All physical objects have 3D boundaries that define their shape. Surface modeling uses points, lines, and faces to define these boundaries mathematically. There are several types of surfaces, including plane, ruled, revolved, and freeform surfaces. Revolved surfaces are created by rotating a profile around an axis, generating surfaces like cylinders and cones. Curves and surfaces are essential for modeling complex shapes encountered in engineering designs.

1. Surface Modeling
All physical objects are 3-dimensional. In a number of cases, it is sufficient to describe the boundary of a solid
object in order to specify its shape without ambiguity. This fact is illustrated in Fig. The boundary is a collection
of faces forming a closed surface. The space is divided into two parts by the boundary - one part containing the
points that lie inside and forming the object and the other the environment in which the object is placed. The
boundary of a solid object may consist of surfaces which are bounded by straight lines and curves, either singly or
in combination.
Fig. Representation of Boundary
Figure is typical of several components, one comes across in engineering. The surface of this component can be
produced by revolving a profile about an axis of rotation. A surface model is defined in terms of points, lines
and faces. This type of modeling is superior to wire frame modeling discussed earlier.
A major advantage of surface modeling is its ability to differentiate flat and curved surfaces. In graphics, this
helps to create shaded image of the product. In manufacture, surface model helps to generate the NC tool path for
complex shaped components that are encountered in aerospace structures, dies and moulds and automobile body
panels. A surface can be created in several ways:
i. Creating a plane surface by the linear sweep of a line or series of lines.
ii. Revolving a straight line about an axis. Cylindrical, conical surfaces etc. can be generated by this technique.
iii. Revolving a curve about an axis.
iv. Combination of plane surfaces.
Fig. A Typical Revolved Surface Model

2. v. Analytic Surfaces: Planes, cylinders, cones, ellipsoid, parabolic hyperboloid etc can be defined by mathematical
equations in terms of X, Y and Z co-ordinates.
vi. Sculptured Surfaces: These are also called Free Form Surfaces. These are created by spline curves in one
or both directions in a 3-D space. These surfaces are used in the manufacture of car body panels, aircraft
structures, mixed flow impellers, telephone instruments, plastic containers and several consumer and engineering
products.
Modeling of curves and surfaces is essential to describe objects that are encountered in several areas of mechanical
engineering design. Curves and surfaces are the basic building blocks in the following designs:
i. Body panels of passenger cars
ii. Aircraft bulk heads and other fuselage structures, slats, flaps, wings etc.
iii. Marine structures
iv. Consumer products like plastic containers, telephones etc.
v. Engineering products like mixed flow impellers, foundry patterns etc
A curve has one degree of freedom while a surface has two degrees of freedom. This means that a point on a
curve can be moved in only one independent direction while on surfaces it has two independent directions to
move. This is shown in Fig.
Fig. Degrees of Freedom
Design of Curved Shapes
Design of curved shapes should satisfy the following requirements:
i. It should be possible to represent the shape mathematically.
ii. The modeling should involve minimum computation.
iii. It should be possible to generate a CNC program to machine the surfaces (2, 3, 4 and 5 axis machining) or to
prepare a mould or die to make the part (as in plastic injection molding or casting or automobile panel pressing).
A component can be designed using the curves and shapes which can be mathematically described e.g. arc, circle,
conics, ellipsoid, hyperbolic paraboloid, sphere, and cone, cylinder, linear, conical - and circular swept surfaces etc.
However, very often the designer starts with specifying a few points which roughly describe the shape.
Two approaches are available to designers to model curves and surfaces in such cases: Interpolation and
Approximation. The interpolation essentially tries to pass a curve on a surface called interpolant through
all these points. Approximation tries to fit a smoother curve on surface which may be close to these points

3. but may not actually pass through each of them. Fig. illustrates the difference between interpolation (a) and
approximation (b).
Fig. Interpolation and Approximation
One of the popular methods of interpolation is to use the Lagrange polynomial, which is the unique polynomial
of degree n passing through n + 1 points. However, Lagrange polynomial is unsuitable in modeling of curves
because of:
i. Large number of computations involved and
ii. Tendency for the curve to oscillate between data points when the data points are large.
Another approach is to look for a polynomial of fewer degrees than W passing through these W + 1 data points.
This is done by combining polynomials of lesser degree passing through several consecutive data points
and smoothly piecing several such curve segments together by blending. In general, the modeling of curves is
done to satisfy the following:
i. Axis Independence
ii. Global and Local Control
iii. Smoothness of curves
iv. Versatility
v. Continuity between adjoining segments of curve.
1. Introduction: Surfaces
Wire Frame Models are unable to represent complex surfaces of objects like car, ship, airplane wing, castings
etc. A Surface Model can be used to represent the surface profile of these objects. Also, surface model can be
used for calculating mass properties, interference between parts, generating cross-sectioned views,
generating finite element mesh, and generating NC tool paths for continuous path machining. Additionally,
surface model can be used to fit experimental data, discretized solutions of differential equations,
construction of pressure surface, construction of stress distribution etc.
Surface creation on a CAD system usually requires wire frame entities: lines, curves, points, etc. All analytical
and synthetic curves can be used to generate surfaces.
In order to visualize surfaces on a graphic display, a mesh, say m x n in size is usually displayed; the mesh size is
controlled by the user. Most CAD systems provide options to set the mesh size. A surface of an object is more
complete and less ambiguous representation than its wire frame model; it is an extension of a wire frame model
with additional information.

