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On Mesh Intersection: exact computation and efficiency

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Bruno Levy
Bruno Levy

A new algorithm to compute intersections between meshes, using arbitrary-precision numbers, constrained Delaunay triangulation and auxiliary combinatorial data structures.

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Lévy, Li, Borgese On Mesh Intersection 1/∞ Bruno Lévy Inria Saclay – Laboratoire de Mathématiques d’Orsay Wan-Chiu Li, Cédric Borgese Tessael ON MESH INTERSECTION ROBUSTNESS AND EFFICIENCY
Lévy, Li, Borgese On Mesh Intersection 2/∞ Lévy, Li, Borgese On Mesh Intersection 2/∞ Introduction IFP Energies Nouvelles
Lévy, Li, Borgese On Mesh Intersection 3/∞ Lévy, Li, Borgese On Mesh Intersection 3/∞ Introduction
Lévy, Li, Borgese On Mesh Intersection 4/∞ Lévy, Li, Borgese On Mesh Intersection 4/∞ Introduction Gocad’s “cut” functionality : [Sword 1996] [Caumon, Sword, Mallet 2003] [Legentil et.al. 2022] [Mallet 2002] A stream of articles : [Cherchi, Livesu, Attene 2020-2022]
Lévy, Li, Borgese On Mesh Intersection 5/∞ Lévy, Li, Borgese On Mesh Intersection 5/∞ Introduction
Lévy, Li, Borgese On Mesh Intersection 6/∞ Lévy, Li, Borgese On Mesh Intersection 6/∞ Introduction
Lévy, Li, Borgese On Mesh Intersection 7/∞ Lévy, Li, Borgese On Mesh Intersection 7/∞ Introduction
Lévy, Li, Borgese On Mesh Intersection 8/∞ Lévy, Li, Borgese On Mesh Intersection 8/∞ Introduction
Lévy, Li, Borgese On Mesh Intersection 9/∞ Lévy, Li, Borgese On Mesh Intersection 9/∞ Introduction
Lévy, Li, Borgese On Mesh Intersection 10/∞ The algorithm Find candidate Δ Δ intersections Compute Δ Δ intersections Retriangulate the Δ’s Find the regions 1 2 3 4
Lévy, Li, Borgese On Mesh Intersection 11/∞ Find candidate Δ Δ intersections Δ Δ intersect Constrained Delaunay 2d Radial Sort 1 2 3 4 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 12/∞ 1. Finding all the pairs of intersecting triangles NVidia
Lévy, Li, Borgese On Mesh Intersection 13/∞ NVidia N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 14/∞ NVidia N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 15/∞ NVidia N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 left(n) = 2*n right(n) = 2*n+1 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 16/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 17/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 18/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 19/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 20/∞ NVidia Triangles array 3. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 21/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 22/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 23/∞ N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 24/∞ N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 25/∞ N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 26/∞ N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Delage Devillers Spatial sort in CGAL std::nth_element() 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 27/∞ 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 28/∞ Node 1, sequence [b1, e1) 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 29/∞ Node 1, sequence [b1, e1) 1. Finding all the pairs of intersecting triangles Node 2, sequence [b2, e2)
Lévy, Li, Borgese On Mesh Intersection 30/∞ Node 1, sequence [b1, e1) 1. Finding all the pairs of intersecting triangles Node 2, sequence [b2, e2) intersect(1, 0, N, 1, 0, N) Find all intersections:
