CLC number: TP391
On-line Access: 2024-08-27
Received: 2023-10-17
Revision Accepted: 2024-05-08
Crosschecked: 0000-00-00
Cited: 4
Clicked: 5810
PAN Yong-juan, WANG Guo-jin. Convexity-preserving interpolation of trigonometric polynomial curves with a shape parameter[J]. Journal of Zhejiang University Science A, 2007, 8(8): 1199-1209.
@article{title="Convexity-preserving interpolation of trigonometric polynomial curves with a shape parameter",
author="PAN Yong-juan, WANG Guo-jin",
journal="Journal of Zhejiang University Science A",
volume="8",
number="8",
pages="1199-1209",
year="2007",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2007.A1199"
}
%0 Journal Article
%T Convexity-preserving interpolation of trigonometric polynomial curves with a shape parameter
%A PAN Yong-juan
%A WANG Guo-jin
%J Journal of Zhejiang University SCIENCE A
%V 8
%N 8
%P 1199-1209
%@ 1673-565X
%D 2007
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.2007.A1199
TY - JOUR
T1 - Convexity-preserving interpolation of trigonometric polynomial curves with a shape parameter
A1 - PAN Yong-juan
A1 - WANG Guo-jin
J0 - Journal of Zhejiang University Science A
VL - 8
IS - 8
SP - 1199
EP - 1209
%@ 1673-565X
Y1 - 2007
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.2007.A1199
Abstract: In computer aided geometric design (CAGD), it is often needed to produce a convexity-preserving interpolating curve according to the given planar data points. However, most existing pertinent methods cannot generate convexity-preserving interpolating transcendental curves; even constructing convexity-preserving interpolating polynomial curves, it is required to solve a system of equations or recur to a complicated iterative process. The method developed in this paper overcomes the above drawbacks. The basic idea is: first to construct a kind of trigonometric polynomial curves with a shape parameter, and interpolating trigonometric polynomial parametric curves with C2 (or G1) continuity can be automatically generated without having to solve any system of equations or do any iterative computation. Then, the convexity of the constructed curves can be guaranteed by the appropriate value of the shape parameter. Performing the method is easy and fast, and the curvature distribution of the resulting interpolating curves is always well-proportioned. Several numerical examples are shown to substantiate that our algorithm is not only correct but also usable.
[1] Chang, G.Z., 1995. The Mathematics of Surfaces. Hunan Education Press, Changsha, p.15 (in Chinese).
[2] Farin, G., 2005. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide (5th Ed.). Morgan Kaufmann Publishers, San Diego, p.420.
[3] Fletcher, Y., McAllister, D.F., 1990. Automatic tension adjustment for interpolation splines. IEEE Computer Graph. Appl., 10(1):10-17.
[4] 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.
[5] Loe, K.F., 1996. α-B-spline: a linear singular blending B-spline. The Visual Computer, 12:18-25.
[6] Lyche, T., Winther, R., 1979. A stable recurrence relation for trigonometric B-splines. J. Approx. Theory, 25:266-279.
[7] Peña, J.M., 1997. Shape preserving representations for trigonometric polynomial curves. Computer Aided Geometric Design, 14(1):5-11.
[8] Schoenberg, I.J., 1964. On trigonometric spline interpolation. J. Math. Mech., 13(5):795-825.
[9] Tai, C.L., Wang, G.J., 2004. Interpolation with slackness and continuity control and convexity-preservation using singular blending. J. Comput. Appl. Math., 172(2):337-361.
[10] Walz, G., 1997a. Some identities for trigonometric B-splines with application to curve design. BIT Numer. Math., 37:189-201.
[11] Walz, G., 1997b. Trigonometric Bézier and stancu polynomials over intervals and triangles. Computer Aided Geometric Design, 14:393-397.
[12] Wang, G.Z., Chen, Q.Y., 2004. NUAT B-spline curves. Computer Aided Geometric Design, 21:193-205.
[13] Wang, G.Z., Li, Y.J., 2006. Optimal properties of the uniform algebraic trigonometric B-splines. Computer Aided Geometric Design, 23:226-238.
[14] Zhang, J.W., 1996. C-curves: an extension of cubic curves. Computer Aided Geometric Design, 13(3):199-217.
[15] Zhang, J.W., 1997. Two different forms of C-B-splines. Computer Aided Geometric Design, 14(1):31-41.
[16] Zhang, J.W., Krause, F.L., 2005. Extend cubic uniform B-splines by unified trigonometric and hyberolic basis. Graph. Models, 67(2):100-119.
[17] Zhang, J.W., Krause, F.L., Zhang, H.U., 2005. Unifying C-curves and H-curves by extending the calculation to complex numbers. Computer Aided Geometric Design, 22:865-883.
Open peer comments: Debate/Discuss/Question/Opinion
<1>