4. A Wire Frame Model can be extracted from a surface model by deleting all surface entities (not the
wireframe entities – point, lines, or curves!). Databases of surface models are centralized and associative;
manipulation of surface entities in one view is automatically reflected in the other views. Surface models can be
shaded and represented with hidden lines.
2. Types of Surfaces
Plane Surface:
This is the simplest surface, requires 3 non-coincidental points to define an infinite plane. The plane surface
can be used to generate cross sectional views by intersecting a surface or solid model with it.
Ruled (Lofted) Surface:
This is a linear surface. It interpolates linearly between two boundary curves that define the surface.
Boundary curves can be any wire frame entity. The surface is ideal to represent surfaces that do not have any
twists or kinks.
Surface of Revolution:
This is an axi-symmetric surface that can model axi-symmetric objects. It is generated by rotating a planar
wire frame entity in space about the axis of symmetry of a given angle.
Tabulated Surface:
This is a surface generated by translating a planar curve a given distance along a specified direction. The
plane of the curve is perpendicular to the axis of the generated cylinder.
Bi-Linear Surface:
This 3-D surface is generated by interpolation of 4 endpoints. Bi-linear surfaces are very useful in finite
element analysis. A mechanical structure is discretized into elements, which are generated by interpolating 4 node
points to form a 2-D solid element.

5. Coons Patch:
Coons patch or surface is generated by the interpolation of 4 edge curves as shown.
Bezier Surface:
This is a synthetic surface similar to the Bezier curve and is obtained by transformation of a Bezier curve. It
permits twists and kinks in the surface. The surface does not pass through all the data points.
B-Spline Surface:
This is a synthetic surface and does not pass through all data points. The surface is capable of giving very smooth
contours, and can be reshaped with local controls. Computer generated surfaces play a very important part in
manufacturing of engineering products. A surface generated by a CAD program provides a very accurate and
smooth surface, which can be generated by NC machines without any room for misinterpretation. Therefore, in
manufacturing, computer generated surfaces are preferred. Since surfaces are mathematical models, we can
quickly find the centroid, surface area, etc. Another advantage of CAD surfaces is that they can be easily modified.
3. Interpolated Surfaces – Bilinear Surface
A bilinear surface is obtained by linear interpolation between four points, which may or may not lie in the
same plane. The four points appear as vertices or corner points and the parameter values u and v create lines at
various intervals to provide the surface visibility, shown in the figure. The parameters u and v are defined as
≤ u ≤ , and ≤ v ≤
The interpolated parametric equation of a bilinear surface is given as:
, = − − , + − , + − , + ,
In matrix form, it can be written as

6. , = [ − − − − ]
[
,
,
,
, ]
Application of Bilinear Surfaces
Bilinear patches are extensively used in 2-D finite element analysis (FEA). In FEA, an engineering structure is
defined by several bilinear surfaces elements , which are created by joining points on the structure’s geometry,
called nodes. The nodes are connected to other nodes to create quadrilateral surfaces. Points not lying on the
nodes are calculated by interpolation. Thus, the entire structure is completely defined by the nodes and the
bilinear surfaces.
Drawbacks of Bilinear Surfaces
Bilinear surfaces have a very limited use, mainly, for FEA. Since only 4 points can be used in the interpolation, the
smoothness of the generated surface is limited. Additionally, there is no flexibility to control shapes of the
surface, unlike the sweeped surfaces.
4. Interpolated Surfaces – Coons Patch
A linear interpolation between four bounded curves is used to generate a Coons surface, also called as Coons patch.
The method is credited to S. Coons who developed this concept for generating a surface.
Linear interpolation between the boundary curves P(0,v), P(u,0), P(1,v) , and P(u,1) gives the equation
, = − , + , + , + − ,
The above equation gives wrong values at the corners (u,v = 0 and 1). For example, substituting the values of u
and v we get,
, = , + , = ,
, = ,
Which are obviously wrong values, therefore, the coons patch is created by modification of the interpolation
equation, where the corners are subtracted. The modified interpolation equation is given as,
, = − , + , − , + − , − − − , − − ,
− − , − ,
For computational purposes, it is more convenient to write this equation as,

