Carlos Hermoso

Detecting symmetries of rational plane and space curves

Abstract This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univariate polynomials. These algorithms have been implemented and tested in the Sage system.

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.

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 (Alcazar et al., 2014b). To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage. The paper presents a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric.The method is significantly faster, simpler, and more general than earlier methods addressing similar problems.An analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage are included.

Involutions of polynomially parametrized surfaces

We provide an algorithm for detecting the involutions leaving a surface defined by a polynomial parametrization invariant. As a consequence, the symmetry axes, symmetry planes and symmetry center of the surface, if any, can be determined directly from the parametrization, without computing or making use of the implicit representation. The algorithm is based on the fact, proven in the paper, that any involution of the surface comes from an involution of the parameter space R2R2; therefore, by determining the latter, the former can be found. The algorithm has been implemented in the computer algebra system Maple 18. Evidence of its efficiency for moderate degrees, examples and a complexity analysis are also given.

Detecting Similarities of Rational Space Curves

We provide an algorithm to check whether two rational space curves are related by a similarity, i.e., whether they are equal up to position, orientation and scale. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Grobner bases for the special case of helical curves.

Algebraic surfaces invariant under scissor shears

Display Omitted Scissor shears are affine transformations in 3-space that, in analogy with the usual rotations, can be understood as hyperbolic rotations about a fixed line, in a fixed coordinate frame. We study algebraic surfaces invariant under scissor shears, and investigate their similarities and differences with the algebraic surfaces invariant under the usual rotations, namely the algebraic surfaces of revolution. In particular, we provide a necessary condition for an algebraic surface to be invariant under scissor shears, and we prove that such shear invariant surfaces can have either one, three, or infinitely many scissor axes. Furthermore, we characterize the surfaces with either three or infinitely many scissor axes. Additionally, in each case we show how to calculate the location of these scissor axes as well as the rest of the coordinate frame.

Recognizing projections of algebraic curves

Abstract Given two irreducible, algebraic space curves C 1 and C 2 , where C 2 is contained in some plane Π , we provide algorithms to check whether or not there exist perspective or parallel projections mapping C 1 onto C 2 , i.e. to recognize C 2 as the projection of C 1 . In the affirmative case, the algorithms provide the eye point(s) of the perspective transformation(s), or the direction(s) of the parallel projection(s). Although the problem is mainly discussed for rational curves, an algorithm for implicit curves is also given. The algorithms presented are mostly symbolic; nevertheless, we include an approximate algorithm for rational curves too.

Symmetry detection of rational space curves

We present a novel, deterministic, and efficient method to detect whether a given rational space curve C is symmetric. The method combines two ideas. On one hand in a similar way to [1], [2], if C is symmetric then the symmetry provides a second parametrization of the curve; furthermore, whenever the first parametrization is proper, i.e. injective except for finitely many parameter values, the latter is also proper and both are related by means of a Mobius transformation [3] that completely determines the symmetry. On the other hand, if C is symmetric then the curvature and torsion of C at corresponding points must coincide. By putting together these two ideas we can give an algorithm to directly find the Mobius transformations defining the symmetries of the curve. From here we can compute these symmetries and its characteristic elements (symmetry axes, symmetry planes, etc.) This completes and improves on an earlier method addressing a similar problem [3]. Keywords Symmetry Detection, Space Curves, Rational Curves References [1] Alcazar J.G. (2014), Efficient detection of symmetries of polynomially parametrized curves, Journal of Computational and Applied Mathematics vol. 255, pp. 715–724. [2] Alcazar J.G., Hermoso C., Muntingh G. (2014), Detecting Similarity of Plane Rational Plane Curves, Journal of Computational and Applied Mathematics, Vol. 269, pp. 1–13. [3] Alcazar J.G., Hermoso C., Muntingh G. (2014), Detecting Symmetries of Rational Plane and Space Curves, to appear in Computer Aided Geometric Design.

The Schubert Triangle Geometry on an Algebraic Surface

For a smooth complex projective surface, and for two families of curves with traditional singularities in it, we enumerate the pairs of curves in each family having two points of contact among them, thus generalizing the double contact formulae known or conjectured by Zeuthen and Schubert in the case of the complex projective plane. The technique we use to this purpose is a particular notion of triangle which can be defined in any smooth surface, thus potentially generalizing to arbitrary surfaces the Schubert technique of triangles.