Lupaş q-Bézier曲线的离散卷积生成与求值算法

计算机工程与科学 ›› 2023, Vol. 45 ›› Issue (01): 104-112.

Lupaş q-Bézier曲线的离散卷积生成与求值算法

耿梦圆1,解滨2,韩力文1,3,4

(1.河北师范大学数学科学学院，河北石家庄 050024；2.河北师范大学计算机与网络空间安全学院，河北石家庄 050024；
3.河北省计算数学与应用重点实验室，河北石家庄 050024；
4.河北省数学与交叉科学国际联合研究中心，河北石家庄 050024)

收稿日期:2022-08-07 修回日期:2022-09-20 接受日期:2023-01-25 出版日期:2023-01-25 发布日期:2023-01-25
基金资助:
国家自然科学基金(62076088);河北省自然科学基金(A2018205103);河北师范大学科研基金(L2020Z02,L2022B30)

A generation and computation algorithm of Lupaş q-Bézier curve via discrete convolution

GENG Meng-yuan1,XIE Bin2,HAN Li-wen1,3,4

(1.School of Mathematical Sciences,Hebei Normal University,Shijiazhuang 050024；
2.College of Computer and Cyber Security,Hebei Normal University,Shijiazhuang 050024；
3.Hebei Key Laboratory of Computational Mathematics and Applications,Shijiazhuang 050024；
4.Hebei International Joint Research Center for Mathematics and Interdisciplinary Science,Shijiazhuang 050024,China)

Received:2022-08-07 Revised:2022-09-20 Accepted:2023-01-25 Online:2023-01-25 Published:2023-01-25

摘要/Abstract

摘要：

Lupaş q-Bernstein算子是最早提出的有理形式下基于q整数的q模拟Bernstein算子。通过Lupaş q-Bernstein基函数的递推关系反向使用金字塔算法,离散卷积生成n次Lupaş q-Bernstein基函数序列。结合离散卷积满足的交换性,针对n次Lupaş q-Bézier曲线推导出其速端曲线及n!种de Casteljau算法。与Bézier曲线de Casteljau算法得到的切点不同,Lupaş q-Bézier曲线的de Casteljau算法得到的曲线上的一点是直线与曲线相交的2个割点之一。针对二次Lupaş q-Bézier曲线,给出了计算左/右割点的充分必要条件，然后通过提出双割点算法,可以同时得到左/右割点。

关键词:

离散卷积, 速端曲线, de Casteljau算法, 割点, 交比不变性

Abstract:

Lupa ş q-Bernstein operator is the first proposed qinteger based qanalogue Bernstein operator in rational form. By using the recurrence formulas in reverse as a pyramid algorithm, the nth degree Lupaş q-Bernstein basis function sequence is generated via discrete convolution. Owing to the commutativity of discrete convolution, for each Lupaş q-Bézier curve of degree n, the hodograph and the collection of n! recursive evaluation algorithms are derived. Unlike the tangent point obtained by de Casteljau algorithm of Bézier curve, de Casteljau algorithm of Lupa ş q-Bézier curve obtains a point on the curve being one of the two cut points where the line intersects the curve. For quadratic Lupa ş q-Bézier curve, sufficient and necessary conditions for computing left and right cut points are obtained. In addition, the left and right cut points can be computed simultaneously by proposing a dual cut point algorithm.

Key words: discrete convolution, hodograph, de Casteljau algorithm, cut point, cross ratio invariant

耿梦圆, 解滨, 韩力文, . Lupaş q-Bézier曲线的离散卷积生成与求值算法[J]. 计算机工程与科学, 2023, 45(01): 104-112.

GENG Meng-yuan, XIE Bin, HAN Li-wen, . A generation and computation algorithm of Lupaş q-Bézier curve via discrete convolution[J]. Computer Engineering & Science, 2023, 45(01): 104-112.