Lévy, Li, Borgese On Mesh Intersection 31/∞ Node 1, sequence [b1, e1) 1. Finding all the pairs of intersecting triangles Node 2, sequence [b2, e2) intersect(1, 0, N, 1, 0, N) Find all intersections: Whole tree Whole tree
Lévy, Li, Borgese On Mesh Intersection 32/∞ 1. Finding all the pairs of intersecting triangles
Lévy, Li, Borgese On Mesh Intersection 33/∞ 1. Finding all the pairs of intersecting triangles Early pruning
Lévy, Li, Borgese On Mesh Intersection 34/∞ 1. Finding all the pairs of intersecting triangles Leaves with 1 triangle
Lévy, Li, Borgese On Mesh Intersection 35/∞ 1. Finding all the pairs of intersecting triangles Recursive descent along the child af the larges node
Lévy, Li, Borgese On Mesh Intersection 36/∞ 1. Finding all the pairs of intersecting triangles Axis Aligned Bounding Box Tree
Lévy, Li, Borgese On Mesh Intersection 37/∞ 1. Finding all the pairs of intersecting triangles Axis Aligned Bounding Box Tree Stream of candidate triangle intersections
Lévy, Li, Borgese On Mesh Intersection 38/∞ Axis Aligned Bounding Box Tree Δ Δ intersect Constrained Delaunay 2d Radial Sort 1 2 3 4 2. Triangle-triangle intersection
Lévy, Li, Borgese On Mesh Intersection 39/∞ 2. Triangle-triangle intersection Generic configuration
Lévy, Li, Borgese On Mesh Intersection 40/∞ 2. Triangle-triangle intersection Degenerate configurations
Lévy, Li, Borgese On Mesh Intersection 41/∞ 2. Triangle-triangle intersection Degenerate configurations
Lévy, Li, Borgese On Mesh Intersection 42/∞ 2. Triangle-triangle intersection Degenerate configurations
Lévy, Li, Borgese On Mesh Intersection 43/∞ 2. Triangle-triangle intersection Degenerate configurations
Lévy, Li, Borgese On Mesh Intersection 44/∞ 2. Triangle-triangle intersection v1 v2 v3 e1 e2 e3 T Σt = { v1, v2, v3, e1, e2, e3, T }
Lévy, Li, Borgese On Mesh Intersection 45/∞ 2. Triangle-triangle intersection v1 v2 v3 e1 e2 e3 T v’1 v’2 v’3 e’1 e’2 e’3 T’ (Naïve algorithm)
Lévy, Li, Borgese On Mesh Intersection 46/∞ 2. Triangle-triangle intersection
Lévy, Li, Borgese On Mesh Intersection 47/∞ 2. Triangle-triangle intersection
Lévy, Li, Borgese On Mesh Intersection 48/∞ Lévy, Li, Borgese On Mesh Intersection 48/∞ 2. Triangle-triangle intersection
Lévy, Li, Borgese On Mesh Intersection 49/∞ Lévy, Li, Borgese On Mesh Intersection 49/∞ 2. Triangle-triangle intersection Floating point coordinates
Lévy, Li, Borgese On Mesh Intersection 50/∞ Lévy, Li, Borgese On Mesh Intersection 50/∞ 2. Triangle-triangle intersection Intersection is not on the grid
Lévy, Li, Borgese On Mesh Intersection 51/∞ Lévy, Li, Borgese On Mesh Intersection 51/∞ 2. Triangle-triangle intersection Computed intersection moves a bit !
Lévy, Li, Borgese On Mesh Intersection 52/∞ Lévy, Li, Borgese On Mesh Intersection 52/∞ 2. Triangle-triangle intersection
Lévy, Li, Borgese On Mesh Intersection 53/∞ Lévy, Li, Borgese On Mesh Intersection 53/∞ 2. Triangle-triangle intersection
Lévy, Li, Borgese On Mesh Intersection 54/∞ 2. Triangle-triangle intersection computed exactly [Shewchuk 96,97]
Lévy, Li, Borgese On Mesh Intersection 55/∞ 2. Triangle-triangle intersection Triangle Intersection Candidate triangle Intersections (from AABB)
Lévy, Li, Borgese On Mesh Intersection 56/∞ 2. Triangle-triangle intersection Triangle Intersection Candidate triangle Intersections (from AABB) Intersection segments in symbolic form
Lévy, Li, Borgese On Mesh Intersection 57/∞ 2. Triangle-triangle intersection Triangle Intersection Candidate triangle Intersections (from AABB) Intersection segments in symbolic form V1.E1 ∩V2.T V1.T ∩ V2.E3
