Dynamic Geometric Computation of Interacting Models*
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Richard RiesenfeldUniversity of Utah
May 2008
Dynamic Geometric Computation of Interacting Models*
* In collaboration with Xianming Chen¹, E Cohen¹, J Damon² _______________________________
1. University of Utah 2. University of North Carolina
May 2008 1Dagstuhl
Today: Intersection of Two Deforming Parametric Surfaces
May 2008 2Dagstuhl
Interactions are Complex
May 2008 Dagstuhl 3
Interactions are Complex
May 2008 Dagstuhl 4
Interactions are Complex
May 2008 Dagstuhl 5
Evolution
Overall Process
May 2008 Dagstuhl 6
Classification
Computation
Identification
Detection
Two Main Ideas
• Construct evolution vector field to follow the gradual change of intersection curve IC
• Use Singularity Theory and Shape Operator to compute topological change of IC
• Formulate locus of IC as 2-manifold in parametric 5-space
• Compute quadric approx at critical points of height function
May 2008 7Dagstuhl
Exchange Event
May 2008 Dagstuhl 8
Deformation as Generalized Offset
May 2008 9Dagstuhl
Curve /Curve IP Under Deformation
May 2008 10Dagstuhl
Tangent Movement
May 2008 Dagstuhl 11
Evolution Vector Field
May 2008 12Dagstuhl
Evolution Algorithm
May 2008 13Dagstuhl
Surface Case
May 2008 14Dagstuhl
Local Basis
May 2008 15Dagstuhl
Evolution Vector Field
May 2008 16Dagstuhl
Evolution Vector Field in Larger Context
• Well-defined actually in a neighborhood of any P in ³, where two surfaces deform to P at t1 and t2
• Vector field is on the tangent planes of level set surfaces defined by f = t1 - t2
• Locus of ICs is one of such level surfaces.May 2008 17Dagstuhl
Topological Change of ICs
May 2008 18Dagstuhl
2-Manifold in Parametric 5-space
May 2008 19Dagstuhl
IC as Height Contour
May 2008 20Dagstuhl
Critical Points of Height Function
May 2008 21Dagstuhl
4 Generic Transition Events
May 2008 22Dagstuhl
Comment
May 2008 23Dagstuhl
Morse theory of height function in augmented parametric space
R5{ s1 , s2 , ŝ1
, ŝ2 , t }
Singularity theory of stable surface mapping in physical space
R3{x, y, z}
Tangent Vector Fields
May 2008 24Dagstuhl
Computing Tangent Vector Fields
May 2008 25Dagstuhl
Computing Transition Events
May 2008 26Dagstuhl
Future Directions
• Application uses• Real models• More complex interactions• More general situations• Better understanding of singularities
May 2008 Dagstuhl 27
Conclusion
• A general mathematical framework for dynamic geometric computation with B-splines– Evolve to neighboring solution by following
tangent– Identify transition points by solving a rational
system– Compute transition events by computing
2nd fundamental form on manifold
May 2008 Dagstuhl 28
Conclusion
General mathematical framework for dynamic geometric computation with B-splines– Encode all solutions as a manifold in product
space of curves/surfaces parametric space and deformation control space
– Construct families of tangent vectors on the manifold
May 2008 Dagstuhl 29
ReferencesTheoretically Based Algorithms for Robustly Tracking
Intersection Curves of Deforming Surfaces, Xianming Chen, R.F. Riesenfeld, Elaine Cohen, James N. Damon, CAD,Vol 39, No 5, May 2007,389-397
May 2008 Dagstuhl 30
Dagstuhl 31May 2008
vielen Dank für die Einladung
Dagstuhl 32May 2008
und auf Wiedersehen
May 2008 33Dagstuhl
Conclusion• Solve dynamic intersection curves of
2 deforming B-spline surfaces• Deformation represented as generalized offset surfaces• Implemented in B-splines, exploiting its symbolic
computation and subdivision-based 0-dimensional root finding.
• Evolve ICs by following evolution vector field• Create, annihilate, merge or split IC by 2nd
order shape computation at critical points of a 2-manifold in a parametric 5-space.
