×

Quadratic trigonometric polynomial curves concerning local control. (English) Zbl 1083.65010

Summary: With a nonuniform knot vector and two local shape parameters, a kind of piecewise quadratic trigonometric polynomial curves is presented. The given curves have similar construction and the same continuity as the quadratic nonuniform B-spline curves. Two local parameters serve to local control tension and local control bias, respectively, in the curves. The changes of a local shape parameter will only affect two curve segments. The given curves can approximate the quadratic nonuniform rational B-spline curves and the quadratic rational Bézier curves well for which the relations of the local shape parameters and the weight numbers of the rational curves are described. The trigonometric polynomial curves can yield tight envelopes for the quadratic rational Bézier curves. The given curve also can be decreased to linear trigonometric polynomial curve which is equal to a quadratic rational Bézier curve and represents ellipse curve.

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)
65D07 Numerical computation using splines
42A05 Trigonometric polynomials, inequalities, extremal problems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Barsky, B.A., Local control of bias and tension, ACM trans. graph., 2, 109-134, (1983) · Zbl 0584.65004
[2] Barsky, B.A., Computer graphics and geometric modeling using beta-spline, (1988), Springer Heidelberg · Zbl 0648.65008
[3] Gregory, J.A.; Sarfraz, M., A rational cubic spline with tension, Comput. aided geom. design, 9, 1-13, (1990) · Zbl 0717.65003
[4] Joe, B., Multiple-knot and rational cubic beta-spline, ACM trans. grahp., 8, 100-120, (1989) · Zbl 0746.68096
[5] Joe, B., Quartic beta-splines, ACM trans. grahp., 9, 301-337, (1990) · Zbl 0729.68086
[6] Farin, G., NURBS curves and surfaces, (1995), Peter Wellesley, MA · Zbl 0835.65020
[7] Piegl, L.; Tiller, W., The NURBS book, (1995), Springer New York · Zbl 0828.68118
[8] Koch, P.E., Multivariate trigonometric B-splines, J. approx. theory, 54, 162-168, (1988) · Zbl 0671.41006
[9] Koch, P.E.; Lyche, T.; Neamtu, M.; Schumaker, L.L., Control curves and knot insertion for trigonometric splines, Adv. comput. math., 3, 405-424, (1995) · Zbl 0925.65251
[10] Lyche, T.; Winther, R., A stable recurrence relation for trigonometric B-splines, J. approx. theory, 25, 266-279, (1979) · Zbl 0414.41005
[11] Peña, J.M., Shape preserving representations for trigonometric polynomial curves, Comput. aided geom. design, 14, 5-11, (1997) · Zbl 0900.68417
[12] Walz, G., Some identities for trigonometric B-splines with application to curve design, Bit, 37, 189-201, (1997) · Zbl 0866.41010
[13] Walz, G., Trigonometric Bézier and Stancu polynomials over intervals and triangles, Comput. aided geom. design, 14, 393-397, (1997) · Zbl 0906.68167
[14] Han, X., Piecewise quadratic trigonometric polynomial curves, Math. comp., 72, 1369-1377, (2003) · Zbl 1072.65019
[15] Han, X., Quadratic trigonometric polynomial curves with a shaper parameter, Comput. aided geom. design, 19, 503-512, (2002) · Zbl 0998.68187
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.