1. 1 Computer Aided Geometric Design Ferienakademie 2004 Bernstein Polynomials, Bézier Curves, de Casteljau‘s Algorithm Shenqiang Wu

2. 2 Computer Aided Geometric Design Ferienakademie 2004 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

3. 3 Computer Aided Geometric Design Ferienakademie 2004 Motivation (1/2) Target : better control over the curve’s 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

4. 4 Computer Aided Geometric Design Ferienakademie 2004 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

5. 5 Computer Aided Geometric Design Ferienakademie 2004 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 Reason: polynom interpolation may oscillate

6. 6 Computer Aided Geometric Design Ferienakademie 2004 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, it‘s due to the interpolation process.  Too high costs for interpolation process: huge amount of necessary operations for constructing and evaluating the interpolant.

7. 7 Computer Aided Geometric Design Ferienakademie 2004 Bernstein Polynomials (1/2)  Preliminaries: Bernstein polynomials Def.: A Bernstein polynomial of grade n has the following description  Method of approximation: Bézier polynomials with binomial coefficients

8. 8 Computer Aided Geometric Design Ferienakademie 2004 Bernstein Polynomials (2/2) Attributes of Bernstein polynomials:  i-times null in t=0, (n-i)-times null in t=1  Proof:   

9. 9 Computer Aided Geometric Design Ferienakademie 2004 Basis functions of Bernstein Polynomials Bernstein-Polynome vom Grad 4

10. 10 Computer Aided Geometric Design Ferienakademie 2004 B é zier Curves (1/2) Def.: The following curve is called B é zier curve of grade n with control points b 0, …,b n b1b1b1b1 b2b2b2b2 b3b3b3b3 b4b4b4b4 Bézier curve Control polygon The complete form of a Bézier polynomial of grade 3, for example, with control points b 0,…,b n looks as follows:

11. 11 Computer Aided Geometric Design Ferienakademie 2004 Bézier Curves (2/2) Different Bézier Curves with its control polygons

12. 12 Computer Aided Geometric Design Ferienakademie 2004 Attributes of B é zier Curves (1/9) Attributes of B é zier curves:  x(0)=b 0 and x(1)=b n, that means the Bézier curve lies on b 0 and b n.  Values x(t) are a convex combination of the control points  The Bézier curve entirely lies in its control polyeder or control polygon  x‘(0)=n(b 1 -b 0 ) and x‘(1)=n(b n -b n-1 ) (tangents in start and end point)

13. 13 Computer Aided Geometric Design Ferienakademie 2004 Attributes of B é zier 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

14. 14 Computer Aided Geometric Design Ferienakademie 2004 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. control polygon line element Bézier curve b0b0 bnbn

15. 15 Computer Aided Geometric Design Ferienakademie 2004 Attributes of Bézier Curves (4/9) Convex hull property: A Bézier curve lies within the convex hull of its control polygon.

16. 16 Computer Aided Geometric Design Ferienakademie 2004 Attributes of Bézier Curves (5/9) Variation diminishing property: Given: Bézier curve, any kind of line or plane A Bézier curve doesn’t change the sides of any line or plane not more often as its control polygon. 1 3 2 1 Sample lines 1 3 2 1 3 2

17. 17 Computer Aided Geometric Design Ferienakademie 2004 Attributes of Bézier Curves (6/9) Linear precision: Are the control points b 0,...,b n of a Bézier curve collinear the Bézier curve itself becomes a line. Control polygon Bézier curve b0b0 bnbn

18. 18 Computer Aided Geometric Design Ferienakademie 2004 c1 c2 c2 c3 c3 c0 Attributes of Bézier Curves (7/9) Subdivision: Given is a Bézier curve with its control polygon (b 0,...,b n ) resp. [0,1]. Sometimes it’s 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 (c 0,...,c n ) and (d 0,...,d n ) of the Bézier curves within the intervals [0,t] and [t,1], resp. b0b0 b1b1 b2b2 b3b3 Example: n=3 d2d2 d1d1 d0d0 d3d3

19. 19 Computer Aided Geometric Design Ferienakademie 2004 Attributes of Bézier Curves (8/9) b0b0 b1b1 b2b2 b3b3 Subdivision: Given is a Bézier curve with its control polygon (b 0,...,b n ) 2. Successively subdivision with de Casteljau’s algorithm leads to a series of polygons fast converging to the curve.

20. 20 Computer Aided Geometric Design Ferienakademie 2004 Attributes of Bézier Curves (9/9) b0b0 b1b1 b2b2 b3b3 Subdivision: Given is a Bézier curve with its control polygon (b 0,...,b n ) 3. Cutting off edges doesn’t lead to further changes of sides.  Variation diminishing property

21. 21 Computer Aided Geometric Design Ferienakademie 2004 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 n+1, thus, the new Bézier points B k can be determined from the old Bézier points b i as follows:

22. 22 Computer Aided Geometric Design Ferienakademie 2004 Increase of Grade of Bézier Curves (2/2) Application:  Design of surfaces  Data exchange between different CAD and graphic systems Increase of grade: both polygons describe the same (cubic) curve

23. 23 Computer Aided Geometric Design Ferienakademie 2004 Evaluation of B é zier Curves Method for determination of single curve points, i.e. determination of x(t) for some t:  de Casteljau‘s algorithm  Recursive calculation of Bernstein polynomials

24. 24 Computer Aided Geometric Design Ferienakademie 2004 Recursive Calculation Recursive calculation of Bernstein polynomials 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

25. 25 Computer Aided Geometric Design Ferienakademie 2004 de Casteljau‘s Algorithm (1/2) Geometric construction according to de Casteljau‘s algorithm for n=3 and t=2/3 b0b0 b1b1 b2b2 b3b3 b01b01 b11b11 b21b21 b02b02 b12b12 b03b03 01t

26. 26 Computer Aided Geometric Design Ferienakademie 2004 de Casteljau‘s Algorithm (2/2) de Casteljau ‘ s algorithm i=0, …,n:It can be described with the following scheme: k=1, …,n: i=k, …,n: This leads to

27. 27 Computer Aided Geometric Design Ferienakademie 2004 Example: De Casteljau‘s Algorithm (1/2)  Given: Bézier curve of grade 4  With Bézier points  Wanted for for

28. 28 Computer Aided Geometric Design Ferienakademie 2004 Example: de Casteljau‘s Algorithm (2/2) de Casteljau scheme for the x-component1 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(t=0.6) de Casteljau scheme for the y-component0 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(t=0.6) Resultat: X(t=0.6)=(x,y)=(3.42,4.174)

29. 29 Computer Aided Geometric Design Ferienakademie 2004 Rating of B é zier Curves (1/2) Rating of B é zier Curves (1/2) Rating of B é zier curves according to controlability and locality: Local changes of control points have global effects, but their influence is only of local interest: 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.

30. 30 Computer Aided Geometric Design Ferienakademie 2004 Rating of B é zier Curves (2/2)  Complex shapes of the desired curves may result in a huge amount of control points that again leads to a high ploynom grade. Problems:  Double points are possible, i.e. the projection is not bijetive

31. 31 Computer Aided Geometric Design Ferienakademie 2004 Further Freeform Curves B-Splines NURBS

32. 32 Computer Aided Geometric Design Ferienakademie 2004 Freeform Surfaces Bézier surface