CSG obj. oper3. obj1 obj2 obj3. obj5. obj4
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1 Solid Modeling Solid: Boundary + Interior Volume occupied by geometry Solid representation schemes Constructive Solid Geometry (CSG) Boundary representations (B-reps) Space-partition representations Operations
2 Constructive Solid Geometry Constructive (procedural) models of complex solids Complex solids as compositions of simpler solids A set of 3D objects (simple primitives) { blocks { pyramids { cylinders { cones { spheres { tori { algebraic halfspaces Other objects as primitives { closed spline surfaces { spline solids { swept solids f(x y z)jf(x y z) <= 0g
3 { others Primitives: point-sets Boolean set operations { union, intersection, dierence Complicated object: a binary tree { leaf nodes: primitive shape { non-leaf nodes: set operators Surfaces are actually dened IMPLICITLY Solids ALSO have B-reps { a set of surfaces { separate object interior from the environment { unambiguous interior and exterior! Examples { polygon facets, spline patches Solid modeling eld
4 { mathematical foundations { representations { algorithms { applications { user interfaces { systems
5 CSG Schemes CSG obj oper1 oper2 obj1 obj2 obj3 oper3 obj4 obj5
6 Object1 (+) Object2 CSG Example
7 Basic Operations Point, curve, and surface classication/neighborhood w.r.t solids Local/global modication Representation conversion Topological validity Approximation Representation redundancy Interaction Manifold/nonmanifold representation Free-form solids Geometric/topological query Large-scale model management
8 B-REPS Faces, edges, and vertices of solid boundary { topological representation (connectivity) { geometric (numerical data) representation Polyhedral (tessellated) models Solids bounded by curved surfaces and edges Separate points inside from outside the solid Interface between the solid and the environment B-reps' conditions { closed, orientable { non-self-intersecting { bounded { connected Internal features of objects
9 { Ray-casting is a fundamental method Determine the new B-reps for composite solids { Surface intersection calculation Determine physical properties { volume { mass, center of mass, { moment of inertia { density Decomposition Voxelization Adaptive subdivision Recursive subdivision Conversion among dierent schemes
10 Ray Casting Method x,y plane object1 object2 a b c d z
11 Spatial Partition Idea: cell decomposition Each piece is easier to describe than the original model Partition the space into a set of nonoverlapping regions (cubes) Uniform subdivision Predetermined resolution Cells are cubical in a xed spatial grid Intersecting cubes Spatial-occupancy enumeration spatial arrays (+) Easy to access points (cells) in arbitrary location
12 (+) Spatial uniqueness (-) No explicit relationship for object parts (-) Large amounts of data storage (-) Limited accuracy (-) Specic coordinate system (-) Hard to manipulate (e.g., rotation)
13 Octree representation Recursive subdivision Hierarchical tree structures Generalization of quadtree schemes Successively divide a 2D region into quadrants Homogeneous quadrants { either full or empty w.r.t. the object Heterogeneous quadrants { partially full (intersection) Terminating conditions { all quadrants are homogeneous { predened resolution level Not restricted to solid geometry
14 Applicable to color, intensity, density, and others Divide 3D space into octants CSG-based set operations Binary space-partition Useful for a lot of graphics and visualization algorithms All computations are based on integer arithmetic Fast and parallel algorithms { translation { rotation { scaling { boolean operations { geometric and material properties { interference analysis Dicult for { curved objects
15 { free-form (sculptured) solids { deformable modeling Irregular shape elements { triangles { tetrahedra
16 Quadtrees
17 Octrees
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