Ball basis was introduced for cubic polynomials by Ball, and was generalized for polynomials of higher degree by Said. An algorithmic approach to degree reduction of interval Said-Ball (SB) curve is presented in this paper. The four fixed Kharitonov's polynomials (four fixed SB curves) associated with the original interval SB curve are obtained. These four fixed SB curves are transformed into f...

The Virtex FPGA introduces new features which support the design of large systems using a synthesis flow. With much improved routing, features like clock DLLs and BlockRAM, a rich collection of resources are available to designers. We investigate how synthesis friendly Virtex is by implementing the Bézier function in a variety of ways, making use of BlockRAM, clock DLLs and shift-register LUTs....

In this paper, we derive a family of fast and stable algorithms for multiplying and inverting n × n Pascal matrices that run in O(n log n) time and are closely related to De Casteljau’s algorithm for Bézier curve evaluation. These algorithms use a recursive factorization of the triangular Pascal matrices and improve upon the cripplingly unstable O(n log n) fast Fourier transform-based algorithm...

In this paper we present an approximation method for the convolution of two planar curves using a pair of two cubic Bézier curves with linear normals (LN). We characterize the necessary and sufficient conditions for two compatible cubic Bézier LN curves to have the same linear normal map. Using this characterization, we obtain the cubic spline approximation of the convolution curve. As illustra...

In this paper we present methods for approximating a helix segment by quintic Bézier curves or quintic rational Bézier curves based on the geometric Hermite interpolation technique in space. The fitting curve interpolates the curvatures as well as the Frenet frames of the original helix at both ends. We achieve a high accuracy of the approximation by giving a proper parametrization of the curve...

The role of differential geometry in describing a curve can not be denied. The differential forms defined for Bézier curves which are widely used in computer aided geometric design, plays a significant role in classification and image processing of curves. For this reason, the definitions such as Serret-Frenet frame, curvature and torsion which are described for Bézier curves are very important...