7. , = [ − − ]
[
,
,
,
, ]
− � .
−[ − − − − ]
[
,
,
,
, ]
− � � �
Which gives,
, = [ − ] [
− , − , ,
− , − , ,
, ,
] [
−
]
Other interpolated surfaces include the Parametric Cubic patches.
Applications
Coon’s Surface is easy to create, and therefore, many 2-D CAD packages utilize it for generating models. However,
it has only a limited application since the surface is inflexible and cannot create very smooth surfaces. It would be
very difficult to produce a smooth automobile fender using the Coons surface. Several CAD software, including,
AutoCAD uses this surface for generating surfaces between 4-bounded edges.
5. Linearly Sweeped Surfaces
A Sweeped Surface is generated when a curve is parametrically translated or rotated. In CAD, a surface is
represented by a series of curves, which are parametrically generated at various instances. For example, a
cylindrical surface is generated when a circular arc is translated up to the given dimension using a parameter t,
where t varies as, ≤ t ≤ .
In the figure shown, the cylindrical surface is generated when a circular arc is translated a distance L, with the
interim instances at t = 0.1, 0.2, 0.3, 1. Here, the parameter t is given 10 values, and therefore, the surface of the
cylinder is represented by 10 circular curves. The appearance of the surface improves as the parameter t varies at
smaller intervals. Thus, if t is varied with Δt = . , there will be 100 circular curves representing the surface.
A surface is an extension of a curve. The parametric representation of a curve is given by a single-vector equation
of the form:
= [ ]
Here, only one parametric variable or one degree of freedom is needed. Whereas, a surface representation requires
two parametric variables, and the equation is given as:
, = [ , , , ]
Tracing a point in the s and t directions, as shown in the figure on the next page, generates a surface. One
parameter variable is kept constant while varying the other one. A series of curves is created along the s and t

8. directions. For example, constraining the parameters s and t between zero and 1, the set of curves generated along
the s direction is,
, , . , , . , … … … . ,
and the other set of curves along the t direction is,
, , , . , , . … … … . ,
Thus, creation of a surface requires creation of the multiple curves that constitute it. This concept can be applied to
both, the surface that has an analytical formulation (conic sections) and to a free-form surface (Bezier, B-spline).
6. Revolved Surfaces (Circular Sweep)
Surface of revolution is obtained by rotating a plane-curve around an axis. In the figure shown, line AB is rotated
about the z-axis through an angle of 2π radians, generating a cylinder. A line or curve when revolved can generate
all kinds of surfaces, based on the condition of rotation. Any point on the surface is a function of two parameters t
and θ. Here, t describes the entity to be rotated and θ represents the angle of rotation. In general, a point on line
AB (lying in the xz-plane) is represented by [x(t), 0, z(t)] and, when rotated by θ radians, it becomes [x t cosθ,
x t sinθ, z t ].
In general, the point matrix gives a point on the surface of revolution obtained by rotation around the z-axis,
, � = [ � � � ]
In matrix form the equation can be written as,
, � = [ ]
[
� � �
]

9. Note: The above rotation matrix is equivalent to the rotational transformation matrix studied earlier, which is,
[
� � �
]
= −
[
� � �
− � � �
]
Thus, the generated surface is a rotational transformation of a line (or curve), except θ is not constant, but has
values, ≤ θ ≤ π.
Example: Generate the conical surface obtained by rotation of the line segment AB around the z-axis with, A = (1,
0, 1) and B = (7, 0, 7).
Solution: Line AB can be represented in parametric form as:
= [ ]
and the parametric equation of a line is,
= + −
based on this equation, the coordinates of a of point on the line are given as,
= + − = + ; = ; = + − = +
The equation of the surface as given above is,
, � = [ � � � ] = [ + � + � � + ]
Any point on the surface can be located by substituting t and θ values in the above equation, e.g.: at t = .4 and θ =
π/2 radians
. , �/ = [ + . �/ + . � �/ + . ] = [ . . ]
which is the point on the surface at (0.4, π/2)
Example: Generate a Torus by rotating a circle of radius r and the center at (a,0,0) about the z-axis.
Solution: Rotating a circle contained in the x z plane around the z-axis can generate a torus. The center of the
circle has coordinates (a, 0, 0) and equation of the circle in parametric form is given as;
� = [ + � , , � � ]
The torus is represented by,
�, � = {[ + � �], [ + � � �], � �}

10. In this case, the parameters are φ and θ.
7. Circular Sweep of a Synthetic Curve
Equation of a synthetic curve (free-form curve), is given as,
= [ ][�][�]
The surface of revolution is then given by,
, � = [ ][�][�][ ]�
= [ ][ ]�
Where, Q (t, θ is the equation of the curve, and [Tr]θ is the rotation matrix about the z-axis.
Note: To rotate the curve about the axis, we will have to use the translation and rotation matrices.
Example: A cubic Bezier curve is defined by the control points: P1 (1,0,2), P2 (3,0,4), P3 (2,0,6), P4 (5,0,7). Find the
surface of revolution obtained by revolving the curve about the z-axis and calculate the point on the surface at t =
0.5, θ = π/4 rad.
Solution: The cubic Bezier curve is given by the equation,
= [ ][�][�] = [ ]
[
− −
−
−
] [ ]
Substituting the coordinates of the points, we get
= [ ]
[
− −
−
−
] [ ]
The surface of revolution is:
, � = [ ]
[
− −
−
−
] [ ] [
� � �
]
≤ � ≤ � ; ≤ ≤
For t = 0.5 and θ = π/4, the surface equation is,
, � = [ . . . ]
[
− −
−
−
] [ ] [
�/ � �/
]
= [ . . . ]
8. Creating a Surface by Parametric Sweeping
In the examples given above, sweeping a curve parametrically generated the surfaces. In parametric sweeping
procedure, a surface is generated through the movement of a line or a curve along or around a defined path. The
curve is sweeped as the sweep parameter is varied from the values of 0 to 1, creating several instances of the curve
along the sweep path. In general, the equation of the surface can be given as,

