Chapter Geometric Modelling

CAD/CAM Principles and Applications by P N Rao, 2nd Ed

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CAD/CAM Principles and Applications

Ch 4 Geometric Modelling


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Objectives• Understand the various requirements for the information that is

generated during the geometric modeling stage.• Study various types of geometric models possible and their

applications • Develop various methodologies used for geometric construction

such as sweep, surface models, solid models, etc.• Recognize the various types of surfaces and their application as

used in geometric modelling • Appreciate the concept of parametric modeling which is the

current mainstay of most of the 3D modeling systems• Develop the various mathematical representations of the curves

used in the geometric construction• Discuss the various CAD system requirements that need to be

considered while selecting a system for a given application• Understand the concept of rapid prototyping and the various

methods available for the purpose.


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4.1 Requirements of Geometric Modelling


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Fig. 4.1 Total product cycle in a

manufacturing environment

Ideas

DesignAnalysis

Production

GeometricModelling


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Functions of Geometric Modelling

• Design analysis:– Evaluation of areas and volumes.– Evaluation of mass and inertia properties.– Interference checking in assemblies.– Analysis of tolerance build-up in assemblies.– Analysis of kinematics — mechanics, robotics.– Automatic mesh generation for finite element

analysis.


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• Drafting– Automatic planar cross sectioning.– Automatic hidden line and surface removal.– Automatic production of shaded images.– Automatic dimensioning.– Automatic creation of exploded views for

technical illustrations.


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• Manufacturing– Parts classification.– Process planning.– Numerical control data generation and

verification.– Robot program generation.


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• Production Engineering– Bill of materials.– Material requirement.– Manufacturing resource requirement.– Scheduling.

• Inspection and Quality Control:– Program generation for inspection machines.– Comparison of produced part with design.


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Requicha and Voelker [1981] specified the following properties to be desired of in any geometric modelling

(solids) system.

• The configuration of solid (geometric model) must stay invariantwith regard to its location and orientation.

• The solid must have an interior and must not have isolated parts.

• The solid must be finite and occupy only a finite shape.• The application of a transformation or other operation that adds

or removes parts must produce another solid.• The model of the solid in E3 (Euler space) may contain infinite

number of points. However, it must have a finite number of surfaces, which can be described.

• The boundary of the solid must uniquely identify which part of the solid is exterior and which is interior.


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4.2 Geometric Models

• Two-dimensional, and• Three-dimensional.• The three principal classifications can

be – The line model,– The surface model, and– The solid or volume model


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Fig. 4.2 3D geometric representation

techniquesP1

P2P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

S6S5

S1

S4

S3

S2S7

S8

V1V2

(a) LINE MODEL (b) SURFACE MODEL

(c) VOLUME MODEL


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Fig. 4.3 A geometric model represented in

wire-frame model


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Fig. 4.4 Ambiguities present in the wire-frame

model


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Fig. 4.5 Impossible objects that can be

modelled using a wire-frame model


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Fig. 4.6 Generation of 3D geometry using

planar surfaces

S8S6

S3

S6S5

S1

S4

S3

S2S7

S8

(b) SURFACE MODEL

S1

S5

S4

S2

S7


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4.3 Geometric Construction Methods

• The three-dimensional geometric construction methods which extend from the 2D that is normally used are:– Linear extrusion or translational sweep,

and– Rotational sweep.


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Fig. 4.8 Component model produced using

translational (linear) sweep (extrusion)


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translational (linear) sweep with taper in sweep direction


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Fig. 4.10 Component model produced using linear

sweep with the sweep direction along a 3D curve


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translational (linear) sweep with an overhanging edge


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Fig. 4.12 Component produced by the

rotational sweep technique


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Fig. 4.13 Various solid modelling primitives


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Fig. 4.14 The Boolean operators and their

effect on model construction


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Fig. 4.15 The Boolean operators and their

effect on model construction


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Fig. 4.16 Creating a solid with the 3D primitives in solid modelling and the model shown in the form of Constructive Solid Geometry

(CSG)


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Fig. 4.17 Model generated using the sculptured surfaces (Image appears with the permission of IBM World Trade Corporation/Dassault

Systems - Model generated using CATIA)


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Fig. 4.18 The various types of surfaces used in geometric modelling

Classification of Surfaces

Planar surfaces Curved surfaces Free form surfaces

Plane Single curved Double curved Coons surface

PolygonPolyhedra

CylindersCones

SpheresEllipsoidsParaboloidTorus

Ruled surfaces Lofted surfaces

B-splineBezier surfaceNURBSFractals


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Fig. 4.19 Ruled surface on the left is shown the curves

from which the ruled surface on the right is formed.


