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On Mesh Intersection: exact computation and efficiency

The document presents research by Lévy, Li, and Borgese on efficient and robust methods for mesh intersection. It covers algorithmic approaches for identifying intersecting triangles, including candidate selection, triangle-triangle intersection handling, and re-triangulation using constrained Delaunay triangulation strategies. The content is technical, referencing various configurations, computational techniques, and their implementation in programming contexts.

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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, BorgeseOn 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, BorgeseOn Mesh Intersection 2/∞ Lévy, Li, Borgese On Mesh Intersection 2/∞ Introduction IFP Energies Nouvelles
  • 3.
    Lévy, Li, BorgeseOn Mesh Intersection 3/∞ Lévy, Li, Borgese On Mesh Intersection 3/∞ Introduction
  • 4.
    Lévy, Li, BorgeseOn 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, BorgeseOn Mesh Intersection 5/∞ Lévy, Li, Borgese On Mesh Intersection 5/∞ Introduction
  • 6.
    Lévy, Li, BorgeseOn Mesh Intersection 6/∞ Lévy, Li, Borgese On Mesh Intersection 6/∞ Introduction
  • 7.
    Lévy, Li, BorgeseOn Mesh Intersection 7/∞ Lévy, Li, Borgese On Mesh Intersection 7/∞ Introduction
  • 8.
    Lévy, Li, BorgeseOn Mesh Intersection 8/∞ Lévy, Li, Borgese On Mesh Intersection 8/∞ Introduction
  • 9.
    Lévy, Li, BorgeseOn Mesh Intersection 9/∞ Lévy, Li, Borgese On Mesh Intersection 9/∞ Introduction
  • 10.
    Lévy, Li, BorgeseOn 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, BorgeseOn 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, BorgeseOn Mesh Intersection 12/∞ 1. Finding all the pairs of intersecting triangles NVidia
  • 13.
    Lévy, Li, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn Mesh Intersection 16/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 17.
    Lévy, Li, BorgeseOn Mesh Intersection 17/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 18.
    Lévy, Li, BorgeseOn Mesh Intersection 18/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 19.
    Lévy, Li, BorgeseOn Mesh Intersection 19/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 20.
    Lévy, Li, BorgeseOn Mesh Intersection 20/∞ NVidia Triangles array 3. Finding all the pairs of intersecting triangles
  • 21.
    Lévy, Li, BorgeseOn Mesh Intersection 21/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 22.
    Lévy, Li, BorgeseOn Mesh Intersection 22/∞ NVidia Triangles array 1. Finding all the pairs of intersecting triangles
  • 23.
    Lévy, Li, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn Mesh Intersection 27/∞ 1. Finding all the pairs of intersecting triangles
  • 28.
    Lévy, Li, BorgeseOn Mesh Intersection 28/∞ Node 1, sequence [b1, e1) 1. Finding all the pairs of intersecting triangles
  • 29.
    Lévy, Li, BorgeseOn Mesh Intersection 29/∞ Node 1, sequence [b1, e1) 1. Finding all the pairs of intersecting triangles Node 2, sequence [b2, e2)
  • 30.
    Lévy, Li, BorgeseOn 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, BorgeseOn 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, BorgeseOn Mesh Intersection 32/∞ 1. Finding all the pairs of intersecting triangles
  • 33.
    Lévy, Li, BorgeseOn Mesh Intersection 33/∞ 1. Finding all the pairs of intersecting triangles Early pruning
  • 34.
    Lévy, Li, BorgeseOn Mesh Intersection 34/∞ 1. Finding all the pairs of intersecting triangles Leaves with 1 triangle
  • 35.
    Lévy, Li, BorgeseOn Mesh Intersection 35/∞ 1. Finding all the pairs of intersecting triangles Recursive descent along the child af the larges node
  • 36.
    Lévy, Li, BorgeseOn Mesh Intersection 36/∞ 1. Finding all the pairs of intersecting triangles Axis Aligned Bounding Box Tree
  • 37.