Lévy, Li, Borgese On Mesh Intersection 58/∞ Axis Aligned Bounding Box Tree Δ Δ intersect Constrained Delaunay 2d Radial Sort 1 2 3 4 3. Re-triangulating the triangles
Lévy, Li, Borgese On Mesh Intersection 59/∞ 3. Re-triangulating the triangles
Lévy, Li, Borgese On Mesh Intersection 60/∞ 3. Re-triangulating the triangles Constrained Delaunay 2d
Lévy, Li, Borgese On Mesh Intersection 61/∞ 3. Re-triangulating the triangles
Lévy, Li, Borgese On Mesh Intersection 62/∞ 3. Re-triangulating the triangles
Lévy, Li, Borgese On Mesh Intersection 63/∞ 3. Re-triangulating the triangles
Lévy, Li, Borgese On Mesh Intersection 64/∞ 3. Re-triangulating the triangles [Sloan 1992]
Lévy, Li, Borgese On Mesh Intersection 65/∞ 3. Re-triangulating the triangles [Sloan 1992]
Lévy, Li, Borgese On Mesh Intersection 66/∞ 3. Re-triangulating the triangles Constrained Delaunay 2d • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p
Lévy, Li, Borgese On Mesh Intersection 67/∞ 3. Re-triangulating the triangles Constrained Delaunay 2d • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p Our points are intersections
Lévy, Li, Borgese On Mesh Intersection 68/∞ 3. Re-triangulating the triangles Constrained Delaunay 2d • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p Our points are intersections Let’s compute them exactly !
Lévy, Li, Borgese On Mesh Intersection 69/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p Our points are intersections Let’s compute them exactly ! Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); };
Lévy, Li, Borgese On Mesh Intersection 70/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p Our points are intersections Let’s compute them exactly ! Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); };
Lévy, Li, Borgese On Mesh Intersection 71/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p Our points are intersections Let’s compute them exactly ! Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); }; How to divide ? Homogeneous coordinates ! (like in OpenGL) [x y z w] ↕ [ x/w y/w z/w ]
Lévy, Li, Borgese On Mesh Intersection 72/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p How to make it faster ? Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); }; How to divide ? Homogeneous coordinates ! (like in OpenGL) [x y z w] ↕ [ x/w y/w z/w ]
Lévy, Li, Borgese On Mesh Intersection 73/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p How to make it faster ? Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); }; How to divide ? Homogeneous coordinates ! (like in OpenGL) [x y z w] ↕ [ x/w y/w z/w ] class interval_nt filter
Lévy, Li, Borgese On Mesh Intersection 74/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p How to make it faster ? Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); }; How to divide ? Homogeneous coordinates ! (like in OpenGL) [x y z w] ↕ [ x/w y/w z/w ] class interval_nt filter Does it always work ? What about overflows / underflows ?
Lévy, Li, Borgese On Mesh Intersection 75/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p How to make it faster ? Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); }; How to divide ? Homogeneous coordinates ! (like in OpenGL) [x y z w] ↕ [ x/w y/w z/w ] class interval_nt filter Does it always work ? What about overflows / underflows ? multiprecision class exact_nt
Lévy, Li, Borgese On Mesh Intersection 76/∞ 3. Re-triangulating the triangles Thingy10K, thing # 996816 “The Nemesis”
Lévy, Li, Borgese On Mesh Intersection 77/∞ 3. Re-triangulating the triangles Thingy10K, thing # 996816 “The Nemesis” Some triangles with 5000 intersections !
