Lupaş q-Bézier曲线的离散卷积生成与求值算法

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

Lupaşq-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Lupaşq-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Lupaşq-Bernstein基函数序列。结合离散卷积满足的交换性,针对n次Lupaşq-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,Lupaşq-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Lupaşq-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。

Lupaşq-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 Lupaşq-Bernstein basis function sequence is generated via discrete convolution.Owing to the commutativity of discrete convolution,for each Lupaşq-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 Lupaşq-Bézier curve obtains a point on the curve being one of the two cut points where the line intersects the curve.For quadratic Lupaşq-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.