a disk wang-bézier type generalized ball curve is a wang-bézier type generalized ball curve whose control points are disks in a plane. it can be viewed as a parametric curve with error tolerances. in this paper, we discuss the problem of degree reduction of disk wang-bézier type generalized ball curve, that is, bounding disk wang-bézier type generalized ball curves with lower degree disk wang-b...

This paper presents a short, simple, and general proof showing that the control polygons generated by subdivision and degree elevation converge to the underlying splines, box-splines, or multivariate B~zier polynomials, respectively. The proof is based only on a Taylor expansion. Then the results are carried over to rational curves and surfaces. Finally, an even shorter but as simple proof is p...

Article history: Received 29 July 2015 Received in revised form 4 May 2016 Accepted 19 May 2016 Available online 2 June 2016

A class of single-valued curves in polar coordinates, which we refer to as p-B ezier curve, has been recently presented by SS anchez-Reyes and independently discovered by P.de Casteljau. From their deenition and expression in terms of the Fourier basis it is obvious that every curve of degree n can be expressed as a curve of degree kn, for any natural value k. In this paper, we provide a formul...

Recently we developed a periodic surface model to assist the construction of nano structures parametrically for computer-aided nano-design. In this paper, we study the properties of periodic surfaces for degree elevation and reduction. Degree elevation approaches are developed to incrementally increase shape complexities, including native, variational, and boundary constrained elevations. A gen...