3D Representation and Solid Modeling

Transcription

1 MCS 585/480 Computer Graphics I 3D Representation and Solid Modeling Week 8, Lecture 16 William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University 1 Overview Implicit solid modeling 3D Mesh File Formats Approximate representations Cell decomposition Spatial occupancy enumeration Quad and oct trees Binary space partition trees Statistical representations Introduction and discussion of homework # Foley/VanDam/Finer/Huges/Phillips ICG Implicit Solid Modeling Computer Algebra meets CAD Idea: Represents solid as the set of points where an implicit global function takes on certain value Primitive solids are combined using CSG Composition operations are implemented by functionals which provide an implicit defining function for the resulting solid Approximate Representations Idea: distretize the world! Surface Models Mesh, facet and polygon representations Volume Models spatial enumeration voxelization From M.Ganter, D. Storti, G. 3 UW 4 Examples 3D Mesh File Formats From exact to facets. Some common formats STL SMF OpenInventor VRML 5 6 Pics/Math courtesy of Dave UMD-CP

2 Vertex + Face No colors, normals, or texture Primarily used to demonstrate geometry algorithms Minimal Full-Featured Colors / Transparency Vertex-Face Normals (optional, can be computed) Scene Graph Lights Textures Views and Navigation 7 8 Simple Mesh Format (SMF) Stereolithography (STL) Michael Garland Triangle data Triangle data + Face Normal The de-facto standard for rapid prototyping Vertex indices begin at How STL Works How STL Works 11 12

3 Open Inventor Virtual Reality Modeling Language (VRML) Predecessor to VRML Scene Graph More minimal SGML Based Scene-Graph Full Featured Issues with 3D mesh formats Easy to acquire Easy to render Harder to model with Error prone split faces, holes, gaps, etc Cell Decomposition Set of primitive cells Parameterized Often curved Compose complex objects by gluing cells together Used in finiteelement analysis Spatial Occupancy Enumeration Brute force A grid Quadtrees 2D representation Octrees 3D representation Extension of quadtrees Voxels Volume pixels 17 Brute Force Spatial Occupancy Enumeration Impose a 2D/3D grid Like graph paper or sugar cubes Enumerate occupied cells Problems High fidelity representations require many cells 18

4 Quadtree Quadtree Example Hierarchically represent spatial occupancy Tree with four regions NE, NW, SE, SW dark if occupied octants 3D space Left, Right, Up, Down, Front, Back Octree Boolean Operations on Octrees S T S T S T Applications for Spatial Occupancy Enumeration Many different applications GIS Medical Engineering Simulation Volume Rendering Video Gaming Approximating real-world data. 23 Issues with Spatial Occupancy Enumeration Approximate Kind of like faceting a surface, discretizing 3D space Operationally, the combinatorics (as opposed the the numerics) can be challenging Not as good for applications wanting exact computation (e.g. tool path programming) 24

5 Binary Space Partition Trees (BSP Trees) Recursively divide space into subspaces Arbitrary orientation and position of planes Homogeneous regions are leafs called in/out cells Statistical Representations Store density (material vs. void) Statistical description of geometry Goal describe the porosity without storing the geometry information Generated by Termite Agents Simulation. Stochastic Functions Need some way of converting a solid into some representative statistical form From each material voxel, calculate the distance to the nearest voxel that is not material 0.2 Repeat for void voxels Cube A, Pore Cube A, Material Store distributions: 0.15 one for empty space 0.1 one for material density value 0.05 Bone tissue MRI data Other biological data Solid modeling Application 0 27 Distance vs. Probability MRI scan of left shoulder Bone matrix from scanned data 28 Programming assignment 8 End Output gray-scale XPM Input SMF file. Extend your previous homework assignment to perform: Backface culling Z-buffering 29 30

6 Other Issues in Boundary Representation Solid Modeling Binary Space Partition Trees (BSP Trees) What s the surface? 31 32