, D. Manocha, J. Cohen and S. Gottschalk,
  Collision Detection Algorithms and Applications.
  Proc. of Algorithms for Robotics Motion and
  Manipulation, pp. 129-142 eds. Jean-Paul Laumond
  and M. Overmars, A.K. Peters.
• Van Den Bergen G. Efficient collision detection
  of complex deformable models using aabb trees.
  Journal of Graphics Tools, 2(4)113, 1998.
• Stefan Gottschalk, Ming Lin, and Dinesh Manocha.
  OBB-Tree A hierarchical structure for rapid
  interference detection. In Holly Rushmeier,
  editor, SIGGRAPH 96 Conference Proceedings,
  Annual Conference Series, pages 171180. ACM
  SIGGRAPH, Addison Wesley, August 1996. held in
  New Orleans, Louisiana, 04-09 August 1996