Approximate Parametrization of Algebraic Curves

The rational Bernstein bases and the multirational blossoms

It is often necessary to represent a hand-drawn shape accurately. Modeling such shapes manually is both cumbersome and commercially expensive. The user's concern for curves that are easy to manipulate has been a major influence on the development of free-form curves. Rational parametric curves have been receiving considerable attention in the area of geometric modeling because any parametric polynomial curve can be expressed as a rational curve and most polynomial splines have rational extensions. Rational parametric curves can be used to model any object, like ships and airplanes, and even in the medical field for modeling the heart or other parts of the body. In object modeling, the rational cubic spline is most popular because it is the lowest degree that can define space curves and curves with points of zero curvature. The foremost objective of this work is to show how conic sections can adequately be used to represent curves and objects that were previously thought to require rational cubic splines. A single rational cubic curve is represented by two conics by splitting the rational cubic at its mid-point. The degree of smoothness has been taken into account to have a visually pleasing effect. It has been realized that the conic representation is advantageous over the rational cubic one in terms of computational requirements and shape control. All of this work is carried out on both 2D curves and 3D objects.

Bezier and rational Bezier curves are important elements in computer aided geometric design (CAGD) due to their capability to represent both the free-form setting and the algebraic one as well. Therefore, such curves are popularly accepted as a standard representation for designing problems. The purpose of this paper is to present the rational Bezier curves as attractors of some iterated function systems.

Algorithms for Intersecting Parametric and Algebraic Curves II: Multiple Intersections

We describe a method for approximate parameterization of a planar algebraic curve by a rational Bezier (spline) curve. After briefly discussing exact methods for parameterization and methods for rational interpolation, we describe a new technique for rational parameterization. Our approach is based on the minimization of a suitable--nonlinear objective function, which takes both the distance from the curve and the positivity of the weight function (i.e., the numerator of the rational parametric representation) into account. The solution is computed by using an SQP-type optimization technique. In addition, we use a region--growing--type approach in order to obtain a good initial solution, which is crucial for the convergence of the nonlinear optimization procedure.

Automatic parametrization of rational curves and surfaces II: cubics and cubicoids

Parametrization of approximate algebraic curves by lines

Vectorization of digital images using algebraic curves

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general (non-planar) algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus. We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation for which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d-1), (ii) the dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of the curve degree alone.

Each rational (projective) Bezier curve is determined by three points in the plane and by positive weights assigned to these points. As is known, any such curve is an arc of either a parabola, an ellipse, or a hyperbola. An equation for a projective Bezier curve in barycentric coordinates is derived. This equation depends on a parameter. A complete classification of the curves under consideration in terms of parameter values is suggested.

In this paper it is shown that cycles of planar dynamical systems with polynomial vector fields are in many ways similar to ovals of algebraic curves, and more generally to "separating solutions" of such systems, on general algebraic curves. For example, for separating solutions the analog of Bezout's theorem is valid. The proofs are straightforward. They are based on the version of Rolle's theorem and Bezout's theorem for planar algebraic curves offered below. We note that according to Hilbert's conjecture the total number of limit cycles of a dynamical system with a polynomial vector field is bounded above by the degree of the field. From the validity of the conjecture and the results of the present note it follows that the curve consisting of all the limit cycles of a polynomial dynamical system resembles an algebraic curve. At the present time, even the finiteness of the number of limit cycles has not been proved: Ii'yashenko proved that Dyudaktsproof [i] contains an unfillable gap (cf. [2]) . I. Rolle's Theorem for Dynamical Systems We consider a smooth dynamical system on the plane

Numerical proper reparametrization of parametric plane curves

This paper is concerned with the re-representation of a G 1 composite rational Bezier curve. Although the rational Bezier curve segments that form the composite curve are G 1 continuous at their jo...

This article discusses the developed practical and homework exercises of the Geometric Modeling course in the direction of graduate training 02.03.01 Mathematics and Computer Science (specialization Mathematical and Computer Modeling). They are devoted to computer geometric modeling techniques using various spline representations and basic logical operations on geometric constructions of complex curved shapes. The spline representations include linear and bilinear splines, Hermite cubic splines and Hermite surfaces, natural cubic and bicubic interpolating splines, Bezier curves, rational Bezier curves, NURBS curves and NURBS surfaces. We consider Boolean operations of the intersection of two spline curves and the intersection of two parametric surfaces. Screenshots of some graphic windows of the developed software are shown. The software has a web interface using the WebGL graphics library in a browser. It is used by students in completing tasks for preparing the initial data and visualizing the results of the constructions. Tasks for four practical exercises in a computer laboratory and sample homework exercises are given.

We study the rationality of each of the components of the conchoid to an irreducible algebraic affine plane curve, excluding the trivial cases of the lines through the focus and the circle centered at the focus and radius the distance involved in the conchoid. We prove that conchoids having all their components rational can only be generated by rational curves. Moreover, we show that reducible conchoids to rational curves have always their two components rational. In addition, we prove that the rationality of the conchoid component, to a rational curve, does depend on the base curve and on the focus but not on the distance. As a consequence, we provide an algorithm that analyzes the rationality of all the components of the conchoid and, in the affirmative case, parametrizes them. The algorithm only uses a proper parametrization of the base curve and the focus and, hence, does not require the previous computation of the conchoid. As a corollary, we show that the conchoid to the irreducible conics, with conchoid-focus on the conic, are rational and we give parametrizations. In particular we parametrize the Limaçons of Pascal. We also parametrize the conchoids of Nicomedes. Finally, we show how to find the foci from where the conchoid is rational or with two rational components.