11. , =
Where, P(t) is the parametric equation of a curve and T(s) is the sweep transformation based on the shape of
the path. The sweep transformation can consist of translation, scaling, rotation or a combined transformation. If
the path is a straight line, the points along the path on the line can be represented by,
= ; = ; =
and T(s) is given as,
=
[ ]
Where, a, b, c are coordinate values, and ≤ s ≤
This is equivalent to a three-dimensional translation of a curve with several traces generated along the path,
controlled by how the parameter s is varied.
Example: Consider the Bezier curve defined by the control points P1 = (0,5,0), P2 = (3,4,0), P3 = (2,0,0), and P4 =
(5,0,0). Translate the curve five units along the z-axis to generate a swept surface.
Solution:
, = [ ][ ]
substituting the numbers, we get,
, = [ ]
[
− −
−
−
] [ ] [ ]
Substituting the value of s and solving the matrices can calculate any point on the surface.
9. Creating a Surface by Sweeping a Polygon
Any polygon can be sweeped around a given path to generate a surface. The equation of the surface is given as,
, = [ ][ ]
Where, [P] is the point matrix, and T(s) is the transformation matrix.
Example: Sweep (rotate) the triangle A (2, 2), B (5, 7), C (-2,-5) around x-axis and generate the surface
Solution:
, = [ ][ ] = [
− −
]
[
� � �
− � � �
]
Note: The value of n locates various positions on the swept surface.
10. Creating a Parametric Cubic Patch
Parametric cubic patch or surface is generated by four boundary curves; the curves are parametric cubic
polynomials. The equation of a parametric cubic curve was defined earlier as:

12. = [ ]
[
−
− − −
] [
′
′
]
[
−
− − −
]
= � � = a�d
[
′
′
]
= � � �
Where P(0) = Coordinates of the first point at t = 0
P(1) = coordinates of the last point at t = 1
P’ = values of the slopes in x, y, z directions at t = 0
P’ = values of the slopes in x, y, z directions at t = 1
Analogous to a cubic curve, a parametric cubic surface can be defined by 16 points:
- 4 points for coordinates of the corner points
- 8 points for slopes in the s & t directions
- 4 points for twist vectors (second derivatives)
Using a procedure similar to the one carried out in the derivation of the cubic curve, we can derive the geometric
coefficient matrix for the surface, which is given as,
Which can be broken into 4 groups as
Twist vectors, not shown here, are the partial derivatives: dPs/dt & dPt/ds. These vectors control the internal
shape of the surface. With the geometric coefficient matrix defined, the equation of the surface can be written as,
. = [ ][�] [ ] [� ] [ ]
Where: [s] = [s3 s2 s1]
[M]H = [Constant matrix for n = 3 ]
[MH]T = Transpose of [M]H
[G]H = Geometry matrix as defined by the 16 points, and

13. [ ] =
[ ]
Example: A parametric cubic surface is defined by its Cartesian components as follows:
, = [ ]
[ − ] [ ]
, = [ ]
[ ] [ ]
, = [ ]
[ ] [ ]
Obtain the normal vector at the point where s = ½, t = ½
Solution:
� , = [ ][�]�[�]�[��] [ ] = [ , , , ]
� , = [ ][�]�[ ]
����� [ ] = [�] [ ] [� ]
N���al ��ct��, =
����� =
�
; =
�
, = [ ]
[ − ] [ ]
, = [ ]
[ − ] [ ]
, = [ ]
[ − ] [ ]
at s = 0.5 & t = 0.5