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Fig. 4.20 Coons surface generation


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Fig. 4.21 The Bézier curve and the associated

control polygon

X

Y

O

ControlPolygon

Curve

Control points


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Fig. 4.22 The various examples of Bézier curves

depending on the associated control polygons

p0p0

p0

p1

p1

p1

p2

p3

p2

p3

p3

p2


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Fig. 4.23 The modification of Bezier curve by

tweaking the control points

X

Y

O

ControlPolygon

Curve

Control points

Curve

O

Control pointsY

X

PolygonControl


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Fig. 4.24 The spline curve

X

Y

O

ControlPolygon

CurveControl points


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Fig. 4.25 The lofted surface


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Fig. 4.26 Example of filleting or blend method

for model generation


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Fig. 4.27 Example of tweaking method for surface modification ((Image appears with the permission of IBM World Trade

Corporation/Dassault Systems - Model generated using CATIA))


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4.4 Constraint Based Modelling


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Fig. 4.28 Example of initial sketch without any

dimensions


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Fig. 4.29 The sketch shown above which is

fully constrained and dimensioned


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Fig. 4.30 The sketch in Fig. 4.29 when swept

along a linear path produces the solid


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Fig. 4.31 The sketch for the new feature (a

cut)


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Fig. 4.32 The solid after executing an

extruded cut of the geometry in Fig. 4.31


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Fig. 4.33 The final solid


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Fig. 4.34 The model tree of the part showing

the modelling process


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Fig. 4.35 A geometric model created following the sequence of features as

Box → Hole → Shell


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Fig. 4.36 A geometric model created following the sequence of features as

Box → Shell → Hole


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Fig. 4.37 Feature based model and its

modified form

Base feature Slots - 2

Base featureSlots - 2

Holes - 5

(A) Original model

Holes - 3

(B) Modified model


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Fig. 4.38 Typical drawing for the variant

method of modelling


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Fig. 4.39 Part model produced using the

symbolic programming

T

C

R

G

N

G CC

SYMBOL KEYSCOMPOSED PART

KEY SEQUENCE


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Fig. 4.40 Examples of form elements used for model generation in the case of axi-symmetric

components

KnurlGroove Taper FilletArc

Chamfer

TurnThread

Face

Blank


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Fig. 4.41 Examples of form features for modelling axi-symmetric

components with milled features

Step

Keyway

Concentric slot

Axial hole

Radial hole

Axial slot

Radial slot

Splines

Turn Taper Face Fillet

Chamfer Groove Thread Knurl


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Fig. 4.42 Example component modelled using

the features shown in Fig. 4.41

R1.5

1.6

Straight KnurlPitch 1mm

All chamfers 1x450

-0.015-0.040

32

75 76

42

187

9042

M36x1

6045

3.2A

0.01 A

+-

0.500.75


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Fig. 4.43 Example component modelled using

the features shown in Fig. 4.41

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Chamfer 2X2 @4510 dia 4 holes

12.5

25 80

Chamfer angle 45

25

4 holes on pcd 37.5

2

2X2 Groove12.5

M8X1 LHT

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Sectional Elevation End view


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4.6 Curve representation

• Implicit form, and• Parametric form.• In parametric form, the curve is

represented as• X = x(t)• Y = y(t)• Z = z(t)


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Fig. 4.44 Circle

X

Y

Oθ

(X, Y)


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Fig. 4.45 Ellipse

X

Y

Oθ

(X, Y)

a

b

12

2

2

2

=+by

ax


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Fig. 4.46 Parametric curve representation in

