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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
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
2
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.
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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4.1 Requirements of 
Geometric Modelling
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.1 Total product cycle in a 
manufacturing environment
Ideas
Design
Analysis
Production
Geometric
Modelling
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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.
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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. 
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Functions of Geometric 
Modelling
• Manufacturing
– Parts classification.
– Process planning.
– Numerical control data generation and 
verification.
– Robot program generation.
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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.
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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 invariant
with 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.
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.2 3D geometric representation 
techniques
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
S6
S5
S1
S4
S3
S2S7
S8
V1
V2
(a) LINE MODEL (b) SURFACE MODEL
(c) VOLUME MODEL
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.3 A geometric model represented in 
wire-frame model
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.4 Ambiguities present in the wire-frame 
model
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.5 Impossible objects that can be 
modelled using a wire-frame model
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.6 Generation of 3D geometry using 
planar surfaces
S8
S6
S3
S6
S5
S1
S4
S3
S2S7
S8
(b) SURFACE MODEL
S1
S5
S4
S2
S7
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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.
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.8 Component model produced using 
translational (linear) sweep (extrusion)
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.9 Component model produced using 
translational (linear) sweep with taper in sweep direction
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.10 Component model produced using linear 
sweep with the sweep direction along a 3D curve
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.11 Component model produced using 
translational (linear) sweep with an overhanging edge
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.12 Component produced by the 
rotational sweep technique
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.13 Various solid modelling primitives
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.14 The Boolean operators and their 
effect on model construction
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.15 The Boolean operators and their 
effect on model construction
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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)
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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)
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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
Polygon
Polyhedra
Cylinders
Cones
Spheres
Ellipsoids
Paraboloid
Torus
Ruled surfaces Lofted surfaces
B-spline
Bezier surface
NURBS
Fractals
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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.
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.20 Coons surface generation
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.21 The Bézier curve and the associated 
control polygon
X
Y
O
Control
Polygon
Curve
Control points
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.22 The various examples of Bézier curves 
depending on the associated control polygons
p0
p0
p0
p1
p1
p1
p2
p3
p2
p3
p3
p2CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.23 The modification of Bezier curve by 
tweaking the control points
X
Y
O
Control
Polygon
Curve
Control points
Curve
O
Control pointsY
X
Polygon
Control
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.24 The spline curve
X
Y
O
Control
Polygon
Curve
Control points
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.25 The lofted surface
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.26 Example of filleting or blend method 
for model generation
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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))
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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4.4 Constraint Based 
Modelling
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.28 Example of initial sketch without any 
dimensions
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Applications by P N Rao, 2nd Ed
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Fig. 4.29 The sketch shown above which is 
fully constrained and dimensioned
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.30 The sketch in Fig. 4.29 when swept 
along a linear path produces the solid
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.31 The sketch for the new feature (a 
cut)
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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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Applications by P N Rao, 2nd Ed
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Fig. 4.33 The final solid
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.34 The model tree of the part showing 
the modelling process
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.35 A geometric model created following 
the sequence of features as 
Box → Hole → Shell
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.36 A geometric model created following 
the sequence of features as 
Box → Shell → Hole
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.37 Feature based model and its 
modified form
Base feature Slots - 2
Base feature
Slots - 2
Holes - 5
(A) Original model
Holes - 3
(B) Modified model
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.38 Typical drawing for the variant 
method of modelling
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.40 Examples of form elements used for 
model generation in the case of axi-symmetric 
components
Knurl
Groove Taper FilletArc
Chamfer
Turn
Thread
Face
Blank
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.42 Example component modelled using 
the features shown in Fig. 4.41
R1.5
1.6
Straight Knurl
Pitch 1mm
All chamfers 1x450
-0.015
-0.040
32
75 76
42
187
90
42
M36x1
6045
3.2
A
0.01 A
+
-
0.50
0.75
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Applications by P N Rao, 2nd Ed
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Fig. 4.43 Example component modelled using 
the features shown in Fig. 4.41
50
Chamfer 2X2 @45
10 dia 4 holes
12.5
25 80
Chamfer angle 45
25
4 holes on pcd 37.5
2
2X2 Groove
12.5
M8X1 LHT
75
Sectional Elevation End view
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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)
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.44 Circle
X
Y
O
θ
(X, Y)
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Applications by P N Rao, 2nd Ed
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Fig. 4.45 Ellipse
X
Y
O
θ
(X, Y)
a
b
12
2
2
2
=+
b
y
a
x
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.46 Parametric curve representation in 
Cartesian space
x
y
z
p0
p1
p2
p3
u
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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4.7 Surface Representation 
Methods
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.48 Typical surface display with the 
parametric variables u and v
x
y
z v
u
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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
p31
p42
p41=p(1,0)
p(0,v), u=0 curve
p12
p22
p32 p43
p33
p21
p24p13 p23
p44=p(1,1)
p(1,v), u=1 curve
p(u,1), v=1 curve
p14=p(0,1)
u
v
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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.
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.50 Elimination of hidden lines in display
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Applications by P N Rao, 2nd Ed
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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))
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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)
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Fig. 4.53 Section view generation from a 
geometric model
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Applications by P N Rao, 2nd Ed
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Fig. 4.54 Exploded view and bill of materials 
of an assembly modelled
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Applications by P N Rao, 2nd Ed
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4.9 Rapid Prototyping (RP)
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Figure 4.55 Schematic of Stereolithography
device
Scanning mirror
Laser
Recoating barLiquid resin
(to form model)
Cured resin
Platform
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Figure 4.56 Schematic of selective laser 
sintering device
Laser
Scanning mirror
Build
powder
Sintered
powder
(to form parts)
Powder feed roller
Platform
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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Figure 4.57 Schematic of Three-dimensional 
printing device
Powder feed roller
(to form parts)
powder
Glued
powder
Build
Binder solution
Printing head
Nozzle
Platform
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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)
PlatformCAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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 actual
cross 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
Excess
laminate
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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.
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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.
CAD/CAM Principles and 
Applications by P N Rao, 2nd Ed
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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.