14. , = . ; , = .
similarly,
, = . ; , = . ; , = . ; , = .
� , = [ . . . ] ; � , = [ . . . ]
= � . , . × � . , . = [ . . .
. . .
] = − . − . + .
11. Bezier Surface
Just as parametric cubic curves are extended to parametric cubic patches, Bezier Curves may be extended to
Bezier Surface Patch. While the surface passes through the four corner points, the control points control all other
points on the surface. Using the placement of these points to specify edge slope is more intuitive than determining
the parametric slopes and twist vectors for the parametric cubic curve surface.
Bezier Surface, as a result, is easier to use because the control points themselves approximate the location of the
desired surface. Bezier surfaces can be generated with any order of the Bezier curve. Two surface patches can be
joined and the two surfaces do not have to be of the same order, one can be cubic and the other a quadratic.
Blending Bezier Patches with slope continuity requires that (1) control points on the common edges be shared
and (2) three control points – one on the edge and ones on the either sides of the edge – form a straight line, as
shown in the figure below.
Figure: Two blended Bezier patches. Control points P41, P42, P43 and P44 are shared by both patches. Slope
continuity between the two patches is maintained by having each group of three control points which cross the
shared edge (P31, P41, P51 etc.) lie on straight line
In Bezier Surface:
 The surface takes the general shape of the control points.
 The surface is contained within the convex hull of the control points.
 The corner of the surface and the corner control points are coincident.
General Equation of the Bezier surface is given as,
, = � � � , , ,
≤ s, t ≤
Vi,j defines the control points
Bi,n(s) & Bj,m(t) are the Bernstein blending functions in the s and t directions.
In matrix form, the Bezier surface can be represented by,

15. , = [ ][�] [�] [� ] [ ]
For a cubic surface this equation reduces to:
, = [ ]
[
− −
− −
]
[
, , , ,
, , , ,
, , , ,
, , , , ]
×
[
− −
− −
] [ ]
Note that, to represent a cubic Bezier surface, 16 control points must be specified, and several Bezier surfaces can
be combined to create a complex surface.
Geometric Modeling Techniques
Computer aided design and drafting (CADD) is a powerful technique to create the drawings. Traditionally, the
components and assemblies are represented in drawings with the help of elevation, plan, and end views and cross
sectional views. In the early stages of development of CADD, several software packages were developed to create
such drawings using computers. Figure shows four views (plan, elevation, end view and isometric view) of a part.
Since any entity in this type of representation requires only two co-ordinates (X and Y) such software packages
were called two-dimensional (2-D) drafting packages. With the evolution of CAD, most of these packages have been
upgraded to enable 3-D representation.
Geometric Modeling
Computer representation of the geometry of a component using software is called a Geometric Model. Geometric
modeling is done in three principal ways. They are:
i. Wire Frame Modeling
ii. Surface Modeling
iii. Solid Modeling
These modeling methods have distinct features and applications.
(i)Wire Frame Modeling
In Wire Frame Modeling the object is represented by its edges. In the initial stages of CAD, wire frame models
were in 2-D. Subsequently 3-D wire frame modeling software was introduced. The wire frame model of a box is
shown in Fig. (a). The object appears as if it is made out of thin wires. Fig. (b), (c) and (d) show three objects which
can have the same wire frame model of the box. Thus in the case of complex parts wire frame models can be
confusing. Some clarity can be obtained through hidden line elimination. Though this type of modeling may not
provide unambiguous understanding of the object, this has been the method traditionally used in the 2-D
representation of the object, where orthographic views like plan, elevation, end view etc are used to describe the
object graphically.

16. Fig. Ambiguity in Wire Frame Modeling
A comparison between 2-D and 3-D models is given below:
2 - D Models 3-D Wire Frame Models
Ends (vertices) of lines are represented by their X and Y
coordinates.
Ends of lines are represented by their X, Y and Z
coordinates.
Curved edges are represented by circles, ellipses,
splines etc. Additional views and sectional views are
necessary to represent a complex object with clarity.
Curved surfaces are represented by suitably spaced
generators. Hidden line or hidden surface elimination is
a must to interpret complex components correctly.
3-D image reconstruction is tedious. 2-D views as well as various pictorial views can be
generated easily.
Uses only one global coordinate system May require the use of several user coordinate systems
to create features on different faces of the component.
(ii) Surface Modeling
In this approach, a component is represented by its surfaces which in turn are represented by their vertices and
edges. For example, eight surfaces are put together to create a box, as shown in Fig.
Fig. Surface Representation
Surface modeling has been very popular in aerospace product design and automotive design. Surface modeling has
been particularly useful in the development of manufacturing codes for automobile panels and the complex doubly
curved shapes of aerospace structures and dies and moulds.
Apart from standard surface types available for surface modeling (box, pyramid, wedge, dome, sphere, cone, torus,
dish and mesh) techniques are available for interactive modeling and editing of curved surface geometry. Surfaces
can be created through an assembly of polygonal meshes or using advanced curve and surface modeling