Lévy, Li, Borgese On Mesh Intersection 78/∞ 3. Re-triangulating the triangles
Lévy, Li, Borgese On Mesh Intersection 79/∞ 3. Re-triangulating the triangles
Lévy, Li, Borgese On Mesh Intersection 80/∞ 3. Re-triangulating the triangles
Lévy, Li, Borgese On Mesh Intersection 81/∞ Axis Aligned Bounding Box Tree Δ Δ intersect Constrained Delaunay 2d Radial Sort 1 2 3 4 4. Building the Weiler model
Lévy, Li, Borgese On Mesh Intersection 82/∞ 4. Building the Weiler model AABB Cnstr. Delaunay
Lévy, Li, Borgese On Mesh Intersection 83/∞ 4. Building the Weiler model Weiler AABB Cnstr. Delaunay
Lévy, Li, Borgese On Mesh Intersection 84/∞ 4. Building the Weiler model Radial edges
Lévy, Li, Borgese On Mesh Intersection 85/∞ 4. Building the Weiler model Radial edges
Lévy, Li, Borgese On Mesh Intersection 86/∞ 4. Building the Weiler model Radial edges
Lévy, Li, Borgese On Mesh Intersection 87/∞ 4. Building the Weiler model Radial edges
Lévy, Li, Borgese On Mesh Intersection 88/∞ 4. Building the Weiler model Radial edges
Lévy, Li, Borgese On Mesh Intersection 89/∞ 4. Building the Weiler model Radial edges Radial sorting (no arccos/arctan, everything with dot and cross product)
Lévy, Li, Borgese On Mesh Intersection 90/∞ 4. Building the Weiler model Regions
Lévy, Li, Borgese On Mesh Intersection 91/∞ Find candidate Δ Δ intersections Compute Δ Δ intersections Retriangulate the Δ’s Find the regions 1 2 3 4 5. Examples Results, Examples 5
Lévy, Li, Borgese On Mesh Intersection 92/∞ 5. Examples
Lévy, Li, Borgese On Mesh Intersection 93/∞ 5. Examples
Lévy, Li, Borgese On Mesh Intersection 94/∞ 5. Examples
Lévy, Li, Borgese On Mesh Intersection 95/∞ 5. Examples Mandaros (ELF)
Lévy, Li, Borgese On Mesh Intersection 96/∞ 5. Examples Tessael HexDom for CCUS coupled flow-geomechanics sim with GEOS, Cooperation with Total Energies
Lévy, Li, Borgese On Mesh Intersection 97/∞ 5. Examples IFP Energies Nouvelles
Lévy, Li, Borgese On Mesh Intersection 98/∞ 5. Examples IFP Energies Nouvelles
Lévy, Li, Borgese On Mesh Intersection 99/∞ 5. Examples Tessael ClipHex hybrid element meshing optimized for IFP Energies Nouvelles Temis flow simulator.
Lévy, Li, Borgese On Mesh Intersection 100/∞ 5. Examples Model courtesy RING Consortium
Lévy, Li, Borgese On Mesh Intersection 101/∞ 5. Examples Model courtesy RING Consortium
Lévy, Li, Borgese On Mesh Intersection 102/∞ When there’s something strange In the (non-manifold) neighborhood, Who you gonna call ? GEOGRAM + GeO2 https://github.com/BrunoLevy/geogram

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On Mesh Intersection: exact computation and efficiency

  • 1. Lévy, Li, Borgese On Mesh Intersection 1/∞ Bruno Lévy Inria Saclay – Laboratoire de Mathématiques d’Orsay Wan-Chiu Li, Cédric Borgese Tessael ON MESH INTERSECTION ROBUSTNESS AND EFFICIENCY
  • 2. Lévy, Li, Borgese On Mesh Intersection 2/∞ Lévy, Li, Borgese On Mesh Intersection 2/∞ Introduction IFP Energies Nouvelles
  • 3. Lévy, Li, Borgese On Mesh Intersection 3/∞ Lévy, Li, Borgese On Mesh Intersection 3/∞ Introduction
  • 4. Lévy, Li, Borgese On Mesh Intersection 4/∞ Lévy, Li, Borgese On Mesh Intersection 4/∞ Introduction Gocad’s “cut” functionality : [Sword 1996] [Caumon, Sword, Mallet 2003] [Legentil et.al. 2022] [Mallet 2002] A stream of articles : [Cherchi, Livesu, Attene 2020-2022]