May 2008 34Dagstuhl
Outline
1. Essential issues2. General mathematical frame3. Point-curve distance tracking4. Surface-surface intersection tracking5. Efficient NURBS symbolic
computation
May 2008 35Dagstuhl
• Evolution• Identification• Detection• Classification• Computation
May 2008 36Dagstuhl
Outline
1. Essential issues
2. General mathematical frame3. Point-curve distance tracking4. Surface-surface intersection tracking5. Efficient NURBS symbolic
computation
May 2008 37Dagstuhl
Singularities of Differential Map• f : Rm → Rn Jacobian matrix singular • f : Rm → R f1 = f2 = … = fm = 0
– Hessian matrix H = ( fij ), nonsingular– Critical points classified by Morse index of H
38May 2008 Dagstuhl
General Mathematical Frame -1
• Construct a manifold in the solution space
May 2008 39Dagstuhl
General Mathematical Frame -2
• Construct d families of tangent vector fields
• Define projection map from the manifold to control space
May 2008 40Dagstuhl
pqdI
ed
e
e
dT
T
T
:
,...)(
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11
11
General Mathematical Frame -1
Construct a manifold in the solution space
May 2008 41Dagstuhl
pccc
psssI
qpqdII
qp}{{ ,...,
21},...,
21
General Mathematical Frame -3
• Singularities of projection map – critical set in the solution
space – transition set in the control
space
May 2008 42Dagstuhl
General Mathematical Frame -3
• Identify singularities– subdivision-based constraint
solver• Robust guarantee for
0-dimensional solution– NURBS algebraic operation
• Just for point-curve distance tracking
– Robustness guarantee even though 1-dimensional
May 2008 43Dagstuhl
General Mathematical Frame -4• Evolution when away
from transition set
• d = 0 is simple
• d > 0 needs extra effort
– Heuristics from front propagation
» Extra d constraints
May 2008 44Dagstuhl
dqq :1
Dagstuhl 45
General Mathematical Frame -5 • Transition when
crossing transition set– Restrict the
projection to perturbation line• Morse function
– Local 2nd order differential computation to catch global topology change
45
qd LL 11:
May 2008
General Mathematical Frame -5 • Transition when
crossing transition set– Restrict the
projection to perturbation line• Morse function
– Local 2nd order differential computation to catch global topology change
May 2008 46Dagstuhl
qd LL 11:
Outline
1. Essential issues2. General mathematical frame
3. Point-curve distance tracking4. Surface-surface intersection tracking5. Efficient NURBS symbolic
computation
May 2008 47Dagstuhl
Critical Distance (CD)
May 2008 48
extremal and perpendicular
extremal and perpendicular
extremal and perpendicular
Dagstuhl
Type Discriminant D
May 2008 49Dagstuhl
Distance Tracking Problem
• Given critical distances of P to the curve
• If P is perturb on the plane by– Create any new CD s if any– Annihilate any old CD s if any– Evolve the rest of CD s
• Distance tracking without global searching
May 2008 50Dagstuhl
CD as a Space Point
May 2008 51Dagstuhl
Normal Bundle
May 2008 52Dagstuhl
Lifted Normal Bundle
May 2008 53
implicit surface =locus of CDs
Dagstuhl
Lifting the Perturbation
May 2008 54Dagstuhl
May 2008 55Dagstuhl
Tangent Vector Field
May 2008 56Dagstuhl
Evolution
May 2008 57Dagstuhl
Transition
May 2008 58Dagstuhl
Transition Type Classification
May 2008 59Dagstuhl
An Example
May 2008 60Dagstuhl
2-Stage Detection Algorithm
May 2008 61
Line hits bounding box of evolute
Line intersect diagonal of hit box
Dagstuhl
Transition Set: Extended Evolute
May 2008 62Dagstuhl
May 2008 63Dagstuhl
May 2008 64Dagstuhl
Outline
1. Essential issues2. General mathematical frame3. Point-curve distance tracking
4. Surface-surface intersection tracking
5. Efficient NURBS symbolic computation
May 2008 65Dagstuhl
Deformation as Generalized Offset
May 2008 66Dagstuhl
Evolution Vector Field
May 2008 67Dagstuhl
Local Basis
May 2008 68Dagstuhl
Evolution Vector Field
May 2008 69Dagstuhl
Local Basis
May 2008 70Dagstuhl
2-Manifold in 5-space
May 2008 71Dagstuhl
Evolution Vector Field
May 2008 72Dagstuhl
IC as Height Contour
May 2008 73Dagstuhl
Critical Points of Height Function
May 2008 74Dagstuhl
4 Generic Transition Events
May 2008 75Dagstuhl
Tangent Vector Fields
May 2008 76Dagstuhl
Computing Tangent Vector Fields
May 2008 77Dagstuhl
Computing Transition Events
May 2008 78Dagstuhl
May 2008 79Dagstuhl
May 2008 80Dagstuhl
May 2008 81Dagstuhl
Extension -1• Extra transition events
– At boundary points
– At boundary vertex points
May 2008 82Dagstuhl
Extension -2
• Triple point events
May 2008 83Dagstuhl
Evolution
Overall Process
May 2008 Dagstuhl 84
Classification
Computation
Identification
Detection
May 2008 Dagstuhl 85
May 2008 86Dagstuhl
New
• Evolution• Identification• Detection• Classification• Computation
May 2008 87Dagstuhl
Outline
1. Essential issues2. General mathematical frame3. Point-curve distance tracking4. Surface-surface intersection tracking
5. Efficient NURBS symbolic computation
May 2008 88Dagstuhl
Simple Equations
May 2008 89Dagstuhl
222
22
222
2
11)(2
1
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Amazing Results
May 2008 90Dagstuhl
D1 = p′ w - p w′D2 = p″ w - p w″D3 = p‴ w - p w‴ D21 = p″ w′ - p′ w″
May 2008 91
Transition Set as Evolute
Dagstuhl
Find Curve Vertices
May 2008 92Dagstuhl
Inflectional and Vertex Points
May 2008 93Dagstuhl
Conclusion
General mathematical framework for dynamic geometric computation with B-splines– Encode all solutions as a manifold in the
product space of curves/surfaces parametric space and deformation control space
– Construct families of tangent vectors on the manifold
May 2008 Dagstuhl 94
Conclusion
• A general mathematical framework for dynamic geometric computation with B-splines– Evolve to neighboring solution by following
tangent– Identify transition points by solving a rational
system– Compute transition events by computing 2nd
fundamental form on the manifold
May 2008 Dagstuhl 95
ReferencesTheoretically Based Algorithms for Robustly Tracking
Intersection Curves of Deforming Surfaces, Xianming Chen, R.F. Riesenfeld, Elaine Cohen, James N. Damon, CAD,Vol 39, No 5, May 2007,389-397
May 2008 Dagstuhl 96
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