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