l32_solidmcsg

8/13/2019 L32_solidmcsg

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Solid Modeling Techniques

Constructive Solid Geometry(CSG)

AML710 CAD LECTURE 32

Computational Solid Geometry Primitive based: It is based on the notion that a physical

object can be divided into a set of primitives basicelements or shapes that can be combined in a certainorder following a set of rules ( Boolean operations) tocreate the object.

Primitives themselves are considered valid CSGmodels. Each primitives is bounded by closed andorientable surfaces.

CSG model is fundamentally and topologically differentfrom a B-rep model in that it does not store faces, edgesand vertices explicitly. Instead it evaluates themwhenever needed by algorithms.

CSG representation is of considerable importance formanufacturing


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Primitive based Vs Half-space basedCSG Schemes

There are two types of CSG schemes Primitive Based CSG: It is based on bounded valid

solid primitives, r-sets. It is the most popular CSGscheme.

Half space Based CSG :This scheme uses unboundedhalf spaces (non r-sets). Bounded solid primitives areconsidered composite half spaces and the boundariesof these are the surfaces of the component half spaces.

Primitive based CSGHalf space based CSG

- +

Data Structure of CSG

Like in B-rep, the database storestopology and geometry.

The validity checking in CSG schemeoccurs indirectly. The each primitive thatis combined using a Boolean operations(r-sets) to build the CSG model ischecked for its validity.

The common data structures used forCSG are graphs and trees


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CSG Data Structure

Graph: A graph is defined as a set of nodes connected by

a set of branches or lines.

Path: Each node in a tree belongs to a path E.g. A to G

Cycle: If starting and ending nodes are the same the pathis called a cycle

A

B C

DEFG

A

B C

DEFG

Graph Digraph

Root node

Leaf nodes

CSG Data Structure

Cyclic Graph: If a graph contains a cycle it is called cyclicotherwise it is acyclic.

Path: Each node in a tree belongs to a path E.g. A to G

Cycle: If starting and ending nodes are the same the pathis called a cycle

A

B C

DEFG

Graph Digraph

Indegree = 3

Outdegree = 2

B


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Tree and Binary Tree

Tree: A tree is an acyclic digraph in which a single node

called root node has a zero indegree and every othernodes has an indegree of one.

Binary Tree: In a tree if the descendent of each node arein order (from left to right) and each node except the leafnode has two decedents (left and right), then the tree iscalled a binary tree.

A

B C

DEFG

Tree Binary Tree

A

B C

DEFG

Binary Tree and Subtree

Subtree: Any binary tree can be thought of as joiningtogether of 2 subtrees. A left subtree and a right subtreerooted at two successor nodes

A

B C

DEFG

Left subtree Right subtree

C

A

E

F

D

B

G

Unbalanced binary treeBalanced binary tree

RL nn = RL nn


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Inverted Binary Tree for CSG Model

Inverted Binary Tree: If the direction of each connector

(arrow) are reversed then we get an inverted binary treewherein each node has one outdegree except the rootnode.

Root node: A node with outdegree = 0. Any node that doesnot have descendent.

Leaf node: Any node that does not have a predecessor orindegree = 0.

Interior node: Any node with outdegree > 0 is an interiornode.

A

B C

DEFGInverted Binary Tree

Inverted Binary Tree as CSG TreeCSG Tree: The inverted binary tree is very convenient to represent the

CSG operations. Here each of the leaf nodes is a valid solidprimitive. The intermediate nodes are the transition states of thesolid modeling (r-set) operations. The root node is the resulting solidfrom the set operations. Here n primitives require (n-1) Booleanoperations to complete the construction of the object.

A

B

D

C

A

B

D

C

A B DC

S=AUBUC

AUB CUD

DCBA

S

D

CSG Tree

Primitives =4

Operations=3

Nodes=2n-1=7


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Tree Traversal

Tree Traversal: Visiting the nodes in sequential or orderly

and efficient way. Many of the sorting and searchingalgorithms need to do the tree traversal

Traversal methods: Two broad methods are:

1. Depth first 2. Breadth first

The depth first is further divided into the following typesdepending on the order in which the root node is visited

a) Preorder b) Inorder c)Postorder

1

32

Preorder

2

31

Inorder

3

21

Postorder

Tree TraversalPreorder: We have the following recursive algorithm

Algorithm

1. Visit the root

2. Traverse the left subtree in preorder

3. Traverse the right subtree

1

2 9

111083

Preorder

Traversal

74

65

1312

1514


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Tree TraversalReverse Preorder: We have the following recursive algorithm

