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...

We want to estimate the chord length Λ of a given rational Bézier curve efficiently. Since rational Bézier are nonlinear function, it is generally impossible to evaluate its length exactly. We approximate the length by using subdivision and we investigate the accuracy of the approximation Λn. In order to improve the efficiency, we use adaptivity with some length estimator. Additionally, we will...

In this paper, the issue of multi-degree reduction of Bézier curves with C and G-continuity at the end points of the curve is considered. An iterative method, which is the first of this type, is derived. It is shown that this algorithm converges and can be applied iteratively to get the required accuracy. Some examples and figures are given to demonstrate the efficiency of this method. 2011 Els...

Arc splines are G continuous curves made of circular arcs and straight-line segments. They have the advantages that the curvature of an arc spline is known and controlled at all but a finite number of points, and that the offset curve of an arc spline is another arc spline. Arc splines are used by computer-controlled machines as a natural curve along which to cut and are used by highway route p...

In this paper we present a local fitting algorithm for converting smooth planar curves to B-splines. For a smooth planar curve a set of points together with their tangent vectors are first sampled from the curve such that the connected polygon approximates the curve with high accuracy and inflexions are detected by the sampled data efficiently. Then, a G1 continuous Bézier spline curve is obtai...