Polygonal Meshes. 3D Object Representations. 3D Object Representations. 3D Polygonal Mesh. 3D Polygonal Mesh. Geometry background

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1 3D Object Representations Polygonal Meshes Adam Finkelstein & Tim Weyrich Princeton University C0S 426, Spring 2008 Points o Range image o Point cloud Surfaces o Polygonal mesh o Subdivision o Parametric o Implicit Solids o Voxels o BSP tree o CSG o Sweep High-level structures o Scene graph o Application specific 1 2 3D Object Representations 3D Polygonal Mesh Points o Range image o Point cloud Surfaces! Polygonal mesh o Subdivision o Parametric o Implicit Solids o Voxels o BSP tree o CSG o Sweep High-level structures o Scene graph o Application specific Set of polygons representing a 2D surface embedded in 3D Isenberg 3 4 3D Polygonal Mesh Geometry & topology Edge Face Vertex (x,y,z) Geometry background Scene is usually approximated by 3D primitives o Point o Vector o Line segment o Ray o Line o Plane o Polygon Zorin & Schroeder 5 6

2 3D Point Specifies a location o Represented by three coordinates o Infinitely small Coordinate x; Coordinate y; Coordinate z; } Point; (x,y,z) 3D Vector Specifies a direction and a magnitude o Represented by three coordinates o Magnitude V = sqrt(dx dx + dy dy + dz dz) o Has no location Coordinate dx; Coordinate dy; Coordinate dz; } Vector; (dx,dy,dz) Origin 7 8 3D Vector Dot product of two 3D vectors o V 1 V 2 = V 1 V 2 cos(") 3D Vector Cross product of two 3D vectors o V 1 xv 2 = (dy 1 dx 2 - dz 1 dy 2, dz 1 dx 2 - dx 1 dz 2, dx 1 dy 2 - dy 1 dx 2 ) (dx 1,dy 1,dz 1 ) o V 1 xv 2 = vector perpendicular to both V 1 and V 2 o V 1 xv 2 = V 1 V 2 sin(") (dx 1,dy 1,dz 1 ) " (dx 2,dy 2,dz 2 ) " (dx 2,dy 2,dz 2 ) V 1 xv D Line Segment Linear path between two points o Parametric representation:» P = + t (P 2 - ), (0! t! 1) Point P1; Point P2; } Segment; P 2 3D Ray Line segment with one endpoint at infinity o Parametric representation:» P = + t V, (0 <= t < #) Point P1; Vector V; } Ray; V Origin Origin 11 12

3 3D Line Line segment with both endpoints at infinity o Parametric representation:» P = + t V, (-# < t < #) Point P1; Vector V; } Line; V 3D Plane A linear combination of three points P 2 P 3 Origin Origin D Plane A linear combination of three points o Implicit representation:» P N + d = 0, or N = (a,b,c)» ax + by + cz + d = 0 Vector N; Distance d; } Plane; o N is the plane normal» Unit-length vector» Perpendicular to plane P 2 d P 3 Origin 3D Polygon Set of points inside a sequence of coplanar points Point *points; int npoints; } Polygon; Points are in counter-clockwise order D Polygonal Mesh Set of polygons representing a 2D surface embedded in 3D 3D Polygonal Meshes Why are they of interest? o Simple, common representation o Rendering with hardware support o Output of many acquisition tools o Input to many simulation/analysis tools Isenberg 17 Viewpoint 18

4 3D Polygonal Meshes Properties + Efficient display + Easy acquisition Accurate Concise Intuitive editing Efficient editing Efficient intersections Guaranteed validity Guaranteed smoothness etc. NVIDIA 9600 GT GPU Outline Acquisition Processing Representation Polygonal Mesh Acquisition o Polygon editors o Interchange formats o Laser range scanners o Geological survey o CAT, MRI, etc. (isosurfaces) o Physical processes Polygonal Mesh Acquisition! Polygon editors o Interchange formats o Laser range scanners o Geological survey o CAT, MRI, etc. (isosurfaces) o Physical processes Sketchup Blender Polygonal Mesh Acquisition o Polygon editors! Interchange formats o Laser range scanners o Geological survey o CAT, MRI, etc. o Physical processes Polygonal Mesh Acquisition o Polygon editors o Interchange formats! Laser range scanners o Geological survey o CAT, MRI, etc. (isosurfaces) o Physical processes Jose Maria De Espona Digital Michelangelo Project Stanford 23 24