    Lévy, Li, BorgeseOn Mesh Intersection 37/∞ 1. Finding all the pairs of intersecting triangles Axis Aligned Bounding Box Tree Stream of candidate triangle intersections
  • 38.
    Lévy, Li, BorgeseOn 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, BorgeseOn Mesh Intersection 39/∞ 2. Triangle-triangle intersection Generic configuration
  • 40.
    Lévy, Li, BorgeseOn Mesh Intersection 40/∞ 2. Triangle-triangle intersection Degenerate configurations
  • 41.
    Lévy, Li, BorgeseOn Mesh Intersection 41/∞ 2. Triangle-triangle intersection Degenerate configurations
  • 42.
    Lévy, Li, BorgeseOn Mesh Intersection 42/∞ 2. Triangle-triangle intersection Degenerate configurations
  • 43.
    Lévy, Li, BorgeseOn Mesh Intersection 43/∞ 2. Triangle-triangle intersection Degenerate configurations
  • 44.
    Lévy, Li, BorgeseOn 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, BorgeseOn 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, BorgeseOn Mesh Intersection 46/∞ 2. Triangle-triangle intersection
  • 47.
    Lévy, Li, BorgeseOn Mesh Intersection 47/∞ 2. Triangle-triangle intersection
  • 48.
    Lévy, Li, BorgeseOn Mesh Intersection 48/∞ Lévy, Li, Borgese On Mesh Intersection 48/∞ 2. Triangle-triangle intersection
  • 49.
    Lévy, Li, BorgeseOn Mesh Intersection 49/∞ Lévy, Li, Borgese On Mesh Intersection 49/∞ 2. Triangle-triangle intersection Floating point coordinates
  • 50.
    Lévy, Li, BorgeseOn Mesh Intersection 50/∞ Lévy, Li, Borgese On Mesh Intersection 50/∞ 2. Triangle-triangle intersection Intersection is not on the grid
  • 51.
    Lévy, Li, BorgeseOn Mesh Intersection 51/∞ Lévy, Li, Borgese On Mesh Intersection 51/∞ 2. Triangle-triangle intersection Computed intersection moves a bit !
  • 52.
    Lévy, Li, BorgeseOn Mesh Intersection 52/∞ Lévy, Li, Borgese On Mesh Intersection 52/∞ 2. Triangle-triangle intersection
  • 53.
    Lévy, Li, BorgeseOn Mesh Intersection 53/∞ Lévy, Li, Borgese On Mesh Intersection 53/∞ 2. Triangle-triangle intersection
  • 54.
    Lévy, Li, BorgeseOn Mesh Intersection 54/∞ 2. Triangle-triangle intersection computed exactly [Shewchuk 96,97]
  • 55.
    Lévy, Li, BorgeseOn Mesh Intersection 55/∞ 2. Triangle-triangle intersection Triangle Intersection Candidate triangle Intersections (from AABB)
  • 56.
    Lévy, Li, BorgeseOn Mesh Intersection 56/∞ 2. Triangle-triangle intersection Triangle Intersection Candidate triangle Intersections (from AABB) Intersection segments in symbolic form
  • 57.
    Lévy, Li, BorgeseOn 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, BorgeseOn 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, BorgeseOn Mesh Intersection 59/∞ 3. Re-triangulating the triangles
  • 60.
    Lévy, Li, BorgeseOn Mesh Intersection 60/∞ 3. Re-triangulating the triangles Constrained Delaunay 2d
  • 61.
    Lévy, Li, BorgeseOn Mesh Intersection 61/∞ 3. Re-triangulating the triangles
  • 62.
    Lévy, Li, BorgeseOn Mesh Intersection 62/∞ 3. Re-triangulating the triangles
  • 63.
    Lévy, Li, BorgeseOn Mesh Intersection 63/∞ 3. Re-triangulating the triangles
  • 64.
    Lévy, Li, BorgeseOn Mesh Intersection 64/∞ 3. Re-triangulating the triangles [Sloan 1992]
  • 65.
