Spline curves with a shape parameter

1. Spline curves with a shape parameter Reporter: Hongguang Zhou April. 2rd, 2008

6. Problem: • To adjust the shape of curves, • To change the position of curves. • Weights in rational Bézier , B-spline curves are used.

7. Problem: • Spline has some deficiencies: • e.g. To adjust the shape of a curve, but the control polygon must be changed.

8. Motivation: • When the control polygons of splines are fixed • Can rectify the shape of curves only by adjusting the shape parameter.

9. Outline • Basis functions • Trigonometric polynomial curves with a shape parameter • Approximability • Interpolation

10. References • Quadratic trigonometric polynomial curves with a shape parameter Xuli Han (CAGD 02) • Cubic trigonometric polynomial curves with a shape parameter Xuli Han (CAGD 04) • Uniform B-Spline with Shape Parameter Wang Wentao, Wang Guozhao (Journal of computer-aided design & computer graphics 04)

11. Quadratic trigonometric polynomial curves with a shape parameter Xuli Han CAGD.(2002) 503–512

12. About the author • Department of Applied Mathematics and Applied Software, Central South University, Changsha • Subdecanal, Professor • Ph.D. in Central South University, 94 • CAGD, Mathematical Modeling

13. Previous work • Lyche, T., Winther, R., 1979. A stable recurrence relation for trigonometric B-splines. J. Approx. Theory 25, 266–279. • Lyche, T., Schumaker, L.L., 1998. Quasi-interpolants based on trigonometric splines. J. Approx. Theory 95, 280–309. • Peña, J.M., 1997. Shape preserving representations for trigonometric polynomial curves. Computer Aided Geometric Design 14,5–11. • Schoenberg, I.J., 1964. On trigonometric spline interpolation. J. Math. Mech. 13, 795–825. • Koch, P.E., 1988. Multivariate trigonometric B-splines. J. Approx. Theory 54, 162–168. • Koch, P.E., Lyche, T., Neamtu, M., Schumaker, L.L., 1995. Control curves and knot insertion for trigonometric splines. Adv. Comp. Math. 3, 405–424. • Sánchez-Reyes, J., 1998. Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials. Computer Aided Geometric Design 15, 909–923. • Walz, G., 1997a. Some identities for trigonometric B-splines with application to curve design. BIT 37, 189–201.

14. Construction of basis functions

15. Basis functions • For equidistant knots, bi(u):uniform basis functions. • Fornon-equidistant knots, bi(u):non-uniform basis functions. • For λ = 0, bi(u):linear trigonometricpolynomial basis functions.

16. Uniform basis function λ = 0 (dashed lines) , λ = 0.5 (solid lines).

17. Properties of basis functions • Has a support on the interval [ui,ui+3]: • Form a partition of unity:

18. The continuity of the basis functions • bi(u) has C1 continuity at each of the knots.

19. The case of multiple knots • knots are considered with multiplicity K=2,3 • Shrink the corresponding intervals to zero; • Drop the corresponding pieces. ui =ui+1 is a double knot

20. Geometric significanceof multiple knots • bi(u) has a knot of multiplicity k (k = 2 or 3) at a parameter value u • At u, the continuity of bi(u) : :discontinuous) • The support interval of bi(u): 3 segments to 4 − k segments • Set : −1 < λ≤1, λ≠-1

21. The case of multiple knots λ = 0 (dashed lines) , λ = 0.5 (solid lines)

22. Trigonometric polynomial curves Quadratic trigonometric polynomial curve with a shape parameter: Given:points Pi (i = 0, 1, . . .,n) in R2 or R3 and a knot vector U = (u0,u1, . . .,un+3). When u ∈ [ui,ui+1], ui≠ui+1(2 ≤ i ≤ n)

23. The continuity of curves When a knot ui:multiplicity k (k=1,2,3) the Trigonometric polynomial curves :continuity, at knot ui.

24. Open trigonometric curves Choose the knot vector: T(U2)=Po, T(Un+1)=Pn;

25. Example: Curves for λ = 0, 0.5, 1(solid lines) and the quadratic B-spline curves (dashed lines), U = (0, 0, 0, 0.5, 1.5, 2, 3, 4, 5, 5, 5).

26. Closed trigonometric curves • Extend points Pi (i=0,1,…,n) by setting: Pn+1=P0,Pn+2=P1 • Let:Un+4=Un+3+∆U2, ∆U1= ∆Un+2,Un+5≥Un+4 • bn+1(u) and bn+2(u)are given by expanding. • T(u2)=T(Un+3), T′(U2)= T′(Un+3)

27. Examples: Closed curves for λ = 0, 0.5 (solid, dashed lines on the left), λ = 0.1, 0.3 (solid, dashed lines on the right) , quadratic B-spline curves (dotted lines)

28. The representation of ellipses When the shape parameterλ = 0,u ∈ [ui,ui+1], Origin:Pi-1, unit vectors:Pi-2-Pi-1, Pi-Pi-1 T (u)is an arc of an ellipse.