17. techniques like B-splines or NURBS (Non-Uniform Rational B-splines). Standard primitives used in a typical
surface modeling software are shown in Fig. Tabulated surfaces, ruled surfaces and edge surfaces and revolved are
simple ways in which curved geometry could be created and edited.
Fig. Typical Approaches in Surface Modeling
(iii) Solid Modeling
The representation of solid models uses the fundamental idea that a physical object divides the 3-D Euclidean
space into two regions, one exterior and one interior, separated by the boundary of the solid. Solid models are:
• Bounded
• Homogeneously three dimensional
• Finite
There are six common representations in solid modeling.
i. Spatial Enumeration: In this simplest form of 3D volumetric raster model, a section of 3D space is described by
a matrix of evenly spaced cubic volume elements called voxels.
ii. Cell Decomposition: This is a hierarchical adaptation of spatial enumeration. 3D space is sub-divided into cells.
Cells could be of different sizes. These simple cells are glued together to describe a solid object.
iii. Boundary Representation: The solid is represented by its boundary which consists of a set of faces, a set of
edges and a set of vertices as well as their topological relations.
iv. Sweep Methods: In this technique a planar shape is moved along a curve. Translational sweep can be used to
create prismatic objects and rotational sweep could be used for axisymmetric components.
v. Primitive Instancing: This modeling scheme provides a set of possible object shapes which are described by a
set of parameters. Instances of object shape can be created by varying these parameters.
vi. Constructive Solid Geometry (CSG): Primitive instances are combined using Boolean set operations to create
complex objects. In most of the modeling packages, the approach used for modeling uses any one of the following
three techniques:

18. i. Constructive Solid Geometry (CSG or C-Rep)
ii. Boundary Representation (B-Rep)
iii. Hybrid Method which is a combination of B-Rep and CSG.
(i) Constructive Solid Geometry (CSG)
In a CSG model, physical objects are created by combining basic elementary shapes known as primitives like
blocks, cylinders, cones, pyramids and spheres. The Boolean operations like union ( ), difference (–) and
intersection ( ) are used to carry out this task. For example, let us assume that we are using two primitives, a
block and a cylinder which are located in space as shown in Fig.
A union operation will combine the two to convert them into a new solid.(Fig. (c)) The difference
operation (A – B) will create a block with a hole (Fig. (D)). An intersection operation ( ) will yield the portion
common to the two primitives.
(ii) Boundary Representation
Boundary representation is built on the concept that a physical object is enclosed by a set of faces which
themselves are closed and orientable surfaces. Fig. shows a B-rep model of an object. In this model, face is
bounded by edges and each edge is bounded by vertices. The entities which constitute a B-rep model are:
Geometric Entities Topological Entities
Point Vertex
Curve, line Edge
Surface Face
A Solid Model is a 3-D representation of an object. It is an accurate geometric description which includes not
only the external surfaces of part, but also the part’s internal structure. A solid model allows the designer to
determine information like the object’s mass properties, interferences, and internal cross sections.

19. Solid models differ from wire frame and surface models in the kind of geometric information they provide. Wire
frame models only show the edge geometry of an object. They say nothing about what is inside an object. Surface
models provide surface information, but they too lack information about an object’s internal structure. Solid
models provide complete geometric descriptions of objects.
Engineers use solid models in different ways at different stages of the design process. They can modify a design as
they develop it. Since computer-based solid models are a lot easier to change and manipulate than the physical
mock-ups or prototypes, more design iterations and modifications can be easily carried out as a part of the design
process. Using solid modeling techniques a design engineer can modify a design several times while optimizing
geometry. This means that designers can produce more finished designs in less time than by using traditional
design methods or 2-D CAD drafting tools.
Solid models can be used for quick and reliable design analysis. Solid models apart from geometric information
provide important data such as volume, mass, mass properties and centre of gravity. The designer can also
export models created to other applications for finite element analysis (FEA), rapid prototyping and other
special engineering applications.
Finally designers can generate detailed production drawings directly from the solid model. This capability
increases design productivity considerably. Another important feature of solid modeling is associativity. Detailed
drawings are linked to solid model through the associativity feature. This is a powerful function - as an engineer
modifies a design, the drawings get updated automatically. In bidirectional associativity, any modifications
made to geometry in the drawing are reflected in the model. In more advanced design and manufacturing
environments, solid models are used for rapid prototyping and automated manufacturing applications.
Salient Features of Solid Modeling
(i) Feature-Based Design:
The most fundamental aspect in creating a solid model is the concept of feature-based design. In typical 2-D CAD
applications, a designer draws a part by adding basic geometric elements such as lines, arcs, circles and splines.

