Chapter: Mechanical : Computer Aided Design : Geometric Modeling

Geometric Modeling

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 Ci(u) and Dj(v) and the corner points pij of the patch with the linear blending functions,

 


 

 

 

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 kij 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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