Geometric modeling

1. Geometric Modelling

2. Learning objectives
• To understand the functions and requirements of
geometric modelling in manufacturing
• To study the different types of geometric
modelling techniques
• To recognize the different types geometric
entities and its mathematical representations

3. Geometric Modelling
Geometric modelling is the representation of
physical objects on computers , allowing both
interactive and automatic analysis of design,
and the expression of design in a form
suitable for manufacturing.

4. Functions of Geometric Modelling
• Design Analysis
• Evaluation of areas and volumes
• Evaluation of inertia properties
• Interference checking and assemblies
• Analysis of tolerance build‐up in assemblies
• Analysis of kinematics – machines, robotics
• Automatic mesh generation for FEA

5. Functions of Geometric Modelling
• Drafting
• Automatic planar cross sectioning
• Automatic hidden line and surface removal
• Automatic production of shaded images
• Automatic dimensioning
• Automatic creation of exploded views for
technical illustrations

6. • Manufacturing
• Parts classification
• Process planning
• Numerical control data generation and
verification
• Robot program generation
Functions of Geometric Modelling

7. Functions of Geometric Modelling
• Production Engineering
• Bill of materials
• Material requirement
• Manufacturing resource requirement
• Scheduling
• Inspection and Quality Control
• Program generation for inspection machines
• Comparison of produced part with design

8. Requirements of Geometric Modelling

9. Geometric Models

10. Wireframe Modelling
• Advantages
• Simple in construction
• Less computer time and memory
• For simple NC tool path generation and
interference detection
• Limitations
• Lack of visual coherence
• Can’t be used for FEA

11. Surface Modelling
• Advantages
• Less ambiguous
• Support hidden line and surface removal
algorithms
• Utilized for mass property calculations,
creation of FE meshes, NC tool path
generation and checking interferences
between mating parts
• Limitations
• Higher computer time and memory
• Don’t specify the topology of objects

12. Solid Modelling
• Advantages
• Represents complete, valid and unambiguous
objects
• Contains both geometrical and topological
information
• Utilized for mass property calculations, FEA,
NC tool path generation and verification,
process planning and checking interferences
between mating parts.
• Limitations
• Higher computer time and memory

13. Wireframe Modelling
A wireframe representation is a 3‐D line drawing
of an object showing only the edges without any
side surface in between.
The image of the object, as the name applies has
the appearance of a frame constructed from thin
wires representing the edges and projected lines
and curves.

14. Wireframe Modelling
A computer representation of a wire‐frame
structure consists essentially of two types of
information:
• The first is termed metric or geometric data
which relate to the 3D coordinate positions of
the wire‐frame node points in space.
• The second is concerned with the
connectivity or topological data, which relate
pairs of points together as edges.

16. Classification of Wireframe Entities
Curve entities are divided into two categories.
1. Analytical curves
Points, lines, arcs, fillets, chamfers, and conics
(ellipses, parabolas, and hyperbolas)
2. Synthetic curves
Includes various types of spline; Hermite cubic
spline, Bezier and B‐spline

17. Curve Representation Methods
The mathematical representation of a curve can be classified
as:
1. Non‐parametric
• Explicit
• Implicit
2. Parametric

18. Non‐parametric Representation
Explicit non‐parametric equation is:
y = c1 + c2 x + c3 x2 + c4x3
‐ Unique single value of the dependent variable for
each value of the independent variable.
Implicit non‐parametric equation is:
(x – xc)2 + (y – yc)2 = r2
‐ No distinction is made between the dependent and
the independent variables.

19. Parametric Representation
In parametric representation, each point on a curve is
expressed as a function of a parameter u. The parameter
acts as a local coordinate for points on the curve.
For 3D Curve
TT
uzuyuxzyxuP )]()()([][)( 
maxmin uuu 

20. Type Form Example Description
1. Explicit Line
2. Implicit Circle
3. Parametric ;
Line
Circle

21. Parametric Representation of Analytic Curves
)(
10)(
)(
10)(
121
121
121
121
zzuzz
uyyuyy
xxuxx
uPPuPP




Vector form
The above equation defines a line bounded by the endpoints P1 and P2 whose
associated parametric value are 0 and 1
. LINE1
Scalar form

22. Parametric Representation of Analytic Curves
. CIRCLE2
• The basic parametric equation of a circle can be written as
c
c
c
zz
uuRyy
uRxx



20sin
cos

23. Parametric Representation of Analytic Curves
• Circular arcs are considered a special case of circles. A
circular arc parametric equation is given as
c
esc
c
zz
uuuuRyy
uRxx



sin
cos
. CIRCULAR ARCS2
Where us and ue are the starting and ending angles of
the arc respectively

24. 1. Determine the parametric representation of line segment
between the position vectors P1 [1 1] and P2 [4 5]. What are the
slope and tangent vector for this line?
A parametric representation is
P(u) = P1 + (P2‐P1)u = [ 1 1] + ([4 5]‐[1 1])u
P(u) = [1 1] + [ 3 4]u
Parametric representation of x and y components are
x(u) = x1 + (x2‐x1) u = 1 + (4‐1) u = 1 + 3u
y(u) = y1+(y2‐y1)u = 1+(5‐1) u = 1+4u
Tangent vector is obtained by differentiating P(u)
P’(u) = [x’(u) Y’(u)] = [ 3 4] (or)
3i + 4j where i,j are unit vectors in the x,y directions
Slope (dy/dx)= y’/x’ = 4/3