20. Then dimensions are added. In solid modeling a 3-D design is created by starting a base feature and then adding
other features, one at a time, until the accurate and complete representation of the part’s geometry is achieved.
A feature is a basic building block that describes the design, like a keyway on a shaft. Each feature indicates how to
add material (like a rib) or remove a portion of material (like a cut or a hole). Features adjust automatically
to changes in the design thereby allowing the capture of design intent. This also saves time when design changes
are made. Because features have the ability to intelligently reference other features, the changes made will
navigate through design, updating the 3-D model in all affected areas. Figure shows a ribbed structure. It consists
of feature like ribs and holes.
Fig. A Ribbed Structure Fig. Flanged Part
Similarly, if a flanged part shown in Fig. (A) is to be created, the one approach is to sketch the cross section as
shown in Fig. (B) and then revolve through 360°.
In typical solid modeling software the designer can create a feature in two basic ways. One is to sketch a section of
the shape to be added and then extrude, revolve, or sweep it to create the shape. These are called Sketched
Features. Another type of feature is the pick-and-place feature. Here the designer simply performs an engineering
operation such as placing a hole, chamfering or rounding a set of edges, or shelling out the model.
An important component of every feature is its dimensions. Dimensions are the variables that one changes in
order to make the design update automatically. When a dimension is changed the solid modeling software
recalculates the geometry. Design of a part always begins with a base feature. This is a basic shape, such as a
block or a cylinder that approximates the shape of the part one wants to design. Then by adding familiar design
features like protrusions, cuts, ribs, keyways, rounds, holes, and others the geometry of a part is created. This
process represents true design. Unlike many CAD applications in which designing means drawing a picture of the
part, working with the feature-based solid modeling method is more like sculpting designs from solid material.
(ii)Modeling Tools:
When a 3-D model is built the designer describes the features that make up a part. Parts are put together to make
an assembly, and then documentation is made.
Sketching:
The first step in creating many 3-D features is sketching a 2-D section. Then by using appropriate instructions the
design conveys the information regarding how far to extend this section in a space. The third dimension is created
this way. For instance, a 30 mm circle extruded 50 mm through space produces a cylinder 30 mm diameter and 50
mm high. The circle is the sketched section of the cylinder.

21. Creating Parts:
As mentioned earlier solid modeler uses features such as cuts, protrusions, holes, chamfers, and other basic shapes
to build part geometry. The designer gets the information about the geometry of a feature (like the size and shape
of cuts and protrusions), from a sketcher window. As the features are added the geometry of the part is enhanced.
By adding to geometry one feature at a time, parts with very complex geometry can be created as shown in Fig.
Fig. Part with Complex Geometry Fig. Exploded View of an Assembly
Many tools for modifying geometry, establishing relations between features, and defining or modifying feature
attributes are provided in solid modeling softwares. Regardless of whether simple parts or complex ones are
designed, part geometry is created and modified in the same way.
Building Assemblies:
Designs usually consist of several parts. Solid modelers can put two or more parts together in an assembly. All the
tools a designer needs to build, modify, and verify assemblies are available in solid modeling softwares.
Documenting Designs:
The final step of designing a part or assembly is communicating it in a medium other than the computer monitor’s
display. For some operations this means plotting out design drawings. Solid modelers provide tools to produce
finished drawings complete with geometric tolerancing and text annotation.
Drawings may not be the final step for everybody. Many users export designs to other applications for analysis,
manufacturing, and other forms of post-processing. Modeling software also will incorporate a variety of tools for
exporting designs to other softwares.
Characteristics of Solid Modeling Packages
Several important characteristics of solid modeler make them capable of creating designs faster. These include
parametric design, the ability to establish relations and the ability to build assemblies.
(i) Parametric Design:
When a feature is created in a solid modeler dimensions are created. These dimensions do more than show the size
of the feature. They define parameters that control part geometry. Since parameters control geometry, the
geometry is said to be dimension driven.
Parameters can be driven by dimensional values, or they can be driven by other parameters using a relation. For
instance, the length parameter of a feature is set up so that it is always twice the width parameter. If the width
changes, the length will change too.

22. There are other kinds of values that can be used as parameters. A formula that relates specific feature geometry to
volume, temperature, stress, weight, and other properties can also be used in parametric designs. When
parameters change, other parameters driven by the modification also change. This is the essence of
parametric design.
(ii) Relations and Relationships:
Solid models provide two fundamental ways to relate elements of geometry to one another within a design.
One is as already mentioned i.e. setting up relations between parameters. The length to width example just
described shows a simple application of parametric relations. Another example that gives a better idea of the
power of relations is shown in Fig. Within an assembly, the designer can define a relation such that the diameter of
a bore in one component part always equals the diameter of a shaft, plus a clearance value, on another part. This
relation ensures that the parts always fit in an assembly, even if the diameter of the shaft changes. If the design
engineer modifies the diameter of the shaft, the diameter of the bore automatically changes to accommodate it.
Fig. Parametric relation between a hole in Fig. Parent-Child Feature
One Component and a Shaft in another
Another way to relate geometry within a design is to create a parent/child relationship. There are many ways
to do this. One is to simply create a dimension parameter between a new feature and an existing one. The
new feature becomes the child of the existing parent feature.
Parent/child relationships can be very useful. If the geometry of a parent changes, the child features are updated
with it. Although parent/child relationships enhance the parametric behavior of the designs, they should only be
used appropriately. Creating parent/child relationships where there should be none can cause the design to
behave in ways the designer did not intend.
Solid Modeling is capable of combining parts into an assembly. These modelers provide all the tools that are
needed to orient, align, and mate parts. The designer can remove parts from assemblies, modify part geometry,
and establish relations between assembly components. In addition, local and global interferences can be checked.
This ability to create a fully parametric assembly makes solid modeling software a very powerful design tool. With
the aid of solid modeling a designer can build extremely complex, multi-component designs.
All designs, whether big or small, begin with an idea. As the idea takes form, so do certain aspects of the idea. What
is the purpose of the design? What are the key features and components that make the design achieve its
objectives? How do the components fit together? And what other questions are to be satisfactorily answered to
make it a viable design? These are some of the issues the designer has to answer while carrying out a design.
(iii) Behavior Modeling:
Behavior modeling is the latest development in mechanical CAD. It gives the designers more efficient and
adaptable ways of creating designs. It helps to synthesize required functional behavior, design context and

