1. IPE-409 CAD/CAM
Dr. Nafis Ahmad
Professor
Department of IPE, BUET
Email: nafis@ipe.buet.ac.bd
Sept-2020
9/21/2020

2. Ch#3.
Techniques for Geometric Modelling
2

3. Techniques for Geometric Modelling
• Representation of curves
– Parametric representation of geometry
– Parametric cubic polynomial curves
– Bézier curves
– Multi-variable curve fitting
– Cubic spline curves
– Rational curves
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4. Techniques for Geometric Modelling
• Techniques for surface modelling
– Surface patch
– The Coons patch
– Bicubic patch
– Bézier surafces
– B-Spline surface
• Techniques for volume modelling
– Boundary models
– Constructive solid geometry
– Other modelling techniques
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5. Representation of curves
• Mathematically straightforward geometries are
curves and their representations are most
complete. Surfaces are extension of curves
• Why we need alternative geometric
representation to classical ones?
y=mx + c ..............................................1
ax+by+c=0 ..........................................2
ax.x+by.y+2kxy+2fx+2gy+d=0 ...........3
Problems??
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6. Cont..
y=mx + c .............1
ax+by+c=0 ...........2
ax.x+by.y+2kxy+2fx+2gy+d=0 ..............3
-Value of m (-infinity to +infinity)
-Unbounded geometry,
-Multi-valued
-Sequence of points not available
-Equation changes with coordinate system
-Other factors: difficulties in faired shapes representation,
intersections between solid or surfaces
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7. 9/21/2020

8. Cont..
Aero foil and Intersection of two cylinders
So, what to do?
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9. 9/21/2020

10. Parametric Representation of
geometry
The parametric representation of geometry
essentially involves expressing relationships for the x,
y and z coordinates of points on a curve or surface or
a solid not in terms of each other but of one or more
independent variables known as parameters.
– For curve a single parameter is used: x, y and z are
express in terms of a single variable typically u
– For surface two parameters u and v
– For solid three parameters u, v and w
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11. Cont..
Position of any point on a space curve can be expressed as
p = p(u), which is same as x=x(u),
y= y(u),
z=z(u)
Similarly position of any point on a surface or solid can
also be expressed by two (u,v) and three (u, v, w)
independent parameters respectively.
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12. 9/21/2020

