l30 solidmod basics

8/12/2019 L30 Solidmod Basics

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Solid Modeling

Concepts

AML710 CAD LECTURE 30

Evolution of Geometric Modeling A wireframe representaion of an object is done using edges (linescurves) and vertices. Surface representation then is the logicalevolution using faces (surfaces), edges and vertices. In thissequence of developments, the solid modeling uses topologicalinformation in addition to the geometrical information to represent theobject unambiguously and completely.

Wireframe Model Solid Model


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Manifold Vs Non-manifoldTwo Manifold Representations:

Two manifold objects are well bound, closed and homomorphic to atopological ball.

Non-manifold Representations:When restrictions of closure and completeness are removed from the soliddefinition, wireframe entities can coexist with volume based solids.

Two-Manifold Object

Danglingface

Non-ManifoldRepresentation

Danglingedge

Disadvantages of Wireframe Models

Ambiguity Subjective human interpretation Complex objects with many edges

become confusing Lengthy and verbose to define Not possible to calculate Volume

and Mass properties, NC toolpath, cross sectioning etc


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Definition of a Solid Model A solid model of an object is a more complete representation than itssurface (wireframe) model. It provides more topological informationin addition to the geometrical information which helps to representthe solid unambiguously.

Wireframe Model Solid Model

History of Geometric Modeling

Univ. of Rochester, Voelcker & RequichaPAP, PADL-I,PADL2

1972

Baumgart, Stanford Univ., USAEuler OpsWinged Edge,B-rep

1975

Evans and Sutherland, 1 st commercialRomulus1981

Eastmans Group in CMU, USAGLIDE-I1975

Hokkaido University, JapanTIPS-I1973

Braids CAD Group inCambridge, UK

Build-IBuild-II

1973

Developer Modeler Year


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Geometric Modeling - Primitives

Primitives and Instances

PRIMITIVES

INSTANCES

Instancing: An instance is a scaled

/ transformed replica of its original.The scaling may be uniform ordifferential. It is a method ofkeeping primitives in their basicminimum condition like unit cubeand unit sphere etc.


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Solid Primitives: Ready starting objects

Origin of MCS: P Orientation Topology and shape

Dimension (size) L,B,H.D,R Perspective on/off

X

Y

Z

L

R

Y

X

Z

R

Z

BH

DX

Y

P

Y

X

Z B

H

D

P

SPHEREWEDGE

CYLINDERBLOCK

Solid Primitives: Ready starting objects

Ro

P

R i

RX

Y

Z

R

H

Z

X

Y

TORUS

CONE

PERSPECTIVE OFF PERSPECTIVE ON


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Operation on Primitives A desired solid can be obtained by combining two or more solidsWhen we use Boolean (set) operations the validity of the third(rusulting) solid is ensured

UNION: BLOCK CYLINDER A B

A

B

A

B

3 Dimensional

Operation on Primitives A desired solid can be obtained by combining two or more solids

When we use Boolean (set) operations the validity of the third(rusulting) solid is ensured

UNION: BLOCK CYLINDER A B

A

B

A

B


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Operation on Primitives

INTERSECTION:

BLOCK CYLINDER

A B

A

B

A

B

A

B

DIFFERENCE:

BLOCK - CYLINDER

A - B

B - A

Eulers formula gives the relationamongst faces, edges and vertices of asimple polyhedron.

V-E+F=2 . This formula tells us thatthere are only 5 regular polyhedra

Consider polyhedra with every facehaving h edges, k edges emanatingfrom each vertex and each edgesharing exactly 2 faces, then we canwrite an invariant for the solid as:

hF=2E=kV Substituting this in Eulersformula we get

Simple Polyhedron Eulers Formula

21111

222

+==+k h E h

E E

k E


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We notice that for a polyhedron h,k 3 What happens if h=k=4 ? If we take h=3, then 3 k 5 Then by symmetry, if k=3 then 3 h 5 Thus the possib le polyhedra can be symbolic ally given as:(h,k,E)=(3,3,6), (3, 4, 12), (4,3,12), (5, 3, 30) and (3, 5, 30);

tetrahedron, oc tahedron, cube, dodecahedron, icosahedron

21111

0222

+=


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The following example will illustrate the conceptsof interior , boundary , closure and exterior . Consider a unit sphere

Its interior is Its Boundary is Solid sphere is

Its closure

Whatever is outside of this closure is exterior

Interior, Exterior and Closure

1

1

1

222

222

222

++

=++


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Rigidity: Shape of the solid is invariant w.r.tlocation/orientation

Homogeneous 3-dimensionality : The solidboundaries must be in contact with the interior.No isolated and dangling edges are permitted.

