Related papers: Rational complex Bezier curves
Matrix weighted rational B\'{e}zier curves can represent complex curve shapes using small numbers of control points and clear geometric definitions of matrix weights. Explicit formulae are derived to convert matrix weighted rational…
We introduce the class of rational plane curves parameterizable by conics as an extension of the family of curves parameterizable by lines (also known as monoid curves). We show that they are the image of monoid curves via suitable…
We give several constructions of bicuspidal rational complex projective plane curves, and list the Newton pairs and the multiplicity sequences of the singularities on the resulting curves. Although the existence of some of the listed cusp…
New geometric methods for fast evaluation of derivatives of polynomial and rational B\'{e}zier curves are proposed. They apply an algorithm for evaluating polynomial or rational B\'{e}zier curves, which was recently given by the authors.…
A new class of rational parametrization has been developed and it was used to generate a new family of rational functions B-splines $\displaystyle{{\left({}^{\alpha}{\mathbf B}_{i}^{k} \right)}_{i=0}^{k}}$ which depends on an index $\alpha…
We present an approach to finding the implicit equation of a planar rational parametric cubic curve, by defining a new basis for the representation. The basis, which contains only four cubic bivariate polynomials, is defined in terms of the…
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical…
A new algorithm for computing a point on a polynomial or rational curve in B\'{e}zier form is proposed. The method has a geometric interpretation and uses only convex combinations of control points. The new algorithm's computational…
How to quickly and stably realize the degree reduction of the rational Bezier curve is an open problem in CAGD. Based on the weighted least squares method and weighted sum method of multi-objective optimization, this paper transforms the…
A method to construct transition curves using a family of the quartic Bezier spiral is described. The transition curves discussed are S-shape and C-shape of contact, between two separated circles. A spiral is a curve of monotone increasing…
We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination…
This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify…
It is still a challenging task of today to recognize the type of a given algebraic surface which is described only by its implicit representation. In~this paper we will investigate in more detail the case of canal surfaces that are often…
In this paper, we propose a method to obtain a constrained approximation of a rational B\'{e}zier curve by a polynomial B\'{e}zier curve. This problem is reformulated as an approximation problem between two polynomial B\'{e}zier curves…
Farin proposed a method for designing Bezier curves with monotonic curvature and torsion. Such curves are relevant in design due to their aesthetic shape. The method relies on applying a matrix M to the first edge of the control polygon of…
An expository description of smooth cubic curves in the real or complex projective plane.
One distinguishing feature of rational curves is that they have algebraic parameterizations. Arc spaces are a way of describing approximations to parameterizations of all curves in some fixed space. Playing on these descriptions, this paper…
This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this…
In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential…
In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with…