  • 5. Lévy, Li, Borgese On Mesh Intersection 5/∞ Lévy, Li, Borgese On Mesh Intersection 5/∞ Introduction
  • 6. Lévy, Li, Borgese On Mesh Intersection 6/∞ Lévy, Li, Borgese On Mesh Intersection 6/∞ Introduction
  • 7. Lévy, Li, Borgese On Mesh Intersection 7/∞ Lévy, Li, Borgese On Mesh Intersection 7/∞ Introduction
  • 8. Lévy, Li, Borgese On Mesh Intersection 8/∞ Lévy, Li, Borgese On Mesh Intersection 8/∞ Introduction
  • 9. Lévy, Li, Borgese On Mesh Intersection 9/∞ Lévy, Li, Borgese On Mesh Intersection 9/∞ Introduction
  • 10. Lévy, Li, Borgese On Mesh Intersection 10/∞ The algorithm Find candidate Δ Δ intersections Compute Δ Δ intersections Retriangulate the Δ’s Find the regions 1 2 3 4
  • 11. Lévy, Li, Borgese On Mesh Intersection 11/∞ Find candidate Δ Δ intersections Δ Δ intersect Constrained Delaunay 2d Radial Sort 1 2 3 4 1. Finding all the pairs of intersecting triangles
  • 12. Lévy, Li, Borgese On Mesh Intersection 12/∞ 1. Finding all the pairs of intersecting triangles NVidia
  • 13. Lévy, Li, Borgese On Mesh Intersection 13/∞ NVidia N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Finding all the pairs of intersecting triangles
  • 14. Lévy, Li, Borgese On Mesh Intersection 14/∞ NVidia N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Finding all the pairs of intersecting triangles
  • 15. Lévy, Li, Borgese On Mesh Intersection 15/∞ NVidia N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 left(n) = 2*n right(n) = 2*n+1 1. Finding all the pairs of intersecting triangles
  • 16. Lévy, Li, Borgese On Mesh Intersection 16/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 17. Lévy, Li, Borgese On Mesh Intersection 17/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 18. Lévy, Li, Borgese On Mesh Intersection 18/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 19. Lévy, Li, Borgese On Mesh Intersection 19/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 20. Lévy, Li, Borgese On Mesh Intersection 20/∞ NVidia Triangles array 3. Finding all the pairs of intersecting triangles
  • 21. Lévy, Li, Borgese On Mesh Intersection 21/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 22. Lévy, Li, Borgese On Mesh Intersection 22/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 23. Lévy, Li, Borgese On Mesh Intersection 23/∞ N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Finding all the pairs of intersecting triangles
  • 24. Lévy, Li, Borgese On Mesh Intersection 24/∞ N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Finding all the pairs of intersecting triangles
  • 25. Lévy, Li, Borgese On Mesh Intersection 25/∞ N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Finding all the pairs of intersecting triangles
  • 26. Lévy, Li, Borgese On Mesh Intersection 26/∞ N1 N2 N3 N4 N5 N6 N7 01 02 03 04 05 06 07 08 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Delage Devillers Spatial sort in CGAL std::nth_element() 1. Finding all the pairs of intersecting triangles
  • 27. Lévy, Li, Borgese On Mesh Intersection 27/∞ 1. Finding all the pairs of intersecting triangles
  • 28. Lévy, Li, Borgese On Mesh Intersection 28/∞ Node 1, sequence [b1, e1) 1. Finding all the pairs of intersecting triangles
