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Solid Modeling

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Title: 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
Introduction
  • Solid modeling

5
Taxonomy
Geometric Modeling
Solid Modeling
Surface Modeling
CSG
Voxels
B-rep
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 sM?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
Information Completeness
  • Able to resolve point inclusion test
    unambiguously
  • Given a point and a solid return In/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
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-EF-(L-F)-2(S-G) 8-126-(6-6)-2(1-0)0
Open Box V-EF-(L-F)-2(S-G) 8-125-(5-5)-2(0-0)
1
Box w/ through hole V-EF-(L-F)-2(S-G)
16-2410-(12-10)-2(1-1)0
Box w/ blind hole V-EF-(L-F)-2(S-G)
16-2411-(12-11)-2(1-0)0
V-EF-(L-F)-2(S-G) 10-157-(7-7)-2(1-0)0
Invalid solid yet still yields ZERO!
25
Count Genus Correctly
G ?
G 3?
G 2!
26
Euler Operators
(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 Vertices Vertices Faces Faces Left Traverse Left Traverse Right Traverse Right Traverse
Name Start End Left Right Pred Succ Pred Succ
a X Y 1 2 d b c e
Edge Table
32
Winged-Edge Data Structure
Edge Vertices Vertices Faces Faces Left Traverse Left Traverse Right Traverse Right Traverse
Name Start End Left Right Pred Succ Pred Succ
a A D 3 1 f e c b
b A B 1 4 a c d f
c B D 1 2 b a e d
d B C 2 4 c e f b
e C D 2 3 d c a f
f A C 4 3 b d e a
33
Winged-Edge Data Structure
  • the vertex table and the face table

Vertex Name Incident Edge
A a
B b
C d
D c
Face Name Incident Edge
1 a
2 c
3 a
4 b
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
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