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图学学报

图学学报 ›› 2021, Vol. 42 ›› Issue (4): 651-658.DOI: 10.11996/JG.j.2095-302X.2021040651

• 计算机图形学与虚拟现实 • 上一篇    下一篇

Lupaş q-Bézier 曲线的几何求值算法及其应用

  

  1. 1. 河北师范大学数学科学学院，河北 石家庄 050024；
    2. 河北省计算数学与应用重点实验室，河北 石家庄 050024
  • 出版日期:2021-08-31 发布日期:2021-08-05
  • 基金资助:
    国家自然科学基金项目(61573127)；河北省自然科学基金项目(A2018205103)；河北师范大学科研基金资助项目(L2020Z02，L2020K09)；河北师范大学研究生创新资助项目(CXZZSS202053)

Geometric evaluation algorithms for Lupaş q-Bézier curve and its applications

  1. 1. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang Hebei 050024, China;
    2. Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang Hebei 050024, China
  • Online:2021-08-31 Published:2021-08-05
  • Supported by:
    National Natural Science Foundation of China (61573127); National Natural Science Foundation of Hebei Porvince (A2018205103);
    Research Fund of Hebei Normal University (L2020Z02, L2020K09); Hebei Normal University Graduate Project (CXZZSS202053)

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摘要： Lupaş q-Bézier 曲线是一种以 q-整数作为形状参数的广义 Bézier 曲线。本文构造了 Lupaş q-Bézier
曲线的一种新型几何求值算法，该算法倒数第二层 2 个节点的仿射组合与曲线相切。利用算法的相切性质得到
Lupaş q-Bézier 曲线导矢的一种新表示，并实现了 Lupaş q-Bézier 曲线的细分。特别地，二次 Lupaş q-Bézier 曲线
分割得到的 2 条子曲线的形状参数的乘积等于原曲线的形状参数。进一步，得到了加权 Lupaş q-Bézier 曲线的一
种新型几何求值算法，该算法具有显式矩阵表示。

关键词: Lupa? q-Bézier 曲线, de Casteljau 算法, 显式矩阵表示, 细分, 计算复杂度

Abstract: The Lupaş q-Bézier curve is a generalized Bézier curve with q-integer as the shape parameter. A new
geometric evaluation algorithm for Lupaş q-Bézier curve was constructed in this paper, in which the affine
combination of two nodes in the penultimate layer of the algorithm is tangent to the curve. A new representation of the
derivative of the Lupaş q-Bézier curve was obtained using the property of algorithm, and the subdivision of curve was
realized. Particularly, a product of the shape parameters of the two subcurves produced by the subdivision of the Lupaş
q-Béziercurves was equal to the shape parameter of the original curve. Furthermore, a new geometric evaluation
algorithm of weighted Lupaş q-Bézier with an explicit matrix representation was gained.

Key words: Lupa? q-Bézier curve, de Casteljau algorithm, explicit matrix representation, subdivision, computational
complexity

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