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Boundary Representations and Topology

Boundary Representations and Topology. Dr. Scott Schaefer. Boundary Representations. Stores the boundary of a solid Geometry: vertex locations Topology: connectivity information Vertices Edges Faces. Boundary Representations. Constant time adjacency information For each vertex,

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Boundary Representations and Topology

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  1. Boundary Representations and Topology Dr. Scott Schaefer

  2. Boundary Representations • Stores the boundary of a solid • Geometry: vertex locations • Topology: connectivity information • Vertices • Edges • Faces

  3. Boundary Representations • Constant time adjacency information • For each vertex, • Find edges/faces touching vertex • For each edge, • Find vertices/faces touching edge • For each face, • Find vertices/edges touching face

  4. Boundary Representations • Typically assume the surface is manifold • A surface is manifold if, for every point, locally the surface is equivalent to an open disk

  5. Boundary Representations • Typically assume the surface is manifold • A surface is manifold if, for every point, locally the surface is equivalent to an open disk Locally manifold

  6. Boundary Representations • Typically assume the surface is manifold • A surface is manifold if, for every point, locally the surface is equivalent to an open disk Non-manifold edge

  7. Boundary Representations • Typically assume the surface is manifold • A surface is manifold if, for every point, locally the surface is equivalent to an open disk Non-manifold vertex

  8. Left face Right face Winged Edge Data Structure

  9. Left face Right face Winged Edge Data Structure WingedEdge { WingedEdge nextLeft, nextRight, prevLeft, prevRight; Face leftFace, rightFace; Vertex prevVert, nextVert; } Face { WingedEdge edge; } Vertex { WingedEdge edge; }

  10. Winged Edge Data Structure • Given a face, find all vertices touching that face • Given a vertex, find all edge-adjacent vertices • Given a face, find all adjacent faces Left face Right face

  11. Half Edge Data Structure face

  12. Half Edge Data Structure HalfEdge { HalfEdge next, prev, flip; Face face; Vertex origin; Edge edge; } Face { HalfEdge edge; // part of this face } face Vertex { HalfEdge edge; // points away } Edge { HalfEdge he; }

  13. Half Edge Data Structure • Very similar to WingedEdge Data Structure, but edges have unique orientation • Slightly more storage • Less conditional operations • Assumes polygons are oriented face

  14. QuadEdge Data Structure

  15. Navigating a QuadEdge Data Structure

  16. Navigating a QuadEdge Data Structure Origin

  17. Navigating a QuadEdge Data Structure Flip

  18. Navigating a QuadEdge Data Structure Rot

  19. Navigating a QuadEdge Data Structure invRot

  20. Navigating a QuadEdge Data Structure Prev

  21. Navigating a QuadEdge Data Structure Next

  22. QuadEdge Data Structure • Treats faces just like vertices • Stores both the shape and its dual • Beneficial for many types of subdivision surfaces

  23. Building a Topological Data Structure • Must connect adjacent edges/faces/vertices • Edges are critical in most data structures • Use a hash table indexed by two vertices face

  24. Euler Characteristic • V: number of vertices • E: number of edges • F: number of faces • G: genus of surface (number of holes)

  25. Euler Characteristic • V: number of vertices • E: number of edges • F: number of faces • G: genus of surface (number of holes)

  26. Euler Characteristic • V: number of vertices • E: number of edges • F: number of faces • G: genus of surface (number of holes)

  27. Euler Characteristic • V: number of vertices • E: number of edges • F: number of faces • G: genus of surface (number of holes)

  28. Euler Characteristic • V: number of vertices • E: number of edges • F: number of faces • G: genus of surface (number of holes)

  29. Topological Operations: Hole Filling • Find a half-edge whose flip is null • Use next and flip points to find next adjacent half-edge with null flip • Repeat until back at original half-edge • Create new half-edges along boundary and face containing those edges face

  30. Topological Operations: Edge Collapse

  31. Topological Operations: Edge Collapse • Set origin of all edges pointing outwards from two vertices to new vertex and vertex to one of the half-edges

  32. Topological Operations: Edge Collapse • Set origin of all edges pointing outwards from two vertices to new vertex and vertex to one of the half-edges • Set flip on outer edges to point to opposite edge

  33. Topological Operations: Edge Collapse • Set origin of all edges pointing outwards from two vertices to new vertex and vertex to one of the half-edges • Set flip on outer edges to point to opposite edge • Make sure half-edge pointer for edges point to correct half-edge and vice versa

  34. Topological Operations: Edge Collapse • Set origin of all edges pointing outwards from two vertices to new vertex and vertex to one of the half-edges • Set flip on outer edges to point to opposite edge • Make sure half-edge pointer for edges point to correct half-edge and vice versa • Update half-edge for wing-vertices

  35. Topological Operations: Edge Collapse • Set origin of all edges pointing outwards from two vertices to new vertex and vertex to one of the half-edges • Set flip on outer edges to point to opposite edge • Make sure half-edge pointer for edges point to correct half-edge and vice versa • Update half-edge for wing-vertices • Delete faces/edges/vertices

  36. Topological Operation: Edge Collapse • Edge collapse preserves topology as long as the local Euler characteristic of the surface remains the same

  37. Topological Operation: Edge Collapse • Edge collapse preserves topology as long as the local Euler characteristic of the surface remains the same

  38. Topological Operation: Edge Collapse • Edge collapse does NOT always preserve topology

  39. Detecting Unsafe Edge Collapses • Let N(v) be the set of vertices edge-adjacent to v • Safe to collapse if

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