  • 29. Lévy, Li, Borgese On Mesh Intersection 29/∞ Node 1, sequence [b1, e1) 1. Finding all the pairs of intersecting triangles Node 2, sequence [b2, e2)
  • 30. Lévy, Li, Borgese On Mesh Intersection 30/∞ Node 1, sequence [b1, e1) 1. Finding all the pairs of intersecting triangles Node 2, sequence [b2, e2) intersect(1, 0, N, 1, 0, N) Find all intersections:
  • 31. Lévy, Li, Borgese On Mesh Intersection 31/∞ Node 1, sequence [b1, e1) 1. Finding all the pairs of intersecting triangles Node 2, sequence [b2, e2) intersect(1, 0, N, 1, 0, N) Find all intersections: Whole tree Whole tree
  • 32. Lévy, Li, Borgese On Mesh Intersection 32/∞ 1. Finding all the pairs of intersecting triangles
  • 33. Lévy, Li, Borgese On Mesh Intersection 33/∞ 1. Finding all the pairs of intersecting triangles Early pruning
  • 34. Lévy, Li, Borgese On Mesh Intersection 34/∞ 1. Finding all the pairs of intersecting triangles Leaves with 1 triangle
  • 35. Lévy, Li, Borgese On Mesh Intersection 35/∞ 1. Finding all the pairs of intersecting triangles Recursive descent along the child af the larges node
  • 36. Lévy, Li, Borgese On Mesh Intersection 36/∞ 1. Finding all the pairs of intersecting triangles Axis Aligned Bounding Box Tree
  • 37. Lévy, Li, Borgese On Mesh Intersection 37/∞ 1. Finding all the pairs of intersecting triangles Axis Aligned Bounding Box Tree Stream of candidate triangle intersections
  • 38. Lévy, Li, Borgese On Mesh Intersection 38/∞ Axis Aligned Bounding Box Tree Δ Δ intersect Constrained Delaunay 2d Radial Sort 1 2 3 4 2. Triangle-triangle intersection
  • 39. Lévy, Li, Borgese On Mesh Intersection 39/∞ 2. Triangle-triangle intersection Generic configuration
  • 40. Lévy, Li, Borgese On Mesh Intersection 40/∞ 2. Triangle-triangle intersection Degenerate configurations
  • 41. Lévy, Li, Borgese On Mesh Intersection 41/∞ 2. Triangle-triangle intersection Degenerate configurations
  • 42. Lévy, Li, Borgese On Mesh Intersection 42/∞ 2. Triangle-triangle intersection Degenerate configurations
  • 43. Lévy, Li, Borgese On Mesh Intersection 43/∞ 2. Triangle-triangle intersection Degenerate configurations
  • 44. Lévy, Li, Borgese On Mesh Intersection 44/∞ 2. Triangle-triangle intersection v1 v2 v3 e1 e2 e3 T Σt = { v1, v2, v3, e1, e2, e3, T }
  • 45. Lévy, Li, Borgese On Mesh Intersection 45/∞ 2. Triangle-triangle intersection v1 v2 v3 e1 e2 e3 T v’1 v’2 v’3 e’1 e’2 e’3 T’ (Naïve algorithm)
  • 46. Lévy, Li, Borgese On Mesh Intersection 46/∞ 2. Triangle-triangle intersection
  • 47. Lévy, Li, Borgese On Mesh Intersection 47/∞ 2. Triangle-triangle intersection
  • 48. Lévy, Li, Borgese On Mesh Intersection 48/∞ Lévy, Li, Borgese On Mesh Intersection 48/∞ 2. Triangle-triangle intersection
  • 49. Lévy, Li, Borgese On Mesh Intersection 49/∞ Lévy, Li, Borgese On Mesh Intersection 49/∞ 2. Triangle-triangle intersection Floating point coordinates
  • 50. Lévy, Li, Borgese On Mesh Intersection 50/∞ Lévy, Li, Borgese On Mesh Intersection 50/∞ 2. Triangle-triangle intersection Intersection is not on the grid
  • 51. Lévy, Li, Borgese On Mesh Intersection 51/∞ Lévy, Li, Borgese On Mesh Intersection 51/∞ 2. Triangle-triangle intersection Computed intersection moves a bit !
