Linear independence of the blending functions of T-splines without multiple knots

Authors:
Aizeng Wang;Gang Zhao;Yong-Dong Li
Affiliations:
School of Mechanical Engineering & Automation, Beihang University, Beijing 100191, PR China and State Key Laboratory of Virtual Reality Technology & Systems, Beihang University, Beijing 100191, PR ...;School of Mechanical Engineering & Automation, Beihang University, Beijing 100191, PR China and State Key Laboratory of Virtual Reality Technology & Systems, Beihang University, Beijing 100191, PR ...;School of Mechanical Engineering & Automation, Beihang University, Beijing 100191, PR China and State Key Laboratory of Virtual Reality Technology & Systems, Beihang University, Beijing 100191, PR ...
Venue:
Expert Systems with Applications: An International Journal
Year:
2014

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Abstract

The purpose of the present paper is to answer the following open question given by Sederberg's research team: are there T-splines with linearly dependent blending functions that do not have multiple knots? First, the mathematical properties of the blending functions of T-splines without multiple knots are analyzed. Then, the linear independence of them is proved, and meanwhile a necessary condition is deduced for the linear dependence of T-spline blending functions. Finally, the answer to the open question is obtained: there are no T-splines with linearly dependent blending functions that do not have multiple knots.