1. Class A Curve Reporter: Cao Juan Date: 2006.12.27

2. Outline: Conclusion and future work Introduction References Application Available definition(2D,3D)

3. References:11 A shape control fitting method for Bézier curves. CAGD,1998. Yves Mineur, Tony Lichah, Jean Marie Castelain, Henri Giaume.22 Class A Bézier curves. CAGD,2006.Gerald Farin 33 Curve fitting for styling application by genetic algorithm. European Journal of Operational Research,2005. Yves Mineur. Marc Sevaux. 44 B-spline to Class-A curves adaptation with k-neighbors LOD processing. CAGD,2006.Giancarlo Amati, Alfredo Liverani.

4. Introduction: Class A surface is a term used in automotive design to describe a set of freeform surfaces of high quality. Although, strictly, it is nothing more than saying the surfaces are aesthetically pleasing, many people interpret class A surfaces have G2 curvature continuity to another one. “Class A” “ A ussenhaut”

5. Examples(1):

6. Examples(2):

7. Introduction: Class A How to make class A? Why it is needed? mathematical challenges?

8. Sectional point from clay model Feature curveDirect control Diagnosis qualitySurpport surface Completed Class A part

9. Definition: 2D (typical curve) by Yves Mineur et al CAGD (1998) 2D (typical curve) by Yves Mineur et al CAGD (1998) 3D by Gerald Farin 2003 CAGD (2006) 3D by Gerald Farin 2003 CAGD (2006)

10. Definition (2D): Bézier,1986

11. Definition (2D):

12. Definition (3D): 2D3D

13. Conditions of M(1): Positive define: Geometric view

14. Monotonity(1):

15. Conditions of M (2): Geometric view

16. Geometry view: non-class A matrix almost class A matrix

17. Subdivision:

18. Monotonity(2006): n=3

19. A query: DO all matrixes satisfy the second condition? t=0.5

20. Application(1.1):11 A shape control fitting method for Bézier curves. CAGD,1998. Yves Mineur, Tony Lichah, Jean Marie Castelain, Henri Giaume. Given a set of 2D date points (P1, …, Pn) Generate a Class-A curve joining the points P1 and Pn and pass close to the intermediate points Problem description

21. Application(1.2) Constraints (order 3,5): x y Regions for a monotonic curvature variation:

22. Application(1.3): Choose an initial values of angles Step 1 Determine a modification point and its displacement vector Step 2

23. Application(1.4): Assumption : The new point is on the starting curve approximation

24. Application(1.5): Reasoning

25. Compare with Least squares (1.1): Original curve 3rd degree curves

26. Compare with Least squares (1.2): 5rd degree curves 4rd degree curves

27. Application(2.1): 33 Curve fitting for styling application by genetic algorithm. European Journal of Operational Research,2005. Yves Mineur. Marc Sevaux. Fitting 2D points with G2 Class-A Bézier curves. Problem description

28. Genetic algorithm(2.1): GA: 1.Randomly generate an initial population M(0) 2.Compute and save the fitness u(m) for each individual m in the current population M(t) 3.Define selection probabilities p(m) for each individual m in M(t) so that p(m) is proportional to u(m) 4.Generate M(t+1) by probabilistically selecting individuals from M(t) to produce offspring via genetic operators (crossover & mutate ) 5.Repeat step 2 until satisfying solution is obtained.

29. Genetic algorithm(2.2): Underlying shape is curvature monotonic 11 22 33 Assumption: Fixed number of segments

30. Genetic algorithm(2.3):

31. Genetic algorithm (2.4)

32. Genetic algorithm (2.5) 3

33. Genetic algorithm (2.6)

34. result (2.1):

35. result (2.2):

36. result (2.3):

37. Comments: Non-sequential or uneven point set Variable segment number

38. Application(3.1): 44 B-spline to Class-A curves adaptation with k-neighbors LOD processing. CAGD,2006.Giancarlo Amati, Alfredo Liverani. lots of B-spline curves don’t hold the property of Class A, i.e., How to transform them to Class A curves as closely as possible to original curves? Problem description

39. Application(3.2) Add your text in here Multi-resolution analysis Approximation spaces: Basis functions:

40. Application(3.3): Add your text in here Multi-resolution analysis Direct sum: Multi-resolution: Minimal requirement:

41. Application(3.4): Add your text in here B-spline wavelet k order B-spline basis functions defined over: define a space of piecewise polynomials

42. Application(3.5): Add your text in here B-spline wavelet k order B-spline basis functions: B-spline wavelet:

43. Application(3.6): Add your text in here B-spline wavelet B-spline curves can be represented as: Control point relation:

44. Class-A adaptation algorithm of k-neighbors LOD methods: Setp(1) control point position scanning: extract bad control point (does not satisfy Class-A properties) at finest level Click to add Title

45. Class-A adaptation algorithm of k-neighbors LOD methods: Setp(2) Curve MRA analysis: extract finer lever details coefficients relate to the bad control point

46. Class-A adaptation algorithm of k-neighbors LOD methods: Setp(3) Recovery step and details manipulation: all details that determine the k-neighbors control points position of the bad point are thresholded. Case k = 0: may still a non-Class-A curve, increase k Case k >0:

47. Algorithm tests(1): original curve result curve

48. Error estimation:

49. Algorithm tests(3): Original Curve

50. Algorithm test(2): K=3

51. Algorithm tests(3): K=5

52. Conclusion and future work: Class-A B-spline curve (construction) 1 More flexible method of construction 2 Application (point set, non-sequence…) 3 Space Class-A (Construction and application) 4

53. Thank you!