  • 52. Lévy, Li, Borgese On Mesh Intersection 52/∞ Lévy, Li, Borgese On Mesh Intersection 52/∞ 2. Triangle-triangle intersection
  • 53. Lévy, Li, Borgese On Mesh Intersection 53/∞ Lévy, Li, Borgese On Mesh Intersection 53/∞ 2. Triangle-triangle intersection
  • 54. Lévy, Li, Borgese On Mesh Intersection 54/∞ 2. Triangle-triangle intersection computed exactly [Shewchuk 96,97]
  • 55. Lévy, Li, Borgese On Mesh Intersection 55/∞ 2. Triangle-triangle intersection Triangle Intersection Candidate triangle Intersections (from AABB)
  • 56. Lévy, Li, Borgese On Mesh Intersection 56/∞ 2. Triangle-triangle intersection Triangle Intersection Candidate triangle Intersections (from AABB) Intersection segments in symbolic form
  • 57. Lévy, Li, Borgese On Mesh Intersection 57/∞ 2. Triangle-triangle intersection Triangle Intersection Candidate triangle Intersections (from AABB) Intersection segments in symbolic form V1.E1 ∩V2.T V1.T ∩ V2.E3
  • 58. Lévy, Li, Borgese On Mesh Intersection 58/∞ Axis Aligned Bounding Box Tree Δ Δ intersect Constrained Delaunay 2d Radial Sort 1 2 3 4 3. Re-triangulating the triangles
  • 59. Lévy, Li, Borgese On Mesh Intersection 59/∞ 3. Re-triangulating the triangles
  • 60. Lévy, Li, Borgese On Mesh Intersection 60/∞ 3. Re-triangulating the triangles Constrained Delaunay 2d
  • 61. Lévy, Li, Borgese On Mesh Intersection 61/∞ 3. Re-triangulating the triangles
  • 62. Lévy, Li, Borgese On Mesh Intersection 62/∞ 3. Re-triangulating the triangles
  • 63. Lévy, Li, Borgese On Mesh Intersection 63/∞ 3. Re-triangulating the triangles
  • 64. Lévy, Li, Borgese On Mesh Intersection 64/∞ 3. Re-triangulating the triangles [Sloan 1992]
  • 65. Lévy, Li, Borgese On Mesh Intersection 65/∞ 3. Re-triangulating the triangles [Sloan 1992]
  • 66. Lévy, Li, Borgese On Mesh Intersection 66/∞ 3. Re-triangulating the triangles Constrained Delaunay 2d • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p
  • 67. Lévy, Li, Borgese On Mesh Intersection 67/∞ 3. Re-triangulating the triangles Constrained Delaunay 2d • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p Our points are intersections
  • 68. Lévy, Li, Borgese On Mesh Intersection 68/∞ 3. Re-triangulating the triangles Constrained Delaunay 2d • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p Our points are intersections Let’s compute them exactly !
  • 69. Lévy, Li, Borgese On Mesh Intersection 69/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p Our points are intersections Let’s compute them exactly ! Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); };
  • 70. Lévy, Li, Borgese On Mesh Intersection 70/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p Our points are intersections Let’s compute them exactly ! Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); };
  • 71. Lévy, Li, Borgese On Mesh Intersection 71/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p Our points are intersections Let’s compute them exactly ! Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); }; How to divide ? Homogeneous coordinates ! (like in OpenGL) [x y z w] ↕ [ x/w y/w z/w ]
  • 72. Lévy, Li, Borgese On Mesh Intersection 72/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p How to make it faster ? Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); }; How to divide ? Homogeneous coordinates ! (like in OpenGL) [x y z w] ↕ [ x/w y/w z/w ]
  • 73. Lévy, Li, Borgese On Mesh Intersection 73/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p How to make it faster ? Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); }; How to divide ? Homogeneous coordinates ! (like in OpenGL) [x y z w] ↕ [ x/w y/w z/w ] class interval_nt filter
  • 74. Lévy, Li, Borgese On Mesh Intersection 74/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p How to make it faster ? Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); }; How to divide ? Homogeneous coordinates ! (like in OpenGL) [x y z w] ↕ [ x/w y/w z/w ] class interval_nt filter Does it always work ? What about overflows / underflows ?
  • 75. Lévy, Li, Borgese On Mesh Intersection 75/∞ 3. Re-triangulating the triangles • Orient2d(p1,p2,p3) • InCircle(p1,p2,p3,p4) • CreateIntersection(p1,p2,p3,p4) → p How to make it faster ? Arithmetic expansions (like in Shewchuk’s predicates) class expansion_nt { expansion_nt(double x); expansion_nt operator+(const expansion_nt& rhs); expansion_nt operator-(const expansion_nt& rhs); expansion_nt operator*(const expansion_nt& rhs); }; How to divide ? Homogeneous coordinates ! (like in OpenGL) [x y z w] ↕ [ x/w y/w z/w ] class interval_nt filter Does it always work ? What about overflows / underflows ? multiprecision class exact_nt
  • 76. Lévy, Li, Borgese On Mesh Intersection 76/∞ 3. Re-triangulating the triangles Thingy10K, thing # 996816 “The Nemesis”
  • 77. Lévy, Li, Borgese On Mesh Intersection 77/∞ 3. Re-triangulating the triangles Thingy10K, thing # 996816 “The Nemesis” Some triangles with 5000 intersections !
