SolidModeling.ppt

History:
CNC: ~1950
Mainframe Computers: ~1960’s
BREP: 1970 (Baumgart)
CSG: 1974 (Ian Braid)
Solid Modeling

Advantages:
Saves on storage/retrieval;
Easy modification, update;
Shortcomings:
Can’t analyse the
strength, shape,
geometry, weight
center of mass, center of inertia
Computerized Drafting
Popular Commercial tools: AutoCAD, CADKEY…

Stores each edge of the object
data:
The vertices
start point, end point
The equation of the edge-curve
3D and Solids Representation
Wireframe Models

Wireframe Problem: Ambiguity
(a) (b) (c)
Wireframe ambiguity:
Is this object (a), (b) or (c) ?
(a) (b) (c)
Wireframe ambiguity:
Is this object (a), (b) or (c) ?

Constructive Solid Geometry (CSG)
Boundary Representation (BREP)
Spatial Enumeration (voxels, octrees)
Other Methods

Introduced: Ian Braid (Cambridge University, ~74)
Concepts:
Primitives: small set of shapes
Transformations: scaling, Rotation, Translation
Set-theoretic Operations Union, Intersection, Difference
[ Euler operators ]
Combinations of these  Solid part
Constructive Solid Geometry (CSG)

U* (regular union)
-* (regular difference)
∩* (regular intersection)
CSG Tree:
Sequence of operators  design
Euler operators

Examples of CSG
x
y
z
y
x
z
box( a, b, c) cylinder( h, r)
primitives
x
y
z
y
x
z
box( a, b, c) cylinder( h, r)
primitives
X
Z
box( 25, 25, 15) Trans( 2,0, 15) Rot(Y, 45) Trans( -5, 0, -5) box( 10, 25, 10)
U* Trans( 20, 12.5, 15) cylinder( 5, 3)
U*
-*
Trans( 10, 0, 0) box( 3, 10, 10)
X
Z
X
Z
box( 25, 25, 15) Trans( 5  -5, 0, -5) box( 10, 25, 10)
U* Trans( 20, 12.5, 15) cylinder( 5, 3)
U*
-*
Trans( 10, 0, 0) box( 3, 10, 10)

Can we use a different set of primitives ?
Questions:
Is the CSG representation unique ?
[how to determine if two solids are identical ?]

Regularized operators
Is the set of 3D solids is closed with respect to ( U, -, ∩ )?
closure of a set S: kS
interior of a set S: iS
A U* B = k i ( A U B)
A -* B = k i ( A –B)
A ∩* B = k i ( A ∩ B)
Why is closure over operations important?
uniform data structures

Regularized Euler Operators
Non 2-Manifold:
Maintain solid as a regular 2-Manifold
2-Manifold regular solids
Open neighborhood of each point is similar to an open disc

Non-Unique representation
Difficulty of performing analysis for some tasks
Problems with CSG

What entities define the
Boundary of a solid ?
Boundary of surfaces?
Boundary of curves (edges) ?
Boundary of points ?
BREP (Boundary REPresentation)

(a) Solid: bounded, connected subset of E3
(b) Faces: boundary of solid
bounded, connected subsets of Surfaces
(c) Edges: boundary of faces
bounded, connected subsets of curves
(a) Solid: bounded, connected subset of E3
(b) Faces: boundary of solid
bounded, connected subsets of Surfaces
(c) Edges: boundary of faces
bounded, connected subsets of curves
Boundary of a solid…
BREP
Boundary of surfaces…
Boundary of curves (edges)…