23. geometry. Through an intelligent process of knowledge capture and iterative solving behavior modeling allows
engineers to pursue highly innovative and robust designs. The process of behavior modeling involves:
(a) Smart Models: These encapsulate engineering intelligence. Designs are created using feature based
techniques, which capture geometry, specifications, design intent, and process knowledge-all at the design level.
(b) Engineering Objective Based Design: Design tools use feature based design specifications within the smart
model to drive and adapt product design. Using objective driven functions, engineers can arrive at an optimum
design, even in designs with several variables and constraints and multiple objective criteria.
(c) Open Extensible Environment: This facilitates associative bi-directional communication to any external
application like analysis, manufacturing etc. at the feature level of the design ensuring that model reflects the
changes incorporated in other applications.
Behavior Modeling helps to create designs which are more innovative, differentiable and more responsive to
customers’ requirements. With the help of objective driven design process engineers can focus on key design
issues, and evolve optimal design solutions for better performance and functionality. Behavior modeling helps to
make electronic product designs more complete.
Behavior modeling strategy advances feature-based modeling to accommodate a set of adaptive process features
that go beyond the traditional core geometric features. These features accommodate a variety of information that
further specifies the intent and performance of the design. There are two distinct categories of adaptive design.
Application features describe process information. Behavioral features contain engineering and functional
specifications. Application features encapsulate product and process information.
Behavioral features define component assembly connectivity, using welds, or pins or slider joints. When
behavioral features consist of assembly connectivity information, including any assembly constraints, the assembly
design process automatically implements that information to execute functional behavior and purpose. By
capturing original design intent, product designs retain their integrity, robustness, and performance while
adapting to market and engineering changes. The adaptive process features make smart models highly flexible.
As the engineers make changes to the smart models, the models regenerate to accommodate all their features and
context. This highly flexible adaptation makes smart models to respond to changes in their environment.
Objective driven design automatically optimizes designs to meet any number of objectives captured in the smart
model by adaptive process features. It can automatically resolve conflicting objectives. In smart models the
specifications can be used to drive the design process.
Color Consideration: Most graphics displays of realistic scenes are in color. But the illumination model we have
described so far considers only monochromatic lighting effects.
Color Models
A color model is a method for explaining the properties or behavior of color within some particular context.
Light or colors are from a narrow frequency band within the electromagnetic spectrum:
Fig: Electromagnetic Spectrum

24. Hue (Color): The dominant frequency reflected by the object.
Brightness: The perceived intensity of the light. This is related to the luminance of the source.
Purity (Saturation): How pure the color of the light appears.
Chromaticity: collectively refer to purity and hue.
Complementary Colors: eg. Red and Cyan, Green and Magenta, Blue and Yellow, which combine and form white
light.
Primary Colors: eg. R,G,B - starting from these colors, a wide range of other colors can be formed.
Shades of a Color: created by adding black pigment to the color.
Tints of a Color: created by adding white pigment to the color.
Tones of a Color: created by adding black or white pigments to the color.
RGB Model
These are used by RGB monitors which separate signals for the red, green, blue components of an image. Based on
the vision system of our eyes, we have 3 visual pigments in the cones of the retina, for red, green, and blue.
CMY Model
Useful for describing color output to hard-copy devices. These devices produce a color picture by coating a paper
with color pigments. We see the colors by reflected light, which is a subtractive process. CMYK means using the ink
of Cyan, Magenta, Yellow, and Black.
Fig: RGB Model Fig: CMY Model
Consider that,
· Magenta ink indeed subtracts the green component from incident light, so the remaining red and blue
components are seen by us, as a resultant color of magenta.

25. · Cyan ink indeed subtracts the red component from incident light, so the remaining green and blue components
are seen by us, as a resultant color of cyan.
· If we mix the ink of magenta and cyan, then, this ink subtracts the green and red component from the incident
light, and the remaining blue component is seen by us, as a resultant color of blue.
HSV Model
In this model, users select a spectral color and apply the amounts of white or black that is to be added to obtain
different shades, tints, and tones. HSV model is derived from the RGB cube. H: Hue; S: Saturation; V: Value
Saturation and value are measured as the horizontal and vertical axes. And a degree (0 - 360) describes the hue.
Complementary colors are 180 degrees apart.
HSL Model: Used by Tektronix.
H: Hue
L: Lightness
S: Saturation
Fig: HSV Model Fig: HSL Model