Computer Aided Geometric Design

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• Bernstein Polynomials, Bézier Curves, de
  Casteljaus Algorithm
• Shenqiang Wu

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Content

• 1. Motivation
• 2. Problems of Polynom Interpolation
• 3. Bézier Curves
• 3.1 Bernstein Polynomials
• 3.2 Definition of Bézier Curves
• 3.3 Evaluation
• 4. Summary

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Motivation (1/2)

• Target better control over the curves
  shape
• Background Computer-supported automobile and
  aircraft design
• Bézier (Renault) and de Casteljau (Citröen) both
  developed independent from each other around
  1960/65 descriptions of curves with the following
  attributes
• Substitutes of pattern drawings by CAD
• Flexible manipulation of curves with guaranteed
  and controllable shape of the resulting curve
• Introduction of control points that not
  necessarily lie itself on the curve

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Motivation (2/2)

• Typical applications are
• Car design, aircraft design, and ship design
• Simulation of movements
• Animations, movie industry and computer graphics

Modelling of objects with free-form-surfaces
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Problems of Polynom Interpolation (1/2)

• Polynom interpolation is an easy and unique
  method for describing curves that also contain
  some nice geometrical attributes.
• Polynom interpolation is not the method of choice
  within CAD applications due to better curve
  descriptions (as we will see later).
• Reason polynom interpolation may oscillate

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Problems of Polynom Interpolation (2/2)

• Problems
• The polynomial interpolant may oscillate even
  when normal data points and paramter values are
  used.
• The polynomial interpolant is not shape
  preserving. This has nothing to do with numerical
  effects, its due to the interpolation process.
• Too high costs for interpolation process huge
  amount of necessary operations for constructing
  and evaluating the interpolant.

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Bernstein Polynomials (1/2)

• Preliminaries Bernstein polynomials
• Def. A Bernstein polynomial of grade n has the
  following description

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Bernstein Polynomials (2/2)

• Attributes of Bernstein polynomials

• i-times null in t0, (n-i)-times null in t1

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Basis functions of Bernstein Polynomials
Bernstein-Polynome vom Grad 4
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Bézier Curves (1/2)

• Def. The following curve
• is called Bézier curve of grade n
• with control points b0,,bn

The complete form of a Bézier polynomial of grade
3, for example, with control points b0,,bn
looks as follows
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Bézier Curves (2/2)
Different Bézier Curves with its control polygons
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Attributes of Bézier Curves (1/9)

• Attributes of Bézier curves

• x(0)b0 and x(1)bn, that means the Bézier curve
  lies on b0 and bn.

• x(0)n(b1-b0) and x(1)n(bn-bn-1) (tangents in
  start and end point)

• Values x(t) are a convex combination of the
  control points

• The Bézier curve entirely lies in its control
  polyeder or control polygon

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Attributes of Bézier curves (2/9)

• Bézier curves are invariant under projections

• Bézier curves are symmetric within their control
  points

• Are all Bézier points collinear the Bézier curve
  becomes a line

• Bézier curves are shape preserving non negative
  (monoton, convex) data leads to a non negative
  (monoton, convex) curve

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Attributes of Bézier Curves (3/9)
Endpoint interpolation and attributes of
tangents A Bézier curve interpolates the first
and the last point of its control polygon and has
the first and last line element of its control
polygon as tangent.
b0
bn
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Attributes of Bézier Curves (4/9)
Convex hull property A Bézier curve lies within
the convex hull of its control polygon.
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Attributes of Bézier Curves (5/9)
Variation diminishing property Given Bézier
curve, any kind of line or plane A Bézier curve
doesnt change the sides of any line or plane not
more often as its control polygon.
Sample lines
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Attributes of Bézier Curves (6/9)
Linear precision Are the control points
b0,...,bn of a Bézier curve collinear the Bézier
curve itself becomes a line.
bn
b0
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Attributes of Bézier Curves (7/9)
Subdivision Given is a Bézier curve with its
control polygon (b0,...,bn) resp.
0,1. Sometimes its necessary to cut a single
Bézier curve into two parts, both together being
identically to the originating curve. 1. The
subdivision algorithm from de Casteljau leads to
the control polygons (c0,...,cn) and (d0,...,dn)
of the Bézier curves within the intervals 0,t
and t,1, resp.
b2
b1
b3
b0
Example n3
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Attributes of Bézier Curves (8/9)
b2
b1
Subdivision Given is a Bézier curve with its
control polygon (b0,...,bn) 2. Successively
subdivision with de Casteljaus algorithm leads
to a series of polygons fast converging to the
curve.
b3
b0
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Attributes of Bézier Curves (9/9)
Subdivision Given is a Bézier curve with its
control polygon (b0,...,bn) 3. Cutting off edges
doesnt lead to further changes of sides. ?
Variation diminishing property
b2
b1
b3
b0
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Increase of Grade of Bézier curves (1/2)

• Problem After a Bézier polygon has been modified
  several times, it can be seen that the curve of
  grade n is not flexible enough to represent the
  desired shape.
• Idea Add one edge without changing the current
  shape of the curve.
• Solution Increase the grade of the Bézier curve
  from n to n1, thus, the new Bézier points Bk
  can be determined from the old Bézier points bi
  as follows

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Increase of Grade of Bézier Curves (2/2)
Increase of grade both polygons describe the
same (cubic) curve

• Application
• Design of surfaces
• Data exchange between different CAD and graphic
  systems

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Evaluation of Bézier Curves

• Method for determination of single curve points,
  i.e. determination of x(t) for some t

• Recursive calculation of Bernstein polynomials

• de Casteljaus algorithm

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Recursive Calculation
Recursive calculation of Bernstein polynomials
According to this definition Bézier curves are
calculated with the help of Bernstein polynomials.
Example of a cubic Bézier curve
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de Casteljaus Algorithm (1/2)
b1
b2
b3
0
1
t
b0

• Geometric construction according to de
  Casteljaus algorithm for n3 and t2/3

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de Casteljaus Algorithm (2/2)

• de Casteljaus algorithm
• i0,,n It can be described with the
• following scheme
• k1,,n
• ik,,n
• This leads to

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Example De Casteljaus Algorithm (1/2)

• Given Bézier curve of grade 4
• With Bézier points
• Wanted
• for

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Example de Casteljaus Algorithm (2/2)

• de Casteljau scheme for the
• x-component
• 1
• 0 0.4
• 1 0.6 0.52
• 6 4.0 2.64 1.8
• 7.5 6.9 5.74 4.5 3.42 x(t0.6)

• de Casteljau scheme for the
• y-component
• 0
• 2 1.2
• 5.5 4.1 2.9
• 5.5 5.5 4.9 4.14
• 0.5 6.7 3.6 4.9 4.174 y(t0.6)

Resultat X(t0.6)(x,y)(3.42,4.174)
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Rating of Bézier Curves (1/2)

• Rating of Bézier curves according to
  controlability and locality

Local changes of control points have global
effects, but their influence is only of local
interest
The change is only significant
within the scope of the control point .
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Rating of Bézier Curves (2/2)
Problems

• Double points are possible, i.e. the
  projection is not bijetive

• Complex shapes of the desired curves may result
  in a huge amount of control points that again
  leads to a high ploynom grade.

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Further Freeform Curves
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Freeform Surfaces
Bézier surface
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End