COmputer Aided design

COMPUTER AIDED DESIGN Unit-2 Geometric Modeling

Prepared by

Er.J.Sankar

Assistant Professor,

Dept of Mechanical Engineering,

A.R. Engineering College,

Villupuram

Representation of curves- Hermite curve- Bezier curve-

B-spline curves-rational curves-Techniques for surface

modeling – surface patch- Coons and bicubic patches-

Bezier and B-spline surfaces. Solid modeling

techniques- CSG and B-rep

A.R. ENGINEERING COLLEGE Department Of Mechanical Engineering

ME6501 Computer Aided Design Prepared by, Er.J.Sankar

Surface modelling

The technique of representation of objects (or)

components by surfaces is called surface modelling. In

this surface modeling, objects (solid objects) can be

clearly interpreted by the user (viewer). The drawback

hear is, no data is available about the difficult. This

surface modeling application is for modelling car-

bodies, ships, aerospace structure, dies etc.

The advanced surface modeling techniques like B-

splines (or) NURBS (Non-Uniform Rational B-splines) are

used for creating surfaces.

The standard surfaces for modeling include

(a) Box

(b) Pyramid wedge

(c) Sphere

(d) Dome

(e) Torus

(f) Mesh

(g) Cones



Box

Box is a three-dimensional solid specified by

length, center and corner.

Wedge

A 3D solid with a shape face tapering along the x-

axis.

Cone

A solid primitive with a circular or elliptical

base tapering symmetrically to a point (vertex)

perpendicular to its base.



Sphere

A 3D solid positioned in such a way that its

central axis is parallel to z-axis of UCS (User Co-

ordinate System)

Torus

A torus is defined by two radius values, one for

the tube and the other foe the torus center to center

of tube.

There are a lot of techniques used for surface

modeling such as B-splines or NURBS.

The techniques available for surface modeling

are,

(a) Surface patch

(b) Coons patch

(c) Bicubic patch

(d) Be’zier surfaces

(e) B-spline surfaces



Surface patch

The building blocks of the surfaces are known as

patch. Generally u and v are the two variables usually

used for representing a patch. A patch is parametric,

values ranging between 0 to 1.

For creating curves patches, one of the parametric

variable is fixed. Fig illustrate a surface patch.

Coons patch

A coons patch is a result of interpolation done on

intersecting mesh of curves, resulting in rectangular

grid of patches. The simplest of coons patch is the

linearly blended coons patch. This patch is defined by

4 boundary curves and corner points of patch. The

figure represents a coons patch.



The Bicubic patch

In order to define the surface using point and

tangent, bicubic patch is referred. The general form

of bicubic patch is

�� (𝑎𝑏) = ∑∑𝐾𝑖𝑗 𝑎𝑖𝑏𝑗

3

𝑗=𝑜

3

𝑖=0

A bicubic patch represented in the following figure.

Bezier surface

A Bezier surface is given by

�� (𝑎, 𝑏) = ∑∑𝐵𝑖,𝑚 (𝑎)𝐵𝑗,𝑛

𝑃𝑖,𝑗 𝑎, 𝑏, 𝜀 [0,1]

𝑛

𝑗=𝑜

𝑚

𝑖=0

𝑃𝑖,𝑗 → Vertices of polygon

𝐵𝑖,𝑗 (𝑎𝑛𝑑)𝐵𝑗,𝑛

→ Blending functions

This Bezier surfaces and polygons are represented

in the fig. in Bezier curves, it is possible to convert

the straight surfaces to smooth surface between

patches,



B-splines

In this surface, a characteristic polygon passes

through center points of polygon with its edges

tangential to the polygon edge. This is illustrated in

fig (a). There is another case, where the central

polygon is closed feature this concept of closed

control polygon is illustrated in fig (b). The general

form for the B-spline surface is:

�� (𝑎, 𝑏) = ∑∑𝑁𝑖,𝑘 (𝑎)𝑁𝑗,𝑖 (𝑏)𝑃𝑖,𝑗 𝑤ℎ𝑒𝑟𝑒,

𝑛

𝑗=𝑜

𝑚

𝑖=0

𝑃𝑖,𝑗 → 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠 𝑜𝑓 𝑝𝑜𝑙𝑦𝑔𝑜𝑛

𝑁𝑖,𝑘 (𝑏)𝑁𝑗,𝑖 → 𝑏𝑙𝑒𝑛𝑑𝑖𝑛𝑔 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠

Solid modeling

Solid modeling is one of the most effective

modelling method. In this approach models are displayed

as solids to viewer, thereby eliminating any chance of

misinterpretation. In solid modelling, to make the

object more realistic, Xn of the object are cut and

different colours are given. Eg. Colour pages in

magazine and animation in pictures.

Solid modeling also allows mass property

determination and helps in interferences checking of

moving parts. Alternate concepts for parts and

assemblies can be developed using solid modelling.



However solid modeling required a great deal of

computational work and it requires.

The various methods by which solid models are

represented are;

1. Half-space method

2. Boundary representation method

3. Constructive solid geometry (CSG)

4. Analytical solid modelling (ASM)

5. Primitive instancing

6. Sweep representation

Boundary representation method (B-rep):

This method is concerned about drawing an outline or

boundary of the object on screen with the help of light

pen. 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 boundary

of an object includes faces, vertices and edges.

The data base for boundary model has topology and

geometry. Euler operations are utilized for creating

topology. Geometry is created by Euclidean

calculations. However, Boolean operations are employed

for manipulating and editing the model, computation of

the volumetric properties is possible in B-rep.

The primitives of B-rep includes the following:

(a) Edge

(b) Vertex

(c) Face

(d) Loop

(e) Body (shell)



The advantages of B-rep are:

(a) This method is very powerful for creating complex

shapes solid models

(b) B-rep model can easily converted into wire frame

model system

(c) B-rep system stores an explicit definition of the

model boundaries

(d) B-rep system is very much compatible with the

other systems

Disadvantages of B-rep are

(a) This requires more storage space

(b) This concept cannot be applied for tool path

generation

Constructive solid geometry (CSG)

This method is known as C-rep. in this method,

solid graphic primitives are employed for constructing

the model. The solid primitive include cubes, spheres,

cylinders, rectangular blocks and pyramids. The shapes

of primitive are depicted in fig.



This method is also known as building block

approach. In this method, Boolean operation like union,

intersection and difference are carried out to create

new objects as well as for any other complex objects.

In some implementations, it is enough to use simple

primitives combination for creating simple solids. But

however in some cases, primitives like half-spaces

(unbounded solids) may be used. But this is quite

complicated and unreliable.

Boolean operation on 2D Model



Boolean operations on 3D solids (Block and Cylinder)

Fig illustrates the use of Boolean operations on

solid models to create new solid shapes.

The advantages of CSG or C-rep are:

(a) This requires less storage space

(b) This method is advantageous in the initial

creation of solid models. Using basic Boolean

operations it is easy to construct solid models

precisely.

Disadvantages of CSG:

(a) This method involves more computational work for

creating a solid model

(b) For complicated solid geometry, this method is

not appropriate.

𝑨 ∩ 𝑩

COmputer Aided design

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