English

Blending implicit interval curves and surfaces

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi-Brauer varieties and mod 2 Galois representations, and prove that the local-global principle holds for conics and cubics. We also construct counterexamples to the local-global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.

In this paper we propose a construction method of the planar cubic algebraic spline curve with endpoint interpolation conditions and a specific analysis of its properties. The piecewise cubic algebraic curve has G 2 continuous contact with the control polygon at two endpoints and is G 2 continuous between each segments of itself. The process of this method is simple and clear, and provides a new way of thinking to design implicit curves.

Symmetries and similarities of planar algebraic curves using harmonic polynomials

Multiscale Representation and Matching of Curves Using Codons

This paper presents an algorithm for approximating planar curves with conjugated circular arcs and straight-line segments. Approximation of digital planar curves using line segments and analytic curve pieces is a problem of fundamental importance in image analysis and pattern recognition. A specific variation of this problem is approximation by conjugated arcs and line segments, which also arises in robotics, particularly in motion planning. We propose an algorithmic solution to this problem, designed for the automatic generation of programs for robotic manipulators in the AS language. The originality of the method lies in employing a special representation of planar curves that is invariant under translation, rotation, and scaling. Such a representation enables the development of an optimal algorithm that automatically determines the number and placement of conjugated arcs and line segments approximating a given curve with a prescribed accuracy. The simplicity, flexibility with respect to input data, and low computational requirements of the proposed method make it suitable not only for image recognition tasks but also for robotic trajectory planning.

Cross-sectional profiles of medieval mouldings are often considered to be a vital form of data for art historians. Mouldings have often been used to highlight salient areas of buildings, and they provide a wealth of information. To date, little use of computer technology has been made in their acquisition, storage and analysis. The paper presents a representation for such planar curves, and an algorithm for its extraction from planar curves. The representation uses geometric primitives, specifically line and ore segments, which allow the curves to be represented in a compact form, approximating the circular sections more effectively than using linear segments alone. These idealised models provide templates which provide a means for the identification of specific features, within profiles, which can then serve as characteristics for use in case-based reasoning.

We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly limit the types of involved exact operations, that is, our algorithms only use resultant and gcd computations as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the use of a certified numerical complex root solver and information derived from the resultant computation. Additionally, we never consider any coordinate transformation and the output is also given with respect to the initial coordinate system.We implemented our algorithm as a prototypical package of the C++-library Cgal. Our implementation exploits graphics hardware to expedite the resultant and gcd computation. We also compared our implementation with the current reference implementation, that is, Cgal's curve analysis and arrangement for algebraic curves. For various series of challenging instances, our experiments show that the new implementation outperforms the existing one.

Chapter 15 - Algebraic Methods for Computer Aided Geometric Design

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

A plane algebraic curve is given as the zeros of a bivariate polynomial. However, this implicit representation is badly suited for many applications, for instance in computer aided geometric design. What we want in many of these applications is a rational parametrization of an algebraic curve. There are several approaches to deciding whether an algebraic curve is rationally parametrizable and if so computing such a parametrization. In all these approaches we ultimately need some simple points on the curve. The field in which we can find such points crucially influences the coefficients in the resulting parametrization. We show how to find simple points over some practically interesting fields.Consequently, we are able to decide whether an algebraic curve defined over the rational numbers can be parametrized over the rationals or the reals. Some of these ideas also apply to parametrization of surfaces. If in the term geometric reasoning we do not only include the process of proving or disproving geometric statements, but also the analysis and manipulation of geometric objects, then algorithms for parametrization play an important role in this wider view of geometric reasoning.KeywordsRational PointAlgebraic CurveAlgebraic CurfRational CaseSimple PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Codes obtained from algebraic curves have attracted much attention from mathematicians and engineers alike since the remarkable work of Tsfasman et al. [17] who showed that the longstAnding Gilbert-Varshamov lower bound can be exceeded for alphabet sizes larger than 49. The Gilbert-Varshamov bound, established in 1952, is a lower bound on the information rate of good codes. This lower bound was not improved until 1982 with the discovery of good algebraic geometric codes. These codes are obtained from modular curves [17], but consideration of these curves is beyond the scope of this book. van Lint and Springer [21] later derived the same results as Tsfasman et al., but by using less complicated concepts from algebraic geometry. Recall that a linear code with parameters [n, k, d] q is a linear subspace of F q n of dimension k and minimum distance d.

An embedding technique for the solution of reaction–diffusion equations on algebraic surfaces with isolated singularities

Determinantal representations of hyperbolic plane curves: An elementary approach

The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sh