29. Approximability Ti(ti) (u ∈ [ui,ui+1]) decrease of ∆ui Merged with: Ti(0)Pi−1 ,Pi−1Ti(π/2). fixed ∆ui-1, ∆ui+1 Increase λ −1 < λ≤ 1 Ti(ti) (u ∈ [ui,ui+1]) The edge of the given control polygon.

30. Examples:

31. Approximability • The associated quadratic B-spline curve: Given points Pi ∈ R2 or R3 (i = 0, 1, . . .,n)and knots u0 <u1 < ···<un+3. u ∈ [uk,uk+1]

32. Approximability The relations of the trigonometric polynomial curves and the quadratic B-spline curves:

33. Approximability

34. Conclusion of Approximability The trigonometric polynomial curvesintersect the quadratic B-spline curves at each of the knots ui (i = 2, 3, . . . , n+1)corresponding to the same control polygon. For λ ∈ (−1, (√2−1)/2], the quadratic B-spline curves are closer to the given control polygon; For λ ∈ [(√2 − 1)/2,√5 − 2], the trigonometric polynomial curves are very close to the quadratic B-spline curves; For λ = (√2 − 1)/2 and λ = √5 − 2, the trigonometric polynomial curves yield a tight envelope for the quadratic B-spline curves; For λ ∈ [√5 − 2, 1], the trigonometric polynomial curves are closer to the given control polygon.

35. Cubic trigonometric polynomial curves with a shape parameter Xuli Han CAGD.(2004) 535–548

36. Related work: • Han, X., 2002. Quadratic trigonometric polynomial curves with a shape parameter. Computer Aided Geometric Design 19,503–512. • Han, X., 2003. Piecewise quadratic trigonometric polynomial curves. Math. Comp. 72, 1369–1377.

37. Construction of basis functions

38. Construction of basis functions

39. Basis functions • For equidistant knots, Bi(u):uniform basis function,simple bi0=bi2=bi3=cio=ci1=ci3=0 • Fornon-equidistant knots, Bi(u):non-uniform basis functions. • For λ = 0, Bi(u):quadratic trigonometricpolynomial basis functions.

40. Properties of basis functions • Has a support on the interval [ui,ui+4]: • If −0.5<λ≤1, Bi(u) > 0 for ui <u<ui+4. • With a uniform knots vector, if −1 ≤λ≤1, Bi(u) > 0 for ui <u<ui+4. • Form a partition of unity:

41. The continuity of the basis functions • With a non-uniform knot vector: • bi(u) has C2 continuity at each of the knots. • With a uniform knot vector: • λ≠1,bi(u) has C3 continuity at each of the knots • λ=1, bi(u) has C5 continuity at each of the knots

42. The case of multiple knots • knots are considered with multiplicity K=2,3,4 • Shrink the corresponding intervals to zero; • Drop the corresponding pieces. ui =ui+1 is a double knot

43. Geometric significanceof multiple knots • bi(u) has a knot of multiplicity k (k = 2,3,4) at a parameter value u • At u, the continuity of bi(u): discontinuous) • The support interval of bi(u): 4 segments to 5 − k segments

44. The case of multiple knots λ= 0.5 λ= 0

45. The case of multiple knots λ= 0.5 λ= 0

46. Trigonometric polynomial curves • Cubic trigonometric polynomial curve with a shape parameter: • Given:points Pi (i = 0, 1, . . .,n) in R2 or R3 and a knot vector U = (u0,u1, . . .,un+4). When u ∈ [ui,ui+1], ui≠ui+1(3 ≤ i ≤ n)

47. Trigonometric polynomial curves • With a uniform knot vector, T(u)=(f0(t),f1(t),f2(t),f3(t)) . (Pi-3,Pi-2,Pi-1,P1)′ .(1/4λ+6) t∈[0,Π/2]

48. The continuity of the curves • With a non-uniform knot vector, ui has multiplicity k (k=1,2,3,4) • The curves have C3-k continuity at ui • The curves have G3continuity at ui, k=1 • With a uniform knot vector: • λ≠1, The curves have C3 continuity at each of the knots • λ=1, The curves have C5 continuity at each of the knots

49. Open trigonometric curves • Choose the knot vector: T(U0)= T(U3)=P0, T(Un+1)= T(Un+4)=Pn;

50. Closed trigonometric curves • Extend points Pi (i=0,1,…,n) by setting: • Pn+1=P0,Pn+2=P1,Pn+3=P2 • Let:∆Uj= ∆Un+j+1, (j=1,2,3,4) • Bn+1(u), Bn+2(u),Bn+3(u)are given by expanding.