solidmodeling

8/12/2019 solidmodeling

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Solid Modeling Techniques

Half Spaces

Boundary Representation (B-rep)

AML710 CAD LECTURE 31

}0)(:{ 3


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Half Spaces

A cylindrical Half Space is a given by:

Cylindrical Half Space

XL

YL

}:),,{( 222 RyxzyxH


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Half Spaces

A conical Half Space is a given by:

Conical Half Space

XL

YL

}))2/(tan(:),,{( 222 zyxzyxH


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Constructing Solids with Half Spaces

Complex objects can be modeled by combining Half Space usingset operations

Constructing solids w ith Half Space

=

=

IUn

i

iHS0

Advantages and Disadvantages of

Half SpacesAdvantages:

The main advantage is its conciseness of representation compared

to other modeling schemes.

It is the lowest level representation available for modeling a solid

object

Disadvantages:

The representation can lead to unbounded solid models as it

depend on user manipulation of half spaces

The modeling scheme is cumbersome for ordinary users / designers


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Adjacency Topology in B-rep

FV

EV

VV

FE

EE

VE

FF

EF

VF

Topological Atlas and Orientability

The simplest data structure keeps track of adjacent

edges. Such a data structure is called an atlas.

A

C

B

A

C

B


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A

A

A

Topological Atlas of a Tetrahedron

Topological Atlas and Orientability

The orientability indicated with arrows or numbers as

shown below. We see that the orienation preserving

arrows are in two opposite rotational directions i.e.,

clockwise and anticlockwise. While orienation reversing

arrows are in the same rotational directions.

1 2 3 4 5 6

1 2 3 4 5 6

6 5 4 3 2 1

1 2 3 4 5 6

Orientation

Preserving

Orientation

Reversing


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Schlegel Diagrams

A common form of embedding graphs on planar faces is calledSchlegel Diagram. It is a projection of its combinatorial equivalent ofthe vertices, edges and faces of the embedded boundary graph onto its surface. Here the edges may not cross except at their incidentvertices and vertices may not coincide.

e1

e2e3

e4

e5

e6

e7

e8

e9 e10

e11e12

Schlegel Diagram of a Cube

Atlas of Cube

An atlas of a cube can also be given by the

arrangement of its faces as shown below

Atl as of Cube

0

1

2 3 4 5

6

1

2

3


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Some Examples of Atlases

While orienation reversing arrows are in the

same rotational directions.

Cylinder Mobius Strip

Can you think of the atlases of Torus ? Klein bottle ?

Boundary Representation (B-rep)

Non-manifold

surfaces

Non-oriented

Manifolds

Moebius strip Klein bottle


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Object Modeling with B-rep

Both polyhedra and curved objects can be modeled

using the following primi tives Vertex: A unique point (ordered triplet) in space.

Edge :A finite, non-selfintersecting directed space curve bounded bytwo vertices that are not necessarily distinct.

Face :Finite, connected, non-selfintersecting region of a closed,orientable surface bounded by one or more loops.

Loop :An ordered alternating sequence of vertices and edges. Aloop defines non-self intersecting piecewise closed space curvewhich may be a boundary of a face.

Body :An independent solid. Sometimes called a shell has a set offaces that bound single connected closed volume. A minimum body

is a point (vortex) which topologically has one face one vortex andno edges. A point is therefore called a seminal or singularbody.

Genus :Hole or handle.

Euler Operations (Euler Poincare Law):

The validity of resulting solids is ensured via Euler operations which

can be built into CAD/CAM systems.

Volumetric Property calculation in B-rep:

It is possible to compute volumetric properties such as mass

properties (assuming uniform density) by virtue of Gauss divergence

theorem which converts volume integrals to surface integrals.

Boundary Representation

Original objectModified objects

Nonsense object

1

4

4

=

=

=

F

V

E

1

5

5

=

=

=

F

V

E

1

5

5

=

=

=

F

V

E


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A connected structure of vertices, edges and faces thatalways satisfies Eulers formula is known as Euler

object.

The process that adds and deletes these boundary

components is called an Euler operation

Applicability of Euler formula to solid objects:

At least three edges must meet at each vertex.

Each edge must share two and only two faces

All faces must be simply connected (homomorphic to disk) with no

holes and bounded by single ring of edges. The solid must be simply connected with no through holes

Euler Operations

SolidsSimple2=+ VEFValidity Checking for Simple Solids

28126

6

8

12

=+

=

=

=

F

V

E

2585

5

5

8

=+

=

=

=

F

V

E

2162410

10

16

24

=+

=

=

=

F

V

E

26106

6

6

10

=+

=

=

=

F

V

E


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SolidsSimple2=+ VEF

Validity Checking for Simple Solids

2233

3

2

3

=+

==

=

F

V

E

2222

2

2

2

=+

==

=

F

V

E

2222

22

2

=+

=

=

=

FV

E

Loops (rings),Genus & Bodies

Genus zero

Genus one

Genus two

One inner loop


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Validity Checking for Polyhedra with

inner loops

2)01(22243616

0

1

2

24

16

36

==+

=

=

=

=

=

=

G

B

L

V

F

E

General)(2 GBLVEF =+


holes

4)02(20162412

0

2

0

16

12

24

==+

=

=

=

=

=

=

G

B

L

V

F

E

General)(2 GBLVEF =+Interiorhole

(void)

