CAD

 The mathematical description of the geometry of an
object using a software is called as geometric
modeling
 There are three basic methods
 Wire – Frame Modeling
 Surface modeling
 Solid modeling

 This is one of the most popular and commonly used
method of geometric modeling.
 In construction of wire frame model, the edges of an
object are presented as lines.
 Wire frame model is used for following
representations
 2D Representation
 Orthographic views representation

2D Wire Frame Model 3D Wire Frame Model
The co-ordinate system is 2D co-ordinate
system i.e. x and y co-ordinates only
3D co-ordinate system is used for
representing objects; x, y and z
coordinates are used
3 Dimensional wire frame system
generation is difficult
Both 2D and 3D wire frame
generation is possible
Hidden lines may not complicate the
figure
Difficult for the viewer to interpret the
figure unless the hidden lines are
removed
Curved surfaces are indicated by circles,
arcs and ellipses
Curved surfaces are represented by
spaced generators.

 Bezier curve was developed by P. Bezier at French car
company “Renault Automobile Company”.
 He used these curves to design automobile bodies.
 It provides the reasonable design flexibility and avoids large
number of calculation.
𝑃 𝑢 =
𝑖=0
𝑛
𝑃𝑖 𝐵𝑖,𝑛 𝑢 , 0 ≤ 𝑢 ≤ 1
 𝐵𝑖,𝑛 𝑢 is the Bernstein function are given by
𝐵𝑖,𝑛 𝑢 = 𝐶 𝑛, 𝑖 𝑢𝑖
(1 − 𝑢)𝑛−𝑖
Where, 𝐶 (𝑛, 𝑖) =
𝑛!
𝑖! 𝑛−1 !

 A Bezier curve is defined on n+1 points 𝑃0, … , 𝑃𝑛 and is
represented as a parametric polynomial curve of degree n.
 It always passes through the first and last control points.
 The Bezier curve is tangent to first and last segments of the
characteristics polygon.
 The curve generally follows the shape of characteristics polygon.
 The degree of polynomial defining the curve segments is one less
that the number defines the polygon points.
 Bezier curve exhibit a symmetry property.
 Each control point is weighted by its blending function for each u
value.
 The curve lies entirely within the convex hull formed by four
control points

 It provide another effective method of generating curve defined
polygons.
 These curves are widely used of approximation splines.
𝑃 𝑢 =
𝑖=0
𝑛
𝑃𝑖 𝐵𝑖,𝑘 𝑢 , 0 ≤ 𝑢 ≤ 𝑢𝑚𝑎𝑥

 The local control of curve can be obtained by changing the
position of control point or using multiple control points by
placing several points at same location.
 A non-periodic B-spline curve passes through the first and last
control points and it is tangent to first and last segment of
control polygon.
 It allows us to vary the number of control points used to design a
curve without changing the degree of polynomial.
 The degree of curve increases, it is more difficult to control and
calculate accurately. Thus, a cubic B-spline curve is sufficient for
many application.

 A rational curve is defined by the algebraic ratio of two polynomials where
as non-rational curve is defined by one polynomial.
 The most widely used rational curves are non-uniform rational b-splines
(NURBS).
 A rational B – spline curve defined by
𝑃 𝑢 =
𝑖=0
𝑛
𝑃𝑖 𝐵𝑖,𝑘 𝑢 , 0 ≤ 𝑢 ≤ 𝑢𝑚𝑎𝑥
 𝐵𝑖,𝑘, 𝑢 are the rational B – spline Basis function are given by
𝐵𝑖,𝑘 𝑢 =
𝑤𝑖 𝑅𝑖,𝑘 (𝑢)
𝑖=0
𝑛
𝑤𝑖 𝑅𝑖,𝑘 𝑢

 The techniques of representation of objects (or)
components by surface is called surface modeling.
 Objects can be clearly interpreted by the user.
 Main draw back here is that, no data is available
about the interior of solid.
 Application is modeling car bodies, ships, aerospace
structure, dies, etc.

