1. Interpolation to Data Points Lizheng Lu Oct. 24, 2007

2. Problem

3. Interpolation VS. Approximation Interpolation Approximation

4. Classification Curve Constraint (piecewise) Bezier curves B-spline curves Rational Bezier/B-spline curves

5. Outline Some classical methods Some recent methods on geometric interpolation Estimate the tangent

6. C 2k-1 Hermite Interpolation Cubic Interpolation

7. C 2 Cubic B-spline Interpolation Given: A set of points and a knot sequence Find: A cubic B-spline curve, s.t.

8. Geometric Hermite Interpolation (GHI) Given: Planar points p i, with positions, tangents and curvatures Result: Piecewise cubic Bezier curves, having G 2 continuity 6th order accuracy Convexity preservation [de Boor et al., 1987]

9. Comments on GHI Independent of parameterization High accuracy But, it usually includes nonlinear problems Questions on the existence of solution and efficient implement Difficult to estimate approximation order, etc …

10. References on GHI

11. High Order Approximation of Rational Curves Given: A rational curve, where f and g are of degree M and N, let k = M+N, with parameters values Find: A polynomial p of degree at most n+k-2, and scalar values satisfying the 2n interpolation conditions: [Floater, 2006]

12. Geometric Interpolation by Planar Cubic Polynomial Curves Comp. Aided Geom. Des. 2007, 24(2): 67-78 Jernej Kozak Marjeta Krajnc FMF&IMFM IMFM Jadranska 19, Ljubljana, Slovenia

13. Problem Given: six points Find: a cubic polynomial parameter curve which satisfies

14. An Alternative Solution: Quintic Interpolating Curves Find a quintic curve s.t., where t i are chosen to be the uniform and chord length parameterization.

15. Essential of Problem Know: t 0, t 5, p 0, p 3 Unknown: t 1, t 2, t 3, t 4, p 1, p 2 Equations: P 3 (t i ) = T i, i = 2, 3, 4

16. Solution of Problem Solved by Newton Iteration with initial values: Know: t 0, t 5, p 0, p 3 Unknown: t 1, t 2, t 3, t 4, p 1, p 2 Equations: P 3 (t i ) = T i, i = 2, 3, 4

17. Existence of Solution Provide two sufficient conditions guaranteeing the existence Summarize cases in a table which does not allow a solution

18. Comparison cubic uniform chord length

19. On Geometric Interpolation by Planar Parametric Polynomial Curves Mathematics of Computation 76(260): 1981-1993

20. Problem Given: 2n points Find: a cubic polynomial parameter curve which satisfies

21. Main Results If the data, sampled from a convex smooth curve, are close enough, then equations that determine the interpolating polynomial curve are derived for general n (Theorem 4.5) if the interpolating polynomial curve exists, the approximation order is 2 n for general n (Theorem 4.6) the interpolating polynomial curve exists for n ≤ 5 (Theorem 4.7)

22. On Geometric Interpolation of Circle-like Curves Comp. Aided Geom. Des. 2007, 24(4): 241-251

23. What is Circle-like Curve? A circular arc of an arclength is defined by Suppose that a convex curve is parameterized by the same parameter as. The curve will be called circle-like, if it satisfies: (1) (2)

24. The Result

25. Outline Some classical methods Some methods on geometric interpolation Estimate the tangent

26. Tangent Estimation Methods FMill, 1974 Circle Method Bessel [Ackland, 1915] Akima, 1970 G. Albrecht, J.-P. B é car, G. Farin, D. Hansford, 2005, 2007

27. Problem ?

28. FMILL

29. Circle Method

30. Bessel Parabola f (t)

31. Bessel

32. Akima ’ s Method

33. Albrecht ’ s Method Albrecht G., B é car J.P. Univ. de Valenciennes et du Hainaut – Cambr é sis, France Farin G., Hansford D. Dep. Comp. Sci., Arizona State Univ. D é termination de tangentes par l ’ emploi de coniques d ’ approximation. On the approximation order of tangent estimators. CAGD, in press

34. Main Idea Method: Estimate the tangent by using the interpolating conic of the given five points Solution: solved by Pascal ’ s theorem in projective geometry Advantages Conic precision Less computations without computing the implicit conic

35. Idea Derivation Any conic section is uniquely determined by five distinct points in the plane, p i =(x i, y i ). [Farin, 2001]

36. Idea Derivation [Pascal, 1640]

37. Projective Geometry in CAGD Express rational forms Implicit representation of rational forms

38. Projective Geometry in CAGD Express rational forms Implicit representation of rational forms Chen, Sederberg Conic section Line conics

39. Projective Geometry

40. A line in is represented by The line joining the two points is The intersection of two lines is

41. Estimate the Tangent

43. Degenerate Cases (b) (c) (a)

44. Examples

45. Experimental results

46. Non-convex Case Conic method Akima Bessel Circle method

47. Approximation order

48. Theoretical Analysis Consider a planar curve:

49. Theoretical Analysis Consider a planar curve: Take five points:

50. Theoretical Analysis Consider a planar curve: Take five points: Let:

51. Theoretical Analysis Taylor expansion: Exact tangent: Exact norm:

52. Theoretical Analysis For a point, with the tangent: Its corresponding tangent in the projective space is:

53. Compute the Approximation Order Taylor expansion Symbolic computation: MAPLE To solve the k in:

54. Numerical Result (1)

55. Numerical Result (2)

56. Summary Obtain order four approximation for the convex case, two for the inflection point Estimate the approximation order with theoretical justification Estimate the direction of the tangent only, not the vector!

57. Thank You!