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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