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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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Functions of Geometric Modelling
• 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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Functions of Geometric Modelling
• Manufacturing– Parts classification.– Process planning.– Numerical control data generation and
verification.– Robot program generation.
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Functions of Geometric Modelling
• 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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Fig. 4.9 Component model produced using
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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Fig. 4.11 Component model produced using
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
50
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
75
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.