 Surface patch
 Coons patch
 Bicubic patch
 Hermite surfaces
 Bezier surfaces
 B-spline surfaces

 A surface patch is defined in terms of point data will
usually be based on a rectangular array data points.
 In computer graphics, the parametric surface are
sometimes called patches, curved surfaces or just
surface.
 The building blocks of the surfaces are known as
surface patch
 Generally u and v are two variables used for
representing a patch.
𝑃 𝑢, 𝑣 = [𝑥 𝑦 𝑧]𝑇
= [𝑥 𝑢, 𝑣 𝑦 𝑢, 𝑣 𝑧 𝑢, 𝑣 ]𝑇
𝑢𝑚𝑖𝑛 ≤ 𝑢 ≤ 𝑢𝑚𝑎𝑥𝑎𝑛𝑑 𝑣𝑚𝑖𝑛 ≤ 𝑣 ≤ 𝑣𝑚𝑎𝑥

 A linear interpolation between four bounded curve
is used to generate a coons surface, which is also
called coons patch.
 The coons formulations interpolate to an infinite
number of control points to generate the surface
and it is referred as a form of transfinite
interpolation.
𝑃 𝑢, 𝑣 = 𝑃 𝑢, 0 1 − 𝑣 + 𝑃 𝑢, 1 𝑣 + {𝑃 0, 𝑣 1 − 𝑢 + 𝑃 1, 𝑣 𝑢}

 Bicubic patch or surface is generated by four boundary curves
which are parametric Bicubic polynomials.
 Bicubic parametric patches are defined over rectangular
domain in uv-space and the boundary curves of patch are
themselves cubic polynomial curves.
 The following are the major types of parametric bi-cubic
surfaces used in CAD
 Hermite surface
 Bezier surface
 B-Spline surface

 Bezier surface is an extension of the Bezier curve in two
parametric directions u and v.
 An orderly set of data or control points is used to build a
topologically rectangular surface as shown in figure.
The surface equation can be written as
𝑃 𝑢, 𝑣 =
𝑖=0
𝑛
𝑗=0
𝑚
𝑃𝑖𝑗𝐵𝑖,𝑛 𝑢 𝐵𝑗,𝑚 𝑣 , 0 ≤ 𝑢 ≤ 1, 0 ≤ 𝑣 ≤ 1
where, P(u,v) is any point on the surface
𝑃𝑖𝑗 are the control points

 B-Spline surface is an extension of the B - Spline curve. A
rectangle set of data points creates the surface.
 A B-Spline surface can approximate or interpolate the vertices
of the polyhedron as shown in figure.
 B-Spline surface equation is defined by
𝑃 𝑢, 𝑣 =
𝑖=0
𝑛
𝑗=0
𝑚
𝑃𝑖𝑗 𝐵𝑖,𝑘 𝑢 𝐵𝑗,𝑙 𝑣 , 0 ≤ 𝑢 ≤ 1, 0 ≤ 𝑣 ≤ 1
where, P(u,v) is any point on the surface
𝑃𝑖𝑗 are the control points

 Solid modeling is one of the most effective
geometric modeling method. In this approach,
models are displayed as solids to viewer, there by
eliminating any chance of misinterpretation.
 The solid modeling is used to make the object
more realistic.

 Boundary representation Method (B-rep)
 Constructive Solid Geometry (CSG)
 Analytical Solid Geometry (ASM)
 Primitive instancing
 Sweep representation
 Spatial portioning representation
 Half – space Method

 Boundary representation is one of the most popular and widely used
schemes to create solid models of physical objects.
 In this method, front view, top view, bottom view, side view of an object
is sketched and connected by means of lines to create a relationship.
 The B-rep polyhedral objects should follows Euler equation given below
F – E + V – L = 2 (B - G)

 Edge (E)
 Vertex (V)
 Face (F)
 Loop (L)
 Genus or Handle (G)
 Body (B)

 This method is very powerful for creating complex shapes
solid models.
 B-rep model can be easily converted into wire frame model
system.
 B-rep system stores an explicit definition of the model
boundaries.
 B-rep system is very much compatible with other systems.

 This requires more storage space.
 This concept cannot be applied for tool path generation.

 This method is also known as C-rep. In this method, solid
graphic primitives are employed for constructing the model.
 The solid primitives include cubes, spheres, cylinders,
rectangle blocks and pyramids.

 The constructive solid model uses building block approach
 The physical objects can be divided into set of elements and
combined in order to form an object.

 This requires less storage space
 This method is advantageous in the initial creation of solid
models. Using Boolean operations, it is easy to construct
solid models precisely.

 This method involves more computational work for creating
a solid model
 For complicated solid geometry, in this method is not
appropriate

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