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Degree elevation of unified and extended spline curves

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Abstract

Unified and extended splines (UE-splines), which unify and extend polynomial, trigonometric, and hyperbolic B-splines, inherit most properties of B-splines and have some advantages over B-splines. The interest of this paper is the degree elevation algorithm of UE-spline curves and its geometric meaning. Our main idea is to elevate the degree of UE-spline curves one knot interval by one knot interval. First, we construct a new class of basis functions, called bi-order UE-spline basis functions which are defined by the integral definition of splines. Then some important properties of bi-order UE-splines are given, especially for the transformation formulae of the basis functions before and after inserting a knot into the knot vector. Finally, we prove that the degree elevation of UE-spline curves can be interpreted as a process of corner cutting on the control polygons, just as in the manner of B-splines. This degree elevation algorithm possesses strong geometric intuition.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Xiao-juan Duan & Guo-zhao Wang

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  1. Xiao-juan Duan
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  2. Guo-zhao Wang
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Correspondence to Xiao-juan Duan.

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Project supported by the National Natural Science Foundation of China (Nos. 60933008 and 61272300)

ORCID: Xiao-juan DUAN, http://orcid.org/0000-0002-7833-6626

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Duan, Xj., Wang, Gz. Degree elevation of unified and extended spline curves. J. Zhejiang Univ. - Sci. C 15, 1098–1105 (2014). https://doi.org/10.1631/jzus.C1400076

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  • DOI: https://doi.org/10.1631/jzus.C1400076

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