13. Parametric cubic polynomial curves
• Two points define a line, three points define a
circle. So, four points can be used to define a cubic
polynomial curve. To find 12 unknown, we need
three equations and four points’ coordinates.
• Vector form
p= p(u) = k0 + k1u1 + k2u2 + k3u3 or
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14. Cubic polynomial curves-Hermite form
• Hermite cubic form: we can also find 12 unknowns
if we know two points and two slopes at the two
ends of the curve.
– ?? (vector form and matrix form)
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15. Cubic polynomial curves-Hermite form
Hermite cubic form: two
points and two slopes at
the two ends of the curve
are known.
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16. Cubic polynomial curves-Hermite form
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17. Cubic polynomial curves-Hermite form
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18. Cubic polynomial curves-Hermite form
Change value of u from 0 to 1 with an increment of 0.1 and plot the curve in excel,
and submit online by tomorrow.
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19. Cubic polynomial curves-Bezier form
• How Bezier cubic form is
different from other
forms.
• Uses four control points
for a 3-degree curve.
• Relation between
tangent vectors at two
end points and other
points is shown below.
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20. Cubic polynomial curves-Bezier form
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21. Cubic polynomial curves-Bezier form
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22. Basis functions Hermite vs. Bezier form
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23. Parametric cubic polynomial curves
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24. Parametric cubic polynomial curves-
Bezier form
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25. Parametric cubic polynomial curves-
Bezier form
Curve for different number of control points or degree (no. of points -1)
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26. Parametric cubic polynomial curves-
Bezier form
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27. Parametric cubic polynomial curves-
Bezier form
P0=(1,1), P1=(3,6), P2=(5,7) and P3=(7,2)
P(u=0.5)=?
P(u=0.5)=(4, 5.25)
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28. Parametric cubic polynomial curves
Sample questions:
 Derive the equation of cubic polynomial of Harmite
and Bezier form.
 Draw and note the differences between blending
functions of Hermite and Bezier form.
 What are the advantages of Bezier curve over
Hermit curve
 Examples of Hermit and Bezier forms of cubic
polynomial curves.
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29. Parametric cubic polynomial curves-
Important considerations
• Local modification vs Global modification
• Modelling faired shapes found in aircraft and ship
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30. Parametric cubic polynomial curves-
Important considerations
• Degree of continuity
• Curve with first degree of continuity C(1) and second degree of
continuity C(2) etc.
• Parametric continuity (C) and Geometric continuity (G)
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31. Parametric cubic polynomial curves
• Degree of continuity
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32. Parametric cubic polynomial curves
• Cubic spline curve/Composite curve: a series of curve(cubic) segments are
jointed end to end.
• Use knot point and boundary conditions continuity at first(tangent) and
second(curvature) derivatives at intermediate points.
• Global modification is a problem, though damped at the remote points.
• For n control point or knot points, (n-1) spans, 4(n-1) co-efficient vectors
• Number of point boundary condition is 2(n-1), plus (n-2) slope conditions
and (n-2) curvature conditions
• Remaining conditions- slope/tangent vectors at two ends.
• Accumulated cord length is used.
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33. Parametric cubic polynomial curves
• B-Spline curve: Local modification and degree of the curve are the
reasons
• Use blending function to combine influence of a series of control
points.
• In Bezier curve degree depends on control points, but here degree is
independent of the number of control points.
• In Bezier curve blending function is non-zero in entire range of u. In B-
Spline curve they may be non-zero for a limited range.
• For a series of n+1 points Pi is:
• Blending functions of order 2(linear), 3(
quadratic) and 4(cubic)
• How these functions define the curve. Curve
do not pass through two end points
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34. Parametric cubic polynomial curves
• Cubic blending functions to approximate eight track points and
localized effect.
• Localized effect can be changed by changing polynomial order
(order reduced localized effect increase) and repeating points.
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35. Parametric cubic polynomial curves
• Non-uniform Rational: Though integer knots
are commonly used, knots can be of arbitrary
ascending numerical value as shown below.
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36. Parametric cubic polynomial curves
• Besides representation of free form curves and data, most of
the engineering design uses standard analytic shapes like arcs,
cylinders, cones, lines and planes.
• To generalize further or include all kind of geometric entities,
Non-uniform Rational B-Spline (NURBS) representation is used.
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37. Surface Modelling
• General form of surface modelling
• Surface patch
• The Coons patch/sculptured surface
• Bicubic patch- extension on cubic spline curve.
Use points and tangent(tensor)
• Bezier surface-extension of Bezier curve
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38. Surface Modelling
• General form of surface modelling: Modelling
of free form surface is the extensions into the
second dimension of polynomial curve
techniques.
• Position of any point on a surface or solid can
also be expressed by two (u,v)
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39. Surface Modelling
• Linearly blended patch.
• A Coons patch, is a type of manifold parametrization to smoothly join
other surfaces together, in finite element method and boundary
element method, to mesh problem domains into elements.
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40. Surface Modelling
General form of surface modelling
• Bezier Surfaces
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41. Surface Modelling
General form of surface modelling
• B-Spline surface
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42. Volume Modelling
• Boundary models/Representation/Graph
based model
• CSG
• Other modelling techniques
– Pure primitive instancing (for part family,
geometrically, topologically similar not
dimensionally similar)
– Cell decomposition-use in FEA
– Special occupancy enumeration
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43. Volume Modelling
• Boundary models/Representation/Graph based model
– Face should be bounded by a single ring or loop
– Each edge should adjoin exactly two faces and have a vertex at each end
– At least three edges should meet at each vertex
– Euler’s rule should apply V – E + F = 2
– Euler-Poincare formula should apply V – E + F – H + 2P = 2B
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44. Volume Modelling
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45. Volume Modelling
– Euler-Poincare formula should apply V – E + F – H + 2P = 2B
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46. Volume Modelling
• CSG
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47. Volume Modelling
• Other modelling techniques
– Pure primitive instancing (for part family, geometrically,
topologically similar not dimensionally similar)
– Cell decomposition-use in FEA
– Special occupancy enumeration
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48. Volume Modelling
Other modelling
techniques
• Pure primitive
instancing (for
part family,
geometrically,
topologically
similar not
dimensionally
similar)
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49. Thank You
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50. Important Links
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Ref: Approximation and Modeling with B-Splines
https://archive.siam.org/books/ot132/ot132-program-collection-index.php