Solid Modeling

1. Solid Modeling Ref. Mantyla

2. Introduction Aim of modeling: • The search of a media of communication

3. Introduction (cont) Geometric modeling • Which parts of the objects are visible to the viewer? Colors?

4. Solid modeling Introduction

5. Taxonomy Geometric Modeling Solid Modeling Surface Modeling Voxels CSG B-rep {Alias Designer} Winged Edge Halfedge OpenMesh

6. Issues of Solid Modeling • (information) Completeness • Integrity • Complexity, Geometric Coverage • What does the object look like? • What is the weight, surface area, of the object • Will the object hit the other object on its path?

7. Representation Schemes Wireframe Surface Modeling Solid Modeling

8. A solid representation is a finite collection of symbols (of a finite alphabet) that designate a solid of M. The representation techniques of a given solid modeler define the representation space R of the modeler. Those representations that actually can be constructed by the solid modeler according to its syntax rules are termed admissible. A representation scheme is a relation s:MR. The domain of s is denoted by D and the image of D under s by V. If any valid representation models exactly one solid under s, s is called unambiguous or informationally complete. A representation scheme s is termed unique if all solids have exactly one representation

9. Solid Modeling CSG Constructive solid geometry Volumetric model B-rep Boundary representation

10. InformationCompleteness • Abletoresolvepointinclusiontestunambiguously • Givenapointandasolid; returnIn/Out/On

11. Constructive Solid Geometry • Point inclusion test for CSG • Classify against leaf primitives • Propagate the result in the true

12. Point Inclusion Test for CSG • Classify against leaf primitives • Propagate the result in the tree out IN IN out IN

13. Volumetric Representation solid otherwise

14. Octree

15. Boundary Model v f e Face, Edge, Vertex

16. Validity of Boundary Model non-manifold (next page) Self-intersecting • Elements of the model • should not self-intersect • should not intersect each other unless at their boundary.

17. Definition of Manifold For every point on the boundary, its neighborhood on the boundary is homeomorphic (topologically equivalent) to an open disc. disc

18. Topologically Equivalent

19. Examples of Non-Manifold Models

20. Plane Models Edge identification Cylinder Torus Mobius strip

21. Plane Model • Each edge (of a polygon) is assigned an orientation from one endpoint to the other • Every edge is identified with exactly to one other edge • For each collection of identified vertices, the polygons identified at that collection can be arranged in a cycle such that each consecutive pair of polygons in a cycle is identified at an edge adjacent to a vertex from the collection.

22. Orientable Solids • A plane model is orientable if the directions of its polygons can be chosen so that for each pair of identifed edges, one edge occcus in its positive orientation, and the other one in its negative orientation

23. Euler-Poincaré Formula (ref) V: the number of vertices E: the number of edges F: the number of faces G: the number of holes that penetrate the solid, usually referred to as genus in topology S: the number of shells. A shell is an internal void of a solid. A shell is bounded by a 2-manifold surface. Note that the solid itself is counted as a shell. Therefore, the value for S is at least 1. L: the number of loops, all outer and inner loops of faces are counted.

24. Examples Box: V-E+F-(L-F)-2(S-G) = 8-12+6-(6-6)-2(1-0)=0 Open Box: V-E+F-(L-F)-2(S-G) = 8-12+5-(5-5)-2(0-0)=1 Box w/ through hole: V-E+F-(L-F)-2(S-G) = 16-24+10-(12-10)-2(1-1)=0 Box w/ blind hole: V-E+F-(L-F)-2(S-G) = 16-24+11-(12-11)-2(1-0)=0 V-E+F-(L-F)-2(S-G) = 10-15+7-(7-7)-2(1-0)=0 Invalid solid yet stillyields ZERO!

25. Count Genus Correctly G = ? G = 3? G = 2!

26. Euler Operators MVFS KEMR MEV MEF (Ring: loop)

27. Global Operators

28. Example: Euler Operators

29. Winged-Edge Data Structure • Commonly used to describe polygon models • Quick traversal between faces, edges, vertices • Linked structure of the network • Assume there is no holes in each face

30. Winged-Edge Data Structure • vertices of this edge • its left and right faces • the predecessor and successor when traversing its left face • the predecessor and successor when traversing its right face.

31. Winged-Edge Data Structure Edge Table

32. Winged-Edge Data Structure

33. Winged-Edge Data Structure • the vertex table and the face table

34. Winged Edge Data Structure (Baumgart 1975)

35. Winged-Edge Data Structure • For a face with inner loops are ordered clockwise. • Adding an auxiliary edge between each inner loop and the outer loop

36. Halfedge Data Structure • Modification of winged edge • Since every edge is used twice, devise “halfedge” for this use • Can have loop to account for multiply connected face (face with multiple boundaries) • Can handle • Manifold models • Face with boundary • OpenMesh: a specialized halfedge implementation (for triangular meshes)

37. Half-Edge Data Structure • Doubly connected edge list

38. Object File Format(OFF) • Storing a description a 2D or 3D object • Simple extension can handle 4D objects • 4D: (x,y,z,w) • OFF File Characteristics • ASCII (there is also a binary version) • Color optional • 3D • No compression

39. Object File Format(OFF)

40. Object File Format(OFF)

41. Polygon File Format • Stanford Triangle Format • Store 3-d data from 3D scanners • Properties can be stored including • color and transparency • surface normals • texture coordinates • data confidence values

42. Stanford 3D Scanning Repository (url) Cyberware 3D Scanners (url) Large models also avaiable at GeogiaTech

43. Polygon File Format • PLY structure • Header • Vertex List • Face List • (lists of other elements)

44. Polygon File Format