Algorithm



3. Visit the root

15

14 7

56813

Reverse PreorderTraversal

912

1011

34

12

Tree TraversalInorder: We have the following recursive algorithm

Algorithm


2. Visit the root


8

6 10

12974

Inorder

Traversal

52

31

1411

1513


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Tree TraversalReverse Inorder: We have the following recursive algorithm

Algorithm


2. Visit the root

3. Traverse the left subtreein preorder

8

10 6

47912

Reverse InorderTraversal

1114

1315

25

13

Tree TraversalPostorder: We have the following recursive algorithm

Algorithm

1. Traverse the left subtreein preorder


3. Visit the root

15

7 14

13865

Postorder Traversal

43

21

129

1110


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Tree TraversalReverse Postorder: We have the following recursive algorithm

Algorithm

1. Visit the root


3. Traverse the left subtree inpreorder

1

9 2

381011

Reverse PostorderTraversal

1213

1415

47

56

Basic Elements

Primitives: Bounded solid primitives are the basic building blocks ofCSG. These (parametric) solids have two sets of geometric data:

1. Configuration Parameters (size information)

2. Rigid motion Parameters (orientation information)

X

Y

Z

L

R

Y

X

Z

R

Z

BH

DX

Y

P

Y

X

Z B

H

D

P

SPHEREWEDGE

CYLINDERBLOCK


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Building Operations

The main building operations are regularised set operatoins

like union (U*), intersection (*) and difference (-*).

Hence the CSG models are known as set-theoretic,Boolean or combinatorial models.

In contrast to Euler operations, the Boolean operations arenot based on any equation or law. They are based onthe set theory and the closure property. Theseoperations are considered higher-level operations thanEuler operations.

Some implementations of solid modelers provide derivedtypes of operations like ASSEMBLE and GLUE

Main algorithms in CSG Operations1. Edge / Solid intersection algorithm

2. Computing set membership classification

a) Divide and conquer: It is like ray tracing. Instead of a ray an edge isused as a reference

b) Neighborhood: It deals with in, on and out decisions

When a point is in the interior of solid face then it is called faceneighborhood

Edge neighborhood occurs when the point lies on the solid edge

When a point is a vertex, vertex neighborhood occurs. This is acomplex case becouse the point is shared between three solid

faces.

P PP


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Summary of a CSG algorithmThe following steps describe a general CSG algorithm based on divide

and conquer approach:1. Generate a sufficient number of t-faces, set of faces of participating primitives, say A

and B.

2. Classify self edges of A w.r.t A including neighborhood.

3. Classify self edges of A w.r.t B using D & C paradigm. If A or B is not primitive thenthis step is followed recursively.

4. Combine the classifications in step 2 and 3 via Boolean operations.

5. Regularize the on segment that result from step 4 discarding the segments thatbelong to only one face of S.

6. Store the final on segments that result from step 5 as part of the boundary of S.Steps 2 to 6 is performed for each of t-edge of a given t-face of A.

7. Utilize the surface/surface intersection to find cross edges that result fromintersecting faces of B (one at a time) with the same t-face mentioned in step 6.

8. Classify each cross edge w.r.t S by repeating steps 2 to 4 with the next self edge ofA.

9. Repeat steps 5 and 6 for each cross edge

10. Repeat steps 2 to 9 for each t-face of A.

11. Repeat stpes 2 to 6 for each t-face of B.

A CSG Example

Create the CSG model of the following solid S.

AU*B-*C

AU*B C

BA

S

Geometry of the primitives

BLOCK A:

BLOCK B:

CYLINDER C:

-+

)2/,,2/(),,(,,

),0,0(),,(,,,

),0,(),,(,,,

cdadPzyxPdHRR

cPzyxPczbydx

cdPzyxPczdydax

CC

BBLLL

AALLL

+===

====

====

AE

E on A

E out B

E on SNull

E on S

U*

*-*

BE E on B

E out A

E in A

E on S

E on S E on S

U* * -*M(E,A)

M(E,B)

M(E,B) M(E,A)

Classify Combine

Classify

Combine

a

B

A

cb

c

d

dx

z

y

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