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Open AccessArticle

A New Quasi Cubic Rational System with Two Parameters

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
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Author to whom correspondence should be addressed.
Symmetry 2020, 12(7), 1070; https://doi.org/10.3390/sym12071070
Received: 14 May 2020 / Revised: 27 June 2020 / Accepted: 28 June 2020 / Published: 30 June 2020
The purpose of this article is to develop a new system for the construction of curves and surfaces. Making the new system not only has excellent properties of the orthodox Bézier and the B-spline method but also has practical properties such as variation diminishing and local shape adjustability. First, a new set of the quasi-cubic rational (QCR) system with two parameters is given, which is verified on an optimal normalized totally positive system (B-system). The related QCR Bézier curve is defined, and the de Casteljau-type algorithm are given. Next, a group of non-uniform QCR B-spline system is shown based on the linear combination of the proposed QCR system, the
relative properties of the B-spline system are analyzed. Then, the definition and properties of nonuniform QCR B-spline curves are discussed in detail. Finally, the proposed QCR system is extended to the triangular domain, which is called the quasi-cubic rational Bernstein-Bézier (QCR-BB) system, and its related definition and properties of patches are given at length. The experimental image obtained by using MATLAB shows that the newly constructed system has excellent properties such
as symmetry, totally positive, and C2 continuity, and its corresponding curve has the properties of local shape adjustability and C2 continuity. These extended systems in the extended triangular
domain have symmetry, linear independence, etc. Hence, the methods in this article are suitable for the modeling design of complex curves and surfaces.
Keywords: optimal normalized totally positive system; de Casteljau-type algorithm; non-uniform B-spline; triangular domain; quasi-cubic rational Bernstein-Bézier system optimal normalized totally positive system; de Casteljau-type algorithm; non-uniform B-spline; triangular domain; quasi-cubic rational Bernstein-Bézier system
MDPI and ACS Style

Tuo, M.-X.; Zhang, G.-C.; Wang, K. A New Quasi Cubic Rational System with Two Parameters. Symmetry 2020, 12, 1070.

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