VEGAS Effective Geometric Algorithms for Surfaces and Visibility Proposed in July 2004 Created in August 2005 Evaluation period: 2005-2006 – p. 1
VEGAS project members Permanent: Sylvain Lazard CR INRIA Sylvain Petitjean CR CNRS Hazel Everett Prof. Univ. Nancy 2 Xavier Goaoc CR INRIA (since May 2005) Laurent Dupont Assist. Prof. Univ. Nancy 2 (since Sept. 2005) Marc Pouget CR INRIA (since Sept. 2006) PhD. Students: Temp. Engineer: (since Sept. 2006) Postdoc: Joint with GEOMETRICA (since Oct. 2006) – p. 2
Scientific Objectives Contribute to the development of effective geometric computing treating complex geometric objects – p. 3
Scientific Objectives: Focus 3D Visibility problems Geometric computing with curved objects – p. 4
Scientific Objectives: Example How to generate high-quality rendered images of scenes modeled with low-degree algebraic surfaces? – p. 5
Scientific Objectives: Originality We attack all the aspects from theory to practice needed for the development of certified and effective geometric computing dedicated to non-discretized non-linear problems – p. 6
Scientific Objectives: Non-linear Objects Need to manipulate non-linear objects Classical approach: discretize into a mesh of triangles • universal, simple algorithms, hardware • numerical error, lots of triangles Our approach: algorithms that take into account the exact geometry of objects – p. 7
Scientific Objectives: Non-linear Objects Need to manipulate non-linear objects both concrete and abstract: – p. 8
Scientific Objectives: Robustness Is R to the left, to the right, or on line PQ ? � � P x Q x R x Q � � R P Compute the sign of P y Q y R y � � � � 1 1 1 � � Finite-precision floating-point computation � Topological incoherences � CRASH ! Robust algorithms: • treat all degenerate situations • implement geometric decisions exactly – p. 9
Scientific Achievements: Plan 3D visibility Geometric computing with curved surfaces – p. 10
3D Visibility: Panorama 3D Visibility problems • surface-to-surface visibility queries • shadows - limits of umbra and penumbra Computations are discretized and extremely costly Objectives: Efficient algorithms (No special use of graphics hardware) Long-term goals: more efficient rendering of higher quality – p. 11
3D Visibility: Research directions • Theory of lines in space • Algorithmic and implementation of 3D vis. data structures • 3D visibility and applications – p. 12
3D Visibility: Theory of lines in space 4 lines in 3D admit at most 2 (cc) transversals 4 segments in 3D admit at most 4 (cc) transversals [BELSW 2005] – p. 13
3D Visibility: Theory of lines in space Maximum number of lines tangent to 4 triangles? – p. 14
3D Visibility: Theory of lines in space Maximum number of lines tangent to 4 triangles? Upper bound: 162 (naive 4 · 3 4 = 324 ) Lower bound: 62 Example with “fat” triangles: 40 [BDLS 2005] – p. 14
3D Visibility: Theory of lines in space 4 spheres admit infinitely many tangents iff aligned centers and at least one common tangent [BGLP 2006] – p. 15
3D Visibility: Theory of lines in space Given a polyhedron P n with n faces approximating a surface in a reasonable way On average over all viewpoints the silhouette of P n is of size O ( √ n ) [G 2006] – p. 16
3D Visibility: Theory of lines in space Given n disjoint unit balls in R d Number of ordering in which a line pierces all the balls? l 3 l 1 : ABC B B A l 2 l 2 : ACB A C l 3 : BAC C l 1 The set of balls admits at most 2 distinct geometric permutations if n > 8 and at most 3 if n � 8 Tight bounds except for n = 4 ,..., 8 [CGN 2005] – p. 17
3D Visibility: Theory of lines in space Given n disjoint unit balls in R d Helly-type theorem for transversals to balls: If every subset of 4 d − 1 balls admit a line transversal then the set of balls admits a line transversal [CGHP 2006] – p. 18
3D Vis.: Complexity, Algorith. & Implem. Sets of free line segments tangent to objects Take into account the structure of the objects For n triangles organized into k convex polytopes • Size Θ ( n 2 k 2 ) in the worst case • Algorithm Θ ( n 2 k 2 log n ) in the worst case [BDD+ 2006] – p. 19
