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Polygon Meshes and other modeling tools

Polygon Meshes and other modeling tools. Object representation Polygon meshes Normals Scene graphs Level of detail Space partitioning. How can we represent objects/scenes. Polygon meshes Scene graphs Parametric representations Equations Bunch o points. Why?. Rendering Manipulate

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Polygon Meshes and other modeling tools

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  1. Polygon Meshes and other modeling tools • Object representation • Polygon meshes • Normals • Scene graphs • Level of detail • Space partitioning

  2. How can we represent objects/scenes • Polygon meshes • Scene graphs • Parametric representations • Equations • Bunch o points

  3. Why? • Rendering • Manipulate • User • System

  4. Point Clouds • Unstructured set of 3D point samples • Why • What for?

  5. Polygon/Triangle Soup • Unstructured set up polygons/triangles • Just a list of vertices for each polygon/triangle • Why? • What for? class Triangle { public: Triangle() {m_material = NULL;} CGrPoint3F m_vertices[3]; CGrPoint3F m_normals[3]; CGrPoint2F m_tvertices[3]; const void *m_material; };

  6. What is tessellation? • Representing a surface with flat approximating polygons • A sampled representation • Geometric Aliasing • Error in the representation due to the sampling with polygons

  7. Polygon/Triangle mesh • Common method for describing geometry • Set of vertices • Coordinates • Normal • Texture coordinates • Etc. • Set of surfaces • Pointers to vertices • Material properties Common: Triangle mesh

  8. class CTriangleMesh { public: CTriangleMesh(void); ~CTriangleMesh(void); // This structure defines a triangle struct Triangle { int m[3]; const CMaterial *material; }; int VertexCnt() const {return (int)m_vertices.size();} int TriangleCnt() const {return (int)m_triangles.size();} const std::vector<CGrPoint3F> &Vertices() const {return m_vertices;} const std::vector<CGrPoint3F> &Normals() const {return m_normals;} const std::vector<CGrPoint2F> &TVertices() const {return m_tvertices;} const std::vector<Triangle> &Triangles() const {return m_triangles;} const CMaterials &Materials() const {return m_materials;} private: CMaterials m_materials; // The materials collection // The vertices, normals, and texture vertices. // These three all correspond with each other. std::vector<CGrPoint3F> m_vertices; std::vector<CGrPoint3F> m_normals; std::vector<CGrPoint2F> m_tvertices; std::vector<Triangle> m_triangles; };

  9. Polygon Mesh • Retains shared-vertex relationships c a Vertex a,b,c,d; Polygon P1(a, b, c); Polygon P2(c, b, d); d b b,c shared by two polygons

  10. Computing a surface normal • Assume the following polygon: p3 p4 N p2 p1

  11. The Method of Projected Areas • Assume this 2D example line: p1 dy p2 dx Projection of length to one axis

  12. Method of Projected Areas • The projection of the area onto the x/y plane is the z part of the normal

  13. Where does this come from? y x Why would we want to use this?

  14. Vertex normals • Add normals for surfaces incident on vertex:

  15. The Winged-Edge Data Structure • Maintains • List of vertices • List of edges • List of faces

  16. Winged-Edge Data Structure Vertex table: Face table:

  17. Edge Table a e D c 1 2 B b d

  18. Voxels • Grid of volumetric samples • CAT, MRI, etc.

  19. Scene Graphs Composite Separator Separator Separator Translate Translate Translate BarbellBar BarbellEnds Color Color Composite Composite … … Polygon Polygon Polygon Polygon

  20. Scene Graphs

  21. Collapsing a Scene Graph • It’s often possible to apply transformations in a scene graph once to the underlying vertices, collapsing some nodes. Example: Translation Collapse to Polygon Rotation Polygon

  22. Cloning • If part of a scene graph has two edges to it, it will be necessary to clone before collapsing Translation Translation Translation Rotation Rotation Rotation Polygon Polygon’ Polygon

  23. Integration of mesheswith scene graphs Composite Separator Separator Separator Translate Translate Translate BarbellBar BarbellEnds Color Color Composite Polygon Mesh … Other possibilities? Polygon Polygon

  24. Bounding Boxes • Bounding boxes describe the X,Y,Z extents (minimum and maximum values). • If we know everything below a certain point in the scene graph is in a box, we can avoid traversal if the box is not on-screen

  25. class CBBoxF { public: class CBBoxF() {}; class CBBoxF(const CGrPoint3F &p) : m_min(p), m_max(p) {} void Set(const CGrPoint3F &p) {m_min = p; m_max = p;} void Set(const CBBoxF &b) {m_min = b.m_min; m_max=b.m_max;} void Include(const CGrPoint3F &p); void Include(const CBBoxF &b) {Include(b.m_min); Include(b.m_max);} const CGrPoint3F &Max() const {return m_max;} const CGrPoint3F &Min() const {return m_min;} const float MinForD(int d) const {return m_min[d];} double Extent(int d) const {return m_max[d] - m_min[d];} private: CGrPoint3F m_min; CGrPoint3F m_max; };

  26. Levels of Detail (lod) • Alternative scene graphs with different resolutions • Varying tessellation • Which we use depends on how far away the camera is. • Significant performance enhancement

  27. BSP Trees • Binary Space Partitioning • Divides space into half-spaces • We can then use this method to • limit how much scene graph we examine • examine the scene graph in certain orders

  28. k-d Trees • k is dimensions • Node • Info • x, y • lt-link • ge-link

  29. Levels • Level 0 – Index X • Level 1 – Index Y • Level 2 – Index X • level = 0 if root, level(parent)+1 otherwise • For each level, index: level mod k • 3D: Index X, Index Y, Index Z, Index X, etc…

  30. Example Banja Luca (19, 45) X • Banja Luca (19, 45) • Derventa (40, 50) • Teslic (38, 38) • Tuzla (54, 40) • Sinj (4, 4) Sinj (4, 4) Derventa (40, 50) Y Teslic (38, 38) X Tuzla (54, 40) Y Insert: East Broko (21, 57) Search for exact? Search for nearest? Range searches?

  31. k-d Trees • Search for exact • Okay, but may be o(k) • Search for nearest • In a moment • Range search?

  32. Range Search • Consider each tree level to be region • Recursively search all regions that overlap search range

  33. Example Regions (-,+) (-,+) Banja Luca (19, 45) (-,19) (-,+) Sinj (4, 4) Derventa (40, 50) [19,+) (-,+) [19,+) [50,+) East Broko (21, 57) Teslic (38, 38) [19,+) (-,50) Tuzla (54, 40) [38,+) (-,50) Range: (10,20) (20, 45)

  34. k-d Trees • Fast and easy • Tend to be rather tall (unbalanced) • How could we extend to disk structure? • What if k=1?

  35. Quadtrees • 2-d only • Split data 4 ways • Node: • info • x,y • nw,sw,ne,se

  36. Example Banja Luca (19, 45) • Banja Luca (19, 45) • Derventa (40, 50) • Teslic (38, 38) • Tuzla (54, 40) • Sinj (4, 4) nw ne se Sinj (4, 4) Derventa (40, 50) Teslic (38, 38) se Tuzla (54, 40) Insert: East Broko (21, 57) Search for exact? Search for nearest? Range searches?

  37. Octtrees • 3-d • Split data 8 ways • Node: • info • x,y,z • up nw,sw,ne,se, down nw,sw,ne,se

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