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Journal of Zhejiang University SCIENCE A 2006 Vol.7 No.9 P.1550-1560

http://doi.org/10.1631/jzus.2006.A1550


A family of quasi-cubic blended splines and applications


Author(s):  SU Ben-yue, TAN Jie-qing

Affiliation(s):  School of Computer & Information, Hefei University of Technology, Hefei 230009, China; more

Corresponding email(s):   subenyue@sohu.com

Key Words:  Blended spline interpolation, C2 continuity, Global parameters, Local parameters, Quasi-cubic spline, Trigonometric polynomials


SU Ben-yue, TAN Jie-qing. A family of quasi-cubic blended splines and applications[J]. Journal of Zhejiang University Science A, 2006, 7(9): 1550-1560.

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T1 - A family of quasi-cubic blended splines and applications
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Abstract: 
A class of quasi-cubic B-spline base functions by trigonometric polynomials are established which inherit properties similar to those of cubic B-spline bases. The corresponding curves with a shape parameter α, defined by the introduced base functions, include the B-spline curves and can approximate the B-spline curves from both sides. The curves can be adjusted easily by using the shape parameter α, where dpi(α,t) is linear with respect to dα for the fixed t. With the shape parameter chosen properly, the defined curves can be used to precisely represent straight line segments, parabola segments, circular arcs and some transcendental curves, and the corresponding tensor product surfaces can also represent spherical surfaces, cylindrical surfaces and some transcendental surfaces exactly. By abandoning positive property, this paper proposes a new C2 continuous blended interpolation spline based on piecewise trigonometric polynomials associated with a sequence of local parameters. Illustration showed that the curves and surfaces constructed by the blended spline can be adjusted easily and freely. The blended interpolation spline curves can be shape-preserving with proper local parameters since these local parameters can be considered to be the magnification ratio to the length of tangent vectors at the interpolating points. The idea is extended to produce blended spline surfaces.

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Reference

[1] Gfrerrer, A., Röchel, O., 2001. Blended Hermite interpolants. Computer Aided Geometric Design, 18(9):865-873.

[2] Grisoni, L., Blanc, C., Schlick, C., 1999. HB-splines: A Blend of Hermite Splines and B-splines. The Eurographics Association, Blackwell Publishers, Oxford, UK.

[3] Kochanek, D., Bartels, R., 1984. Interpolating splines with local tension, continuity, and bias control. ACM SIGGRAPH Computer Graphics, 18(3):33-41.

[4] Mainar, E., Peña, J.M., Sánchez-Reyes, J., 2001. Shape preserving alternatives to the rational Bézier model. Computer Aided Geometric Design, 18(1):37-60.

[5] Peña, J.M., 1997. Shape preserving representations for trigonometric polynomial curves. Computer Aided Geometric Design, 14(1):5-11.

[6] Piegl, L., Ma, W., Tiller, W., 2005. An alternative method of curve interpolation. The Visual Computer, 21(1-2):104-117.

[7] Sánchez-Reyes, J., 1998. Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials. Computer Aided Geometric Design, 15(9):909-923.

[8] Tai, C.L., Loe, K.F., 1999. Alpha-spline: A C2 Continuous Spline with Weights and Tension Control. Proc. of International Conference on Shape Modeling and Applications, p.138-145.

[9] Walz, G., 1997. Trigonometric Bézier and Stancu polynomials over intervals and triangles. Computer Aided Geometric Design, 14(4):393-397.

[10] Zhang, J.W., 1996. C-curves: An extension of cubic curves. Computer Aided Geometric Design, 13(3):199-217.

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