3D Visibility: Conclusion • Many fundamental results on the properties of free lines and line segments in space • Practical algorithm for computing 3D visibility global data structures • Ongoing implementation – p. 20
Scientific Achievements: Plan 3D visibility Geometric computing with curved surfaces – p. 21
Geometric comput. with curved surfaces Low-degree surfaces are everywhere 95 % of surfaces of mechanical objects are made up of quadrics and torii – p. 22
Intersection of quadrics Compute an exact parametric form of the intersection of two arbitrary implicit quadric surfaces Input: Q 1 : 4 x 2 + z 2 − 1 = 0 Q 2 : x 2 + 4 y 2 − z 2 − 1 = 0 Output: Smooth quartic 2 u 3 − 6 u − 2 7 u 2 + 3 u ∆ ( u ) � ± · 10 u 2 − 6 2 u 2 u 3 + 18 u 2 ∆ ( u ) = − 3 u 4 + 26 u 2 − 3 – p. 23
Intersection of quadrics Compute an exact parametric form of the intersection of two arbitrary implicit quadric surfaces Input: Q 1 : z 2 + xy = 0 Q 2 : x + yz = 0 Output: Line Cubic − u 3 0 u 4 u − 2 u 2 0 1 − 8 – p. 23
Intersection of quadrics Algorithm • Classification of the type of intersection in P 3 ( R ) (smooth quartic, cubic & line, 2 conics, etc.) • Parameterization of each component • Rational parameterization if one exists • Optimal or almost optimal parameterization in the degree of the extension of Z of the coefficients [DDLP03, D04, DLLP 05a, DLLP 05b, DLLP 05c] – p. 24
Intersection of quadrics Implementation [LPP 2006] • Efficient C++ implementation ( ∼ 20 000 lines) • On-line web server • Code distributed: QI (INRIA License) – p. 25
Intersection of quadrics Implementation [LPP 2006] • Efficient C++ implementation ( ∼ 20 000 lines) • On-line web server • Code distributed: QI (INRIA License) Applications • Interactions between potential energy surfaces Dept. of Chemistry, Imperial College, London • Image of conics seen by a catadioptric camera with a paraboloidal mirror IRIT (CNRS - Univ) Toulouse • Spacecraft thermal radiation analysis – p. 25
Intersection of quadrics Experiments [LPP 2006] • Random quadrics, coeffs up to 10 digits: < 50ms • Chess set: - 6 pieces (108 quadrics), 971 intersections - 3.4 ms on average – p. 26
Comput. with curved surf.: Conclusions Major leap forward on certified computations with quadrics – p. 27
Objectives of the project proposal • 3D visibility • Theory of lines in space • Algorithmics and implementation • Visibility and applications • Geometric computing with curved objects – p. 28
Objectives for the next four years • Theory of lines in space • 3D visibility • Theory of lines in space • Algorithmics and implementation • Visibility and applications (longer term) • Geometric computing with curved objects – p. 28
Objectives for the next four years Theory of lines in space • Discrete properties of sets of lines Combinatorial geometry: Helly-type theorems, geometric permutations • Combinatorial complexity Complexity of sets of lines in space • Efficient and effective computing on sets of lines Predicates for line transversals – p. 29
Objectives for the next four years 3D visibility • Theory & Algorithms Visibility skeleton for non-convex polyhedra Shadows • Implementation Visqueux : Visibility skeleton – p. 30
Objectives for the next four years Geometric computing with curved objects • Certified computation with low-degree surfaces Boundary evaluation, Medial axis of polyhedra Geometric computing with algebraic tools Improving QI • Math. investigation of geometric features on surfaces Differential geometry for smooth and discrete objects “Visual event” curves on smooth surfaces – p. 31
Collaborations GEOMETRICA, SALSA McGill (Montréal), Poly. Univ (New-York), KAIST (Korea) and many more punctual collaborations Also close contacts with CACAO, ALICE, ARTIS – p. 32
Conclusions Robust and efficient geometric algorithms dealing with curved objects may be obtained by using the right mathematical tools making thorough treatment of degenerate cases and a careful implementation of the primitive operations – p. 33
Recommend
More recommend
