Types Of Curves Used In Geometric Modelling:
• Hermite curves
• Bezeir curves
• B-spline curves
• NURBS curves

**GEOMETRIC MODELING**

**PRE-REQUISITE DISCUSSION**

**CURVE REPRESENTATION**

(1) Parametric
equation x, y, z coordinates are related by a parametric var iable (*u* or
θ)

(2) Nonparametric
equation x, y, z coordinates are related by a function

**Example: Circle (2-D)**

**TYPES OF CURVES USED IN GEOMETRIC MODELLING**

• Hermite
curves

• Bezeir
curves

• B-spline
curves

• NURBS
curves

**HERMITE CURVES**

**Effect of tangent vector on t he curve’s shape**

**BEZIER CURVE**

**Two Drawbacks of Bezier Curves**

**B-SPLINE CURVES**

**NURBS curve**

**Advantages of B-spline curves and NURBS curve**

**TECHNIQUES IN SURFACE MODELLING**

i.
Surface Patch

ii.
Coons Patch

iii.
Bicubic Patch

iv.
Be’zier Surface

v.
B-Spline Surface

**i.
****Surface Patch **

The patch is the fundamental building block for surfaces. The
two variables *u* and *v* vary across the patch; the patch may be
termed *biparametric.* The parametric variables often lie in the range 0
to 1. Fixing the value of one of the parametric variables results in a curve on
the patch in terms of the other variable (*Isoperimetric curve*). Figure
shows a surface with curves at intervals of *u* and *v* of 0 : 1.

**ii. Coons Patch**

The sculptured surface often
involve interpolation across an intersecting mesh of curves that in effect
comprise a rectangular grid of patches, each bounded by four boundary curves.
The linearly blended coons patch is the simplest for interpolating between such
boundary curves. This patch definition technique blends for four boundary
curves C* _{i}*(

**iii.
****Bicubic Patch **

The bi-cubic patch is used for surface
descriptions defined in terms of point and tangent vector information. The
general form of the expressions for a bi-cubic patch is given by:

This is a vector equation with 16 unknown
parameters *k _{ij}* which can be found by Lagrange interpolation
through 4 x 4 grid.

**iv.
****Be’zier Surface **

The Be’zier surface formulation use
a characteristic polygon

Points the Bezier surface are given
by

**v.
****B-Spline Surfaces **

The B-spline surface approximates a
characteristics polygon as shown and passes through the corner points of the
polygon, where its edges are tangential to the edges of the polygon

This may not happen when the control
polygon is closed

A control point of the surface
influences the surface only over a limited portion of the parametric space of
variables u and v.

The expression for the B-spline
surfaces is given by

**GEOMETRIC MODELLING**

Geometric modeling is the starting point of the
product design and manufacture process. Functions of Geometric Modeling are:

**Design
Analysis**

Evaluation
of area, volume, mass and inertia properties

Interference
checking in assemblies

Analysis
of tolerance build-up in assemblies

Kinematic
analysis of mechanisms and robots

Automatic
mesh generation for finite element analysis

**Drafting**

Automatic
planar cross-sectioning

Automatic
hidden lines and surface removal

Automatic
production of shaded images

Automatic
dimensioning

Automatic
creation of exploded views of assemblies

**Manufacturing**

Parts
classification

Process
planning

NC data
generation and verification

Robot
program generation

**Production
Engineering**

Bill of
materials

Material requirement

Manufacturing
resource requirement

Scheduling

**Inspection
and quality control**

Program
generation for inspection machines

Comparison
of produced parts with design

**PROPERTIES OF A GEOMETRIC MODELING SYSTEM**

The 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 Boolean operation must produce another solid The solid must
have a finite number of surfaces which can be described

The boundary of the solid must not be ambiguous

**WIRE FRAME MODELING**

It uses networks of
interconnected lines (wires) to represent the edges of the physical objects
being modeled

Also called ‘Edge-vertex’ or ‘stick-figure’ models Two types
of wire frame modeling:

1. 2 ½ - D
modeling

2. 3 – D
modeling

3-D Wire frame models: These are

Simple and easy to create, and
they require relatively little computer time and memory; however they do not
give a complete description of the part. They contain little information about
the surface and volume of the part and cannot distinguish the inside from the
outside of part surfaces. They are visually ambiguous as the model can be interpreted
in many different ways because in many wire frame models hidden lines cannot be
removed. Section property and mass calculations are impossible, since the
object has no faces attached to it. It has limited values a basis for
manufacture and analysis

2 ½ - D Wire frame models:

Two classes of shape for which a
simple wire-frame representation is often adequate are those shapes defined by
projecting a plane profile along its normal or by rotating a planar profile
about an axis. Such shapes are not two-dimensional, but neither do they require
sophisticated three-dimensional schemes for their representation. Such
representation is called 2 ½ - D.

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Mechanical : Computer Aided Design : Geometric Modeling : Geometric Modeling |

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