Properties of Solid Models

=

Danglingedge

Danglingface

Non-ManifoldRepresentation

Finiteness and finite describability: Size isfinite and a finite amount of information candescribe the solid model.

Closure under rig id motion and regularisedBoolean operations : Movement and Booleanoperations should produce other valid solids.

Boundary determinism: The boundary mustcontain the solid and hence determine the soliddistinctly.

Any valid so lid must be bounded , closed,regular and semi-analytic subsets of E 3



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The regularized point sets which representthe resulting solids from Booleanoperations are called r-sets (regularizedsets .

R-sets can be imagined as curvedpolyhedra with well behaved boundaries .


The schemes operate on r-sets or point sets to produce thedesired solid model, which is automatically a valid solid

For unambiguous, complete and unique representation thescheme maps points form point set (Euclidian) into a validmodel to represent the physical object.

Properties of Solid Modeling Schemes

Unambiguous,complete, unique

O P

O P 1 P 2

O1 O 2P

Unambiguous,complete, non-unique

Amb iguous ,incomplete, non-unique

Unl imited Po in t set , W Mod el space , S

UCU

UCN

AINRepresentation

Scheme


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Domain : The type of objects that can be represented or thegeometric coverage.

Validity : The resulting solid model when subjected toalgorithms should succeed as valid solid.

Completeness and Unambiguousness .

Uniqueness : Positional and Permutational uniqueness.

Common Features of Modeling Schemes

Positionally non-unique

A B C AC B

a

b ba

Permutationally non-unique

S=A U B U C S=C U B U A

Conciseness, ease of creation and efficiency inapplication.

Conciseness means compact database torepresent the object.

Ease of operation implies user-friendl iness.Most of the packages use CSG approach tocreate the solids

The representation must be usable for lateroperations like CAM / CNS etc

Desirable Properties of ModelingSchemes


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Classification of Solid Modeling

Boundary based or Volume based Object based or spatially based Evaluated or unevaluated

Evaluated

Unevaluated

Boundarybased

VolumeBased

SpatialbasedObjectbased

E v al u a t e d

Un ev al u a t e d

OctreeHalf spaceSpatial

Non-parametic

BoundaryRep

Object

Cell EnumBoundaryCell Enum

Spatial

CSGEuler OpsObject

Boundary | Volume

1. Half Spaces.

2. Boundary Representation (B-Rep)

3. Constructive Solid Geometry (CSG)

4. Sweeping

5. Analytical Solid Modeling (ASM)

6. Cell decomposition

7. Spatial Ennumeration

8. Octree encoding

9. Primitive Instancing

The 3 most popular schemes: B-rep, CSG,Sweeping

Major Modeling Schemes


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Some representations can be converted to otherrepresentations.

CSG to B-REP (but not B-rep to CSG)

B-Rep

2. Sweep CSGCell Decomposition

Conversion Among Representations

Some Solid Modelers in Practice

BREP+CSGBREPPARAMETRICPRO-E

BREP+CSGBREPINTERGRAPHSOL. MOD. SYS

BREP+CSGCSGMcDONELLDOUGLAS

UNISOLIDS /UNIGRAPHICS

BREP+CSGBREPCOMPUTERVISION

SOLIDESIGN

CSGCSGCORNELL UNI.PADL-2

HYPERPATCHES+CSG ASMPDA ENGG.PATRAN-G

BREP+CSGBREPSDRC/EDSGEOMOD /I-DEAS

BREP+CSGCSGIBMCATIA

User InputPrimaryScheme

Developer Modeler


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Three types of algorithms are in use in manysolid modelers according to the kind of inputand output

1. a; data -> representation

most common build, maintain, managerepresentations

Algorithms in Solid Modelers

Data Representation

Algori thm

Some algorithms operate on existing models toproduce data.

2. a; representation -> data

E.g.. Mass property calculation


DataRepresentation

Algori thm


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Some representations use the availablerepresentations to produce anotherrepresentation.

3. a: representation -> representation

E.g. Conversion from CSG to B-rep


Representation

Algori thm

Representation

Closure and Regularized setOperations

The interior of the solid is geometrically closed by itsboundaries

S bS iS =U

bS iS kiS S U==

If the closure of the interior of a given set yields that samegiven set then the set is regular

Mathematically a given set is regular if and only if:

)(*);(*

)(*);(*

cQkiQcQPkiQP

QPkiQPQPkiQP

bS iS kiS S

==

==

== U


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l30 solidmod basics

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