  • 78. Lévy, Li, Borgese On Mesh Intersection 78/∞ 3. Re-triangulating the triangles
  • 79. Lévy, Li, Borgese On Mesh Intersection 79/∞ 3. Re-triangulating the triangles
  • 80. Lévy, Li, Borgese On Mesh Intersection 80/∞ 3. Re-triangulating the triangles
  • 81. Lévy, Li, Borgese On Mesh Intersection 81/∞ Axis Aligned Bounding Box Tree Δ Δ intersect Constrained Delaunay 2d Radial Sort 1 2 3 4 4. Building the Weiler model
  • 82. Lévy, Li, Borgese On Mesh Intersection 82/∞ 4. Building the Weiler model AABB Cnstr. Delaunay
  • 83. Lévy, Li, Borgese On Mesh Intersection 83/∞ 4. Building the Weiler model Weiler AABB Cnstr. Delaunay
  • 84. Lévy, Li, Borgese On Mesh Intersection 84/∞ 4. Building the Weiler model Radial edges
  • 85. Lévy, Li, Borgese On Mesh Intersection 85/∞ 4. Building the Weiler model Radial edges
  • 86. Lévy, Li, Borgese On Mesh Intersection 86/∞ 4. Building the Weiler model Radial edges
  • 87. Lévy, Li, Borgese On Mesh Intersection 87/∞ 4. Building the Weiler model Radial edges
  • 88. Lévy, Li, Borgese On Mesh Intersection 88/∞ 4. Building the Weiler model Radial edges
  • 89. Lévy, Li, Borgese On Mesh Intersection 89/∞ 4. Building the Weiler model Radial edges Radial sorting (no arccos/arctan, everything with dot and cross product)
  • 90. Lévy, Li, Borgese On Mesh Intersection 90/∞ 4. Building the Weiler model Regions
  • 91. Lévy, Li, Borgese On Mesh Intersection 91/∞ Find candidate Δ Δ intersections Compute Δ Δ intersections Retriangulate the Δ’s Find the regions 1 2 3 4 5. Examples Results, Examples 5
  • 92. Lévy, Li, Borgese On Mesh Intersection 92/∞ 5. Examples
  • 93. Lévy, Li, Borgese On Mesh Intersection 93/∞ 5. Examples
  • 94. Lévy, Li, Borgese On Mesh Intersection 94/∞ 5. Examples
  • 95. Lévy, Li, Borgese On Mesh Intersection 95/∞ 5. Examples Mandaros (ELF)
  • 96. Lévy, Li, Borgese On Mesh Intersection 96/∞ 5. Examples Tessael HexDom for CCUS coupled flow-geomechanics sim with GEOS, Cooperation with Total Energies
  • 97. Lévy, Li, Borgese On Mesh Intersection 97/∞ 5. Examples IFP Energies Nouvelles
  • 98. Lévy, Li, Borgese On Mesh Intersection 98/∞ 5. Examples IFP Energies Nouvelles
  • 99. Lévy, Li, Borgese On Mesh Intersection 99/∞ 5. Examples Tessael ClipHex hybrid element meshing optimized for IFP Energies Nouvelles Temis flow simulator.
  • 100. Lévy, Li, Borgese On Mesh Intersection 100/∞ 5. Examples Model courtesy RING Consortium
  • 101. Lévy, Li, Borgese On Mesh Intersection 101/∞ 5. Examples Model courtesy RING Consortium
  • 102. Lévy, Li, Borgese On Mesh Intersection 102/∞ When there’s something strange In the (non-manifold) neighborhood, Who you gonna call ? GEOGRAM + GeO2 https://github.com/BrunoLevy/geogram