Rational Algebraic Curves

Sendra, J. Rafael; Winkler, Franz; Pérez–Díaz, Sonia

doi:10.1007/978-3-540-73725-4

Cited by 144 publications

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“…This comes as a consequence of the more general and well-known result for rational curves [17], stating that two proper parameterizations of the same curve are necessarily related by a Möbius (rational linear) reparameterization.…”

Section: Remarkmentioning

confidence: 95%

The conditions for the coincidence or overlapping of two Bézier curves

Sánchez-Reyes

2014

Applied Mathematics and Computation

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Section: Remarkmentioning

confidence: 95%

The conditions for the coincidence or overlapping of two Bézier curves

Sánchez-Reyes

2014

Applied Mathematics and Computation

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“…Now we recall some definitions about the parameterizations of algebraic curve C over C defined by f (x, y) = 0 [24,7,15,16]. Definition 2.1.…”

Section: Describes §2 Preliminariesmentioning

confidence: 99%

“…Definition 2.4. [16] An equivalence class of irreducible parameterizations of algebraic curve is called a place of the algebraic curve. The common center of the equivalent parameterizations is called the center of the place.…”

Section: Definition 23mentioning

confidence: 99%

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Numerical realization of the conditions of Max Nöther’s residual intersection theorem

Luo

Feng

Zhang

2014

Appl. Math. J. Chin. Univ.

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The aim of this paper is to study numerical realization of the conditions of Max Nöther's residual intersection theorem. The numerical realization relies on obtaining the intersection of two algebraic curves by homotopy continuation method, computing the approximate places of an algebraic curve, getting the exact orders of a polynomial at the places, and determining the multiplicity and character of a point of an algebraic curve. The numerical experiments show that our method is accurate, effective and robust without using multiprecision arithmetic, even if the coefficients of algebraic curves are inexact. We also conclude that the computational complexity of the numerical realization is polynomial time. §1 Introduction There are many achievements in mathematics, which condense human intelligence and represent the progress of culture, such as algebra, geometry, topology. Once the theorems and conclusions can be applied in practice, it must make themselves not just for the theoretical research. With the development of the computer, this idea not only becomes a possible reality, but also leads to new research fields. For example, when a matrix A is nonsingular, the solution of the linear system AX = b is X = A −1 b, which is a perfect theoretical result. However, when the matrix A is dense and of large order, the result is not practical. To solve the linear system efficiently, an important research subject called Numerical Algebra has arisen.Max Nöther's residual intersection theorem plays very important roles in algebraic geometry [8], such as algebraic curves and polynomial ideal theories. It is a special case of CayleyBacharach theorem and also an aided tool to prove the generalizations of Cayley-Bacharach theorem [8]. Its comprehensive application in linear series shows its importance [24]. This paper presents the numerical realization of the conditions of this theorem, which involves the computations of some fundamental quantities of algebraic curve, such as the places of an algebraic curve, the orders of a bivariate polynomial at the places, and the intersection of two algebraic curves. Obviously, it is difficult to compute these quantities for the algebraic curve of large degree by manual calculation.The places of an algebraic curve can be computed from its Puiseux expansions. The earlier work on the computation of Puiseux expansions can be found in [24,5,6,11]. Duval [7] describes

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“…Observe that for lines and circles the shape of the offsets can be trivially deduced. Furthermore, since C is real and irreducible we can assume that it is defined by an irreducible polynomial f ∈ R[x, y] (see [19]). Also, we will denote the offset to C at distance d, as O d (C); along the paper we always work with real, positive distances, i.e.…”

Section: Introductionmentioning

confidence: 99%

Good Local Behavior of Offsets to Implicit Algebraic Curves

Alcázar

2009

Math.Comput.Sci.

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The notion of good local behavior of an offset curve was introduced in [2, 4] to denote that the behavior of an offset was locally good from a topological point of view. Also, in these papers the problem of checking whether good local behavior holds for a particular offsetting distance, and of computing intervals of distances with this nice property, was addressed for the case of rational algebraic curves. Thus, here we generalize the results and techniques of these papers to the case when the curve to work with is an implicitly given, possibly singular, non-necessarily rational algebraic curve. Furthermore, a generalization of the (already known) results relating offsets and evolutes of regularly parametrized curves, is presented for the case of possibly singular, implicit algebraic curves.

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Rational Algebraic Curves

Cited by 144 publications

References 0 publications

The conditions for the coincidence or overlapping of two Bézier curves

The conditions for the coincidence or overlapping of two Bézier curves

Numerical realization of the conditions of Max Nöther’s residual intersection theorem

Good Local Behavior of Offsets to Implicit Algebraic Curves

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