Cartesian space

x

y

z

p0

p1

p2

p3

u


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Fig. 4.47 Two cubic Bézier curves joined at p3

x

y

z

p0

p1

p2

p3

u p4

p5

p6


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4.7 Surface Representation Methods


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Fig. 4.48 Typical surface display with the

parametric variables u and v

x

y

z v

u


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Fig. 4.49 A bi-cubic Bézier surface patch

x

y

z

p(u,0), v=0 curve

p11=p(0,0)

p21

p31p42

p41=p(1,0)

p(0,v), u=0 curve

p12p22

p32 p43

p33

p21p24p13 p23

p44=p(1,1)

p(1,v), u=1 curve

p(u,1), v=1 curve

p14=p(0,1)

u

v


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4.8 Modelling Facilities Desired

• The geometric modelling features.• The editing or manipulation features.• The display control facilities.• The drafting features.• The programming facility.• The analysis features.• The connecting features.


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Fig. 4.50 Elimination of hidden lines in display


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Fig. 4.51 Shaded image of a CAD geometric model ((Image appears with the permission of IBM World Trade Corporation/Dassault Systems -

Model generated using CATIA))


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Fig. 4.52 Orthographic views from a geometric model (Image appears with the permission of IBM World Trade Corporation/Dassault Systems -

Model generated using CATIA)


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Fig. 4.53 Section view generation from a

geometric model


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Fig. 4.54 Exploded view and bill of materials

of an assembly modelled


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4.9 Rapid Prototyping (RP)


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Figure 4.55 Schematic of Stereolithography

deviceScanning mirror

Laser

Recoating barLiquid resin(to form model)Cured resin

Platform


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Figure 4.56 Schematic of selective laser

sintering device

LaserScanning mirror

Buildpowder

Sinteredpowder(to form parts)

Powder feed roller

Platform


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Figure 4.57 Schematic of Three-dimensional

printing device

Powder feed roller

(to form parts)powderGlued

powderBuild

Binder solution

Printing head

Nozzle

Platform


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Fig. 4.58 Schematic of Fused deposition

modelling device

Filament from a coil

Feeder

Melter

Extrusion nozzle

Solidified plaster(to form model)

Platform


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Fig. 4.59 Schematic of Laminated Object

Manufacturing device

material

Take-up roll Material supply roll

Top view

Laminating roller Contour of actualcross section of the model

Band of build material

Splits in excess material(for ease of removal)

Platform

Laser

Scanning mirror

Laminating roller Laminate model Band of build material

Excesslaminate


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Summary• Information entered through geometric modeling is utilized in a

number of downstream applications such as drafting, manufacturing, inspection and planning.

• Geometric models are three types, viz line model, surface model and solid model. Line model though simple is rarely used because of the ambiguity present. Surface and solid models are extensively used in industrial applications.

• Among the geometric construction methods sweep or extrusion is most widely used, because of its simplicity and elegance in developing 3D models.

• Solid modeling provides the most unambiguous representation of the solid model, but is more computing intensive. However to get the correct geometric model, it is essential to utilize solid modeling approach.


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Summary• Surfaces are more widely used and it is necessary to use

different types of surfaces such as b-splines, Bezier, NURB, lofted, to get the user requirements fulfilled.

• Constraint or parametric based modeling is the main methodology used by most of the 3D CAD systems. This system helps in grasping the designer’s intent and would greatly facilitate the modification and reuse of the existing designs.

• Some variant modeling systems are used based on tabular data for specific applications.

• Form features is another form of modeling system that helps in designing CAD systems with more intelligence built into the geometric entities that is possible by purely geometric systems discussed thus far.


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Summary• The mathematical representation of the geometric entities can

be in implicit or parametric form, the latter being the preferred method used in CAD systems because of its easier adaptation in software development.

• The curve representation methods can be extended for surface representations such as used in free form surfaces.

• A number of modeling facilities need to be considered while selecting a CAD/CAM system for any given application.

• Rapid prototyping is used to generate the product directly from the 3D CAD model data. A number of different processes such as stereo lithography, selective laser sintering, 3D printing, fused deposition modeling, laminated object manufacturing, are used for this purpose.