CAD/CAM Principles and
Applications
Ch 4 Geometric Modelling
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
1
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
Geometric
Modelling
Ideas
Design
Analysis
Production
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
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Functions of Geometric
Modelling
• Design analysis:
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–
–
–
–
–
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.
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•
•
•
•
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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
P2
P3
P1
S8
P10
S6
S5
P4
P9
P11
P12
S4
P5
P8
P6
S3
S1
S7
S2
P7
(a) LINE MODEL
(b) SURFACE MODEL
V1
V2
(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
S5
S3
S6
S6
S5
S8
S8
S4
S3
S4
S1
S7
S2
S1
S2
S7
(b) SURFACE MODEL
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
Plane
Polygon
Polyhedra
Curved surfaces
Free form surfaces
Single curved
Double curved
Cylinders
Cones
Spheres
Ellipsoids
Paraboloid
Torus
Ruled surfaces
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
Coons surface
B-spline
Bezier surface
NURBS
Fractals
Lofted surfaces
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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
Y
Control
Polygon
Control points
Curve
X
O
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
p2
p2
p1
p3
p3
p0
p0
p1
p1
p3
p0
p2
CAD/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
Y
Control
Polygon
Control points
Y
Curve
Curve
O
Control
Polygon
Control points
X
X
O
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
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Fig. 4.24 The spline curve
Y
Control points
Control
Polygon
Curve
X
O
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
CAD/CAM Principles and
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
CAD/CAM Principles and
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
Holes - 3
Slots - 2
(A) Original model
Base feature
Slots - 2
Holes - 5
(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
COMPOSED PART
SYMBOL KEYS
G
C
C
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
Thread
Arc
Groove
Taper
Turn
Fillet
Knurl
Chamfer
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
Taper
Turn
Chamfer
Groove
Step
Keyway
Splines
Concentric slot
Fillet
Face
Thread
Knurl
Axial hole
Radial hole
Axial slot
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
Radial slot
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Fig. 4.42 Example component modelled using
the features shown in Fig. 4.41
A
3.2
All chamfers 1x45
0
Straight Knurl
Pitch 1mm
M36x1
1.6
45
90
60
42
-0.015
-0.040
0.01 A
R1.5
42
32
75
187
+
-
0.50
0.75
76
CAD/CAM Principles and
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
Chamfer 2X2 @45
10 dia 4 holes
2X2 Groove
M8X1 LHT
12.5
75
50
Chamfer angle 45
25
2
12.5
25
80
Sectional Elevation
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
4 holes on pcd 37.5
End view
53
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
Y
(X, Y)
θ
X
O
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
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Fig. 4.45 Ellipse
Y
2
2
x
y
+
=
1
2
2
a
b
(X, Y)
b
θ
O
a
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
X
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Fig. 4.46 Parametric curve representation in
Cartesian space
p3
z
p2
y
u
x
p1
p0
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
p2
p3
p1
u
p4
p0
z
p5
y
p6
x
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
z
v
u
y
x
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
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Fig. 4.49 A bi-cubic Bézier surface patch
p(u,1), v=1 curve
p(0,v), u=0 curve
p23
p13
p24
p21
p12
p22
p44=p(1,1)
v
p21
z
p11=p(0,0)
p33
p14=p(0,1)
u
p32
p43
p31
p42
y
p(1,v), u=1 curve
p(u,0), v=0 curve
x
p41=p(1,0)
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
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4.8 Modelling Facilities
Desired
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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
Cured resin
(to form model)
Liquid resin
Laser
Recoating bar
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
Scanning mirror
Laser
Powder feed roller
Platform
Build
powder
Sintered
powder
(to form parts)
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
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Figure 4.57 Schematic of Three-dimensional
printing device
Binder solution
Powder feed roller
Printing head
Nozzle
Platform
Build
powder
Glued
powder
(to form parts)
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)
Platform
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
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Fig. 4.59 Schematic of Laminated Object
Manufacturing device
Top view
Splits in excess
material
(for ease of removal)
Band of
build material
Contour of actual
cross section of the model
Laminating roller
Scanning mirror
Laser
Band of
build material
Laminate model
Laminating roller
Excess
laminate
material
Take-up roll
CAD/CAM Principles and
Applications by P N Rao, 2nd Ed
Platform
Material supply roll
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Summary
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•
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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.
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Applications by P N Rao, 2nd Ed
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