Surface hole

2)01(21162411

01

116

1124

==+

==

==

==

GB

LV

FE


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through holes (handles)

0)11(22162410

1

1

2

16

10

24

==+

=

=

=

=

=

=

G

B

L

V

F

E

General)(2 GBLVEF =+Throughhole

Handles/through ho le 0)11(24324820

11

432

2048

==+

====

==

GB

LV

FE

Validity Checking for Open ObjectsGBLVEF =+

Lamina polyhedra

Wireframe polyhedra Shell polyhedra

Open three dimensional polyhedra


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Exact Vs Faceted B-rep Schemes

Exact B-rep: If the curved objects are represented byway of equations of the underlying curves and sufraces,then the scheme is Exact B-rep.

Approximate or faceted B-rep :In this scheme ofboundary representation any curved face divided intoplanar faces. It is also know as tessellationrepresentation.

Exact B-rep: Cylinder and

SphereFaceted cylinder and sphere

Data structure for B-rep modelsGeneral)(2 GBLVEF =+

Topology Topology

Object

Body

Genus

Face

Loop

Edge

Vertex

Underlying surface equation

Underlying curve equation

Point coordinates


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Winged Edge Data structure

Successor 1

Predecessor 1

Predecessor 2

Successor 2

E F2

v2

F1

v1

F1 F2E

All the adjacency relations of each edge are described explicitly. Anedge is adjacent to exactly two faces and hence it is component in

two loops, one for each face.

As each face is orientable, edges of the loops are traversed in a

given direction. The wingded edge data structure is efficient in object

modifications (addition, deletion of edges, Euler operations).

Building Operations

The basis of the Euler operations is the above equation. M and K

stand for Make and Kill respectively.

Make Multiple Edges

Edge Split

Kill Vertex Edge

KME

ESQUEEZE

MME

ESPLIT

KVE

Composite

Operations

Kill Face Edge Vertex Make GenusKill Face Edge Vertex Body

MFEVKGMFEVB

KFEVMGKFEVB

Glue

Make Edge Vertex

Make Edge Kill Loop

Make Edge Face

Make Edge Kill Body, Face Loop

Make Edge Kill Loop Genus

KEV

KEML

KEF

KEMBFL

KFMLG

MEV

MEKL

MEF

MEKBFL

MFKLG

Create edges and

vertices

Create edges and

faces

Make Body Face VertexKBFVMBFVInitiate Database

and begin creation

DescriptionComplementOperatorOperation



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Transition States of Euler Operations

While creating B-rep models at each stage we use Euler operatorsand ensure the validity.


0

0

0

0

0

-1

1

0

9

9

9

1

0

0

0

-1

0

0

-1

0

0

0

0

0

-1

0

-1

-1

0

0

0

0

0

1

1

0

0

0

0

-n

-n

n

1

-1

0

1

1

1

1

0

-n

-n

n

1

-n

1

0

0

1

-1

1

-2

-2

0

0

-(n-1)

MBFV

MEV

MEKL

MEF

MEKBFL

MFKLG

KFEVMG

KFEVB

MME

ESPLIT

KVE

GBLVEFOperator

Euler Operations)(2 GBLVEF =+

MBFV

MEV

KEV

MEKL

KEML

MEF

KEF


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Euler Operations

)(2 GBLVEF =+

MEKBFL

KEMBFL

GLUE(KFEVMG)

MFKLG

KFMLG

UNGLUE(MFEVKG)

Euler Operations)(2 GBLVEF =+

MME

KME

A B

GLUE(KFEVB)

UNGLUE(MFEVB)

MMEKME

KVE


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MEF, MEF, MEF,MEF, MEF

MMEMEF

MME

MBFV

Building operations

Merits and Demerits of Euler Operations

If the operator acts on a valid topology and the statetransition it generates is valid, then the resulting topologyis a valid solid. Therefore, Eulers law is never verifiedexplicitly by the modeling system.

Merits:

They ensure creating valid topology

They provide full generality and reasonable simplicity

They achieve a higher semantic level than that of manipulatingfaces, edges and vertices directly

Demerits : They do not provide any geometrical information to define a solid

polyhedron

They do not impose any restriction on surface orientation, faceplanarity, or surface self intersection


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Advantages and Disadvantages of B-rep

Advantages:

It is historically a popular modeling scheme related closely totraditional drafting

It is very appropriate tool to construct quite unusual shapes likeaircraft fuselage and automobile bodies that are difficult to buildusing primitives

It is relatively simple to convert a B-rep model into a wireframemodel because its boundary definition is similar to the wireframedefinitions

In applications B-rep algorithms are reliable and competitive toCSG based algorithms

Disadvantages :

It requires large storage space as it stores the explicit definitionsof the model boundaries

It is more verbose than CSG

Faceted B-rep is not suitable for manufacturing applications

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