1. Graphics Graphics Lab @ Korea University cgvr.korea.ac.kr Solid Modeling 고려대학교 컴퓨터 그래픽스 연구실

2. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Solid Modeling Represent Solid Interiors of Objects Surface may not be described explicitly

3. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Motivation Some Acquisition Methods Generate Solids Example: CAT scan

4. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Motivation Some Applications Require Solids Example: CAD/CAM

5. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Motivation Some Algorithms Require Solids Example: ray tracing with refraction

6. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Solid Modeling Representation Voxels Quadtrees & Octrees Binary Space Partitions Constructive Solid Geometry

7. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Voxels Partition Space into Uniform Grid Grid cells are called a voxels (like pixels) Store Properties of Solid Object with Each Voxel Occupancy Color Density Temperature Etc.

8. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Voxel Acquisition Scanning Devices MRI CAT Simulation FEM

9. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Voxel Storage O( n 3 ) Storage for n x n x n Grid 1 billion voxels for 1000 x 1000 x 1000

10. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Voxel Boolean Operations Compare Objects Voxel by Voxel

11. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Voxel Display Isosurface Rendering Render surfaces bounding volumetric regions of constant value (e.g., density)

12. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Voxel Display Slicing Draw 2D image resulting from intersecting voxels with a plane

13. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Voxel Display Ray Casting Integrate density along rays through pixels

14. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Voxels Advantages Simple, intuitive, unambiguous Same complexity for all objects Natural acquisition for some applications Trivial boolean operations Disadvantages Approximate Large storage requirements Expensive display

15. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Solid Modeling Representation Voxels Quadtrees & Octrees Binary Space Partitions Constructive Solid Geometry

16. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Quadtrees & Octrees Refine Resolution of Voxels Hierarchically More concise and efficient for non-uniform objects Uniform Voxel Quadtree

17. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Quadtree Boolean Operations

18. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Solid Modeling Representation Voxels Quadtrees & Octrees Binary Space Partitions Constructive Solid Geometry

19. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Binary Space Partitions (BSPs) Recursive Partition of Space by Planes Mark leaf cells as inside or outside object a b cd e f g Object a b cd e f g Binary Spatial Partition 1 2 3 4 5 6 7 1 2 a 3 b c 4 d 5 6 e 7 f BSP Tree

20. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr BSP Fundamentals Single Geometric Operation Partition a convex region by a hyperplane Single Combinatorial Operation Two child nodes added as leaf nodes

21. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr BSP Display Visibility Ordering Determine on which side of plane the viewer lies  Near-subtree -> polygons on split -> far-subtree A B C o2 o1 o3 o4 Viewer Partitioning Tree A BC o1o2o3o4 3rd4th1st2nd Viewer

22. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Solid Modeling Representation Voxels Quadtrees & Octrees Binary Space Partitions Constructive Solid Geometry

23. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Constructive Solid Geometry (CSG) Represent Solid Object as Hierarchy of Boolean Operations Union Intersection Difference

24. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr CSG Acquisition Interactive Modeling Programs CAD/CAM

25. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr CSG Display & Analysis Ray Casting Union CircleBox

26. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Summary VoxelsOctreeBSPCSG No Some No Yes No Yes Some NoSome Yes No Yes No YesNo Accurate Concise Affine Invariant Easy Acquisition Guaranteed Validity Efficient Boolean Operations Efficient Display