5 Polygonal Mesh Acquisition o Polygon editors o Interchange formats o Laser range scanners! Geological survey o CAT, MRI, etc. (isosurfaces) o Physical processes Large Geometric Model Repository Georgia Tech Polygonal Mesh Acquisition o Polygon editors o Interchange formats o Laser range scanners o Geological survey! CAT, MRI, etc. (isosurfaces) o Physical processes Large Geometric Model Repository Georgia Tech Polygonal Mesh Acquisition o Polygon editors o Interchange formats o Laser range scanners o Geological survey o CAT, MRI, etc. (isosurfaces)! Physical processes SGI Outline Acquisition Processing Representation MIT

6 Sheffer Sheffer Sheffer Sheffer Sheffer Thouis Ray Jones Weighted Average of Neighbor Vertices Olga Sorkine 35 36

7 Desbrun Weighted Average of Neighbor Vertices Olga Sorkine 0.35 Conway Jarek Rossignac 0.40 Conway Szymon Rusinkiewicz 41 42

8 Original Resampled Zorin & Schroeder Sorkine Vertex Clustering Garland Rossignac Podolak Borodin 47 48

9 Boundary P 3 P 2 P 4 P 5 Inside Outside P 2 Boundary P! P!! Inside P 3 P 4 P 5 Outside Boundary P 2 P! P!! Inside Outside Procedural generation o Surface of revolution o Sweep o Fractalize FvDFH Figure

10 Procedural generation o Surface of revolution o Sweep o Fractalize Procedural generation o Surface of revolution! Sweep o Fractalize Blinn Fowler Procedural generation o Surface of revolution o Sweep! Fractalize Most operations use a few low-level operations: face edge o Collapse edge o Merge vertices o Remove vertex Dirk Balfanz, Igor Guskov, Sanjeev Kumar, & Rudro Samanta, Most operations use a few low-level operations: face edge o Collapse edge o Merge vertices o Remove vertex Most operations use a few low-level operations: face edge o Collapse edge o Merge vertices o Remove vertex Subdivide face Subdivide edge 59 60

11 Most operations use a few low-level operations: face edge o Collapse edge o Merge vertices o Remove vertex Most operations use a few low-level operations: face edge o Collapse edge o Merge vertices o Remove vertex Collapse edge Merge Vertices Most operations use a few low-level operations: face edge o Collapse edge o Merge vertices o Remove vertex Outline Acquisition Processing Representation Remove Vertex Polygon Mesh Representation Data structures determine algorithms o Data structure must support key operations of algorithm efficiently Polygon Mesh Representation Important properties of mesh representation? Examples: o Drawing a mesh o Removing a vertex ing a region o Intersecting polyhedra Different data structures for different algorithms 65 66

12 Polygon Mesh Representation Important properties of mesh representation? o Efficient traversal of topology o Efficient use of memory o Efficient updates Polygon Mesh Representation Possible data structures o List of independent faces o Vertex and face tables o Adjacency lists o Winged edge o Half edge o etc. Large Geometric Model Repository Georgia Tech Independent Faces Each face lists vertex coordinates o Redundant vertices o No adjacency information Vertex and Face Tables Each face lists vertex references o Shared vertices o Still no adjacency information Adjacency Lists Store all vertex, edge, and face adjacencies o Efficient adjacency traversal o Extra storage Partial Adjacency Lists Can we store only some adjacency relationships and derive others? 71 72

13 Winged Edge Adjacency encoded in edges o All adjacencies in O(1) time o Little extra storage (fixed records) o Arbitrary polygons Winged Edge Example: Half Edge Adjacency encoded in edges o All adjacencies in O(1) time o Little extra storage (fixed records) o Arbitrary polygons Similar to winged-edge, except adjacency encoded in half-edges he next F left e he inv v begin Simple Triangle Mesh Do not store edges at all o All faces have 3 vertices and 3 neighbors Store adjacency in vertices and faces o For each face: 3 vertices and 3 faces o For each vertex: N faces NULL Summary Polygonal meshes o Easy acquisition o Fast rendering Processing operations o Must consider irregular vertex sampling o Must handle/avoid topological degeneracies Representation o Which adjacency relationships to store depend on which operations must be efficient 77