Compute Volume, Weight
Compute Surface area
Point inside/outside solid
Intersection of two faces
…
Using a Boundary Model

v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
Faces:
f1 e1 e4 e5
f2 e2 e6 e4
f3 e3 e5 e6
f4 e3 e2 e1
Edges:
e1 v1 v2
e2 v2 v3
e3 v3 v1
e4 v2 v4
e5 v1 v4
e6 v3 v4
Vertices:
v1 x1 y1 z1
v2 x2 y2 z2
v3 x3 y3 z3
v4 x4 y4 z4
v5 x5 y5 z5
v6 x6 y6 z6
v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
Faces:
f1 e1 e4 e5
f2 e2 e6 e4
f3 e3 e5 e6
f4 e3 e2 e1
Edges:
e1 v1 v2
e2 v2 v3
e3 v3 v1
e4 v2 v4
e5 v1 v4
e6 v3 v4
Vertices:
v1 x1 y1 z1
v2 x2 y2 z2
v3 x3 y3 z3
v4 x4 y4 z4
v5 x5 y5 z5
v6 x6 y6 z6
An Edge-Based Model

Edge-Based Models: inefficient algorithms
face
face
Compute Surface Area:
1. Identify Loops
2. Compute area of each loop
3. Compute area of face

Efficient implementation of often-used algorithms
Area of Face
Hidden surface removal
Find neighbor-faces of a face
The Winged-Edge Data Structure

Observations
e1
e4
e5
e2
e6
e4
e1
e4
e5
e2
e6
e4
Face CCW convention =>
Each edge is once +ve, once -ve
2-Manifold => Each edge is shared by exactly 2 faces
co-edges

v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
v5 v6
v8
e7
e8
e9
e10
e11
e12
v7
v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
v5 v6
v8
e7
e8
e9
e10
e11
e12
v7
BREP Example

v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
v5 v6
v8
e7
e8
e9
e10
e11
e12
v7
v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
v5 v6
v8
e7
e8
e9
e10
e11
e12
v7
Vertices:
v1 x1 y1 z1
v2 x2 y2 z2
v3 x3 y3 z3
v4 x4 y4 z4
v5 x5 y5 z5
v6 x6 y6 z6
v7 x7 y7 z7
v8 x8 y8 z8
v9 x9 y9 z9
v10 x10 y10 z10
v11 x11 y11 z11
v12 x12 y12 z12
BREP Example

v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
v5 v6
v8
e7
e8
e9
e10
e11
e12
v7
v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
v5 v6
v8
e7
e8
e9
e10
e11
e12
v7
Edges:
e1 v1 v2
e2 v2 v3
e3 v3 v1
e4 v2 v4
e5 v1 v4
e6 v3 v4
e7 v5 v6
e8 v6 v7
e9 v7 v5
e10 v6 v8
e11 v5 v8
e12 v7 v8
BREP Example..

v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
v5 v6
v8
e7
e8
e9
e10
e11
e12
v7
v1 v2
v3
v4
e1
e2
e3
e4
e6
e5
v5 v6
v8
e7
e8
e9
e10
e11
e12
v7
Faces:
f1 l1 l2
f2 l3
f3 l4
f4 l5
f5 l6
f6 l7
f7 l8
Loops:
l1 +e1 +e4 -e5
l2 -e7 +e11 -e10
l3 +e2 +e6 -e4
l4 +e5 -e6 +e3
l5 -e1 -e3 -e2
l6 +e7 +e8 +e9
l7 +e10 -e12 -e8
l8 -e11 -e9 +e12
BREP Example…

BREP: Winged edge data structure
Solid
Face
Loop
coEdge
Vertex
s_faces f_solid
f_loops l_face
l_coedge c_loop
v_start v_end
prev_f next_f
prev_l next_l
prev_ce next_ce
prev_v next_v
Solid
Face
Loop
coEdge
Vertex
s_faces f_solid
f_loops l_face
l_coedge c_loop
v_start v_end
prev_f next_f
prev_l next_l
prev_ce next_ce
prev_v next_v

BREP or CSG ?
Using: CSG is more intuitive
Computing: BREP is more convenient
Modern CAD Systems:
CSG for GUI (feature tree)
BREP for internal storage and API’s

BREP: non-polyhedral models?
Same Data Structure, plus
For each edge, store equation
For each curved face, store equation
Why do we need to learn all this ?
(a) To anticipate when an operation will fail
(b) To allow us to write API’s

SolidModeling.ppt

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SolidModeling.ppt