    Lévy, Li, BorgeseOn Mesh Intersection 65/∞ 3. Re-triangulating the triangles [Sloan 1992]
  • 66.
    Lévy, Li, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn 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, BorgeseOn Mesh Intersection 76/∞ 3. Re-triangulating the triangles Thingy10K, thing # 996816 “The Nemesis”
  • 77.
    Lévy, Li, BorgeseOn Mesh Intersection 77/∞ 3. Re-triangulating the triangles Thingy10K, thing # 996816 “The Nemesis” Some triangles with 5000 intersections !
  • 78.
    Lévy, Li, BorgeseOn Mesh Intersection 78/∞ 3. Re-triangulating the triangles
  • 79.
    Lévy, Li, BorgeseOn Mesh Intersection 79/∞ 3. Re-triangulating the triangles
  • 80.
    Lévy, Li, BorgeseOn Mesh Intersection 80/∞ 3. Re-triangulating the triangles
  • 81.
    Lévy, Li, BorgeseOn 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, BorgeseOn Mesh Intersection 82/∞ 4. Building the Weiler model AABB Cnstr. Delaunay
  • 83.
    Lévy, Li, BorgeseOn Mesh Intersection 83/∞ 4. Building the Weiler model Weiler AABB Cnstr. Delaunay
  • 84.
    Lévy, Li, BorgeseOn Mesh Intersection 84/∞ 4. Building the Weiler model Radial edges
  • 85.
    Lévy, Li, BorgeseOn Mesh Intersection 85/∞ 4. Building the Weiler model Radial edges
  • 86.
    Lévy, Li, BorgeseOn Mesh Intersection 86/∞ 4. Building the Weiler model Radial edges
  • 87.
    Lévy, Li, BorgeseOn Mesh Intersection 87/∞ 4. Building the Weiler model Radial edges
  • 88.
    Lévy, Li, BorgeseOn Mesh Intersection 88/∞ 4. Building the Weiler model Radial edges
  • 89.
    Lévy, Li, BorgeseOn Mesh Intersection 89/∞ 4. Building the Weiler model Radial edges Radial sorting (no arccos/arctan, everything with dot and cross product)
  • 90.
    Lévy, Li, BorgeseOn Mesh Intersection 90/∞ 4. Building the Weiler model Regions
  • 91.
    Lévy, Li, BorgeseOn 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, BorgeseOn Mesh Intersection 92/∞ 5. Examples
  • 93.
    Lévy, Li, BorgeseOn Mesh Intersection 93/∞ 5. Examples
  • 94.
    Lévy, Li, BorgeseOn Mesh Intersection 94/∞ 5. Examples
  • 95.
    Lévy, Li, BorgeseOn Mesh Intersection 95/∞ 5. Examples Mandaros (ELF)
  • 96.
    Lévy, Li, BorgeseOn Mesh Intersection 96/∞ 5. Examples Tessael HexDom for CCUS coupled flow-geomechanics sim with GEOS, Cooperation with Total Energies
  • 97.
    Lévy, Li, BorgeseOn Mesh Intersection 97/∞ 5. Examples IFP Energies Nouvelles
  • 98.
    Lévy, Li, BorgeseOn Mesh Intersection 98/∞ 5. Examples IFP Energies Nouvelles
  • 99.
    Lévy, Li, BorgeseOn Mesh Intersection 99/∞ 5. Examples Tessael ClipHex hybrid element meshing optimized for IFP Energies Nouvelles Temis flow simulator.
  • 100.
    Lévy, Li, BorgeseOn Mesh Intersection 100/∞ 5. Examples Model courtesy RING Consortium
  • 101.
    Lévy, Li, BorgeseOn Mesh Intersection 101/∞ 5. Examples Model courtesy RING Consortium
  • 102.
    Lévy, Li, BorgeseOn Mesh Intersection 102/∞ When there’s something strange In the (non-manifold) neighborhood, Who you gonna call ? GEOGRAM + GeO2 https://github.com/BrunoLevy/geogram