Article ID: MTJPAM-D-22-00011

Title: Homogeneous q-blossoming and Bézier curves


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-22-00011; Volume 4 / Issue 2 / Year 2022, Pages 86-102

Document Type: Research Paper

Author(s): Çetin Dişibüyük a

aDepartment of Mathematics, Dokuz Eylül University, Faculty of Science, Tınaztepe Campus, 35390 Buca, İzmir, Turkey

Received: 25 March 2022, Accepted: 3 September 2022, Published: 11 October 2022.

Corresponding Author: Çetin Dişibüyük (Email address: cetin.disibuyuk@deu.edu.tr)

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Abstract

Homogeneous q-blossom is introduced by altering the diagonal property of classical homogeneous blossom. We apply this new blossom to define two parameter family of Bernstein basis functions and Bézier curves. A special case of homogeneous q-blossom gives infinitely many de Casteljau type algorithms for classical Bézier curves. An analogue of Marsden’s identity is also derived by applying homogeneous q-blossom. Properties and identities of new Bernstein basis functions and Bézier curves including affine invariance, linear precision and end point interpolation derived. De Casteljau type evaluation algorithm is used to develop a subdivision procedure for (q1, q2)-Bézier curves. Finally, it is shown that the control polygons generated by recursive midpoint subdivision converge uniformly to the original (q1, q2)-Bézier curve.

Keywords: q-blossom, homogeneous q-blossom, (q1, q2)-Bernstein basis functions, (q1, q2)-Bézier curves, de Casteljau algorithm, subdivision

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