Symmetry Detection of Rational Space Curves from their Curvature and Torsion

We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem. To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage.Comment: 25 page

Detecting Similarity of Rational Plane Curves

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.Comment: 22 page

Topological Behavior of Families of Algebraic Curves Continuously Depending on a Parameter Under Certain Conditions.

En trabajos previos del autor, se considera el problema de determinar los tipos topológicos en una familia de curvas algebraicas planas, algebraicamente dependientes de un parámetro. En este trabajo, estos resultados se generalizan, bajo ciertas condiciones, al caso de una familia de curvas algebraicas planas dependientes de forma continua de un parámetro t, que toma valores en un subconjunto de la recta real que es unión de una cantidad finita de intervalos abiertos. Los resultados conducen al cálculo de un polinomio R(t) con la propiedad de que para todo intervalo contenido en U, que no contenga ninguna raíz de R(t), el tipo topológico de la familia no varía. Un caso importante en el que los resultados son aplicables, es el caso en que los coeficientes de las curvas son algebraicamente independientes. Si el número de raíces de R(t) es finito, los tipos topológicos presentes en la familia pueden calcularse mediante métodos bien conocidos.In previous works of the author, the question of computing the different shapes arising in a family of algebraic curves\ud algebraically depending on a real parameter was addressed. In this work we show how the ideas in these papers can be used to extend the results to\ud a more general class of families of algebraic curves, namely families not algebraically but just continuously depending on a parameter. These\ud families correspond to polynomials in the variables x,y whose coefficients are continuous functions of a parameter t taking values in U, where U is in general the union of finitely many open intervals. Under certain conditions, here we provide an algorithm for computing a\ud univariate real function R(t), with the property that the topology of the family stays invariant along every real interval I contained in U, and \ud not containing any real root of R(t). In that situation, a partition of the real line where each\ud element gives rise to a same shape arising in the family, can be computed. Then, these shapes can be described by using well-known methods. An important situation when the method is applicable is the case when the coefficients are algebraically\ud independent, or can be expressed in terms of algebraically independent functions

Efficient detection of symmetries polynomially parametrized curves

We present efficient algorithms for detecting central and mirror symmetry for the case of algebraic curves defined by means of polynomial parametrizations. The algorithms are based on an algebraic relationship between proper parametrizations of a same curve, which leads to a triangular polynomial system that can be solved in a very fast way; in particular, curves parametrized by polynomials of serious degrees/coefficients can be analyzed in a few seconds. In our analysis we provide a good number of theoretical results on symmetries of polynomial curves, algorithms for detecting rotation and mirror symmetry, and closed formulae to determine the symmetry center and the symmetry axis, when they exist. Some observations and empiric results for the case of polynomial parametrizations with floating point coefficients are also reported.Presentamos algoritmos eficientes para detectar simetría central y simetría especular en curvas algebraicas definidas por medio de parametrizaciones polinomiales. Los algoritmos se basan en la relación existente entre dos parametrizaciones propias de una misma curva. Esta relación conduce a un sistema polinómico triangular, que puede resolverse de forma muy rápida. En particular, curvas parametrizadas por polinomios de grados y coeficientes muy elevados pueden analizarse en unos pocos secgundos. En el análisis propuesto se presentan varios resultados teóricos sobre simetrías de curvas polinomiales, algoritmos para detectar simetría rotacional y simetría especular, y fórmulas cerradas para detectar el centro de simetría y el eje de simetría, en caso de que existan. Se discute también el caso de parametrizaciones en coma flotante