Efficient Detection of Symmetries of Polynomially Parametrized Curves

We formulate a simple algorithm for computing global exact symmetries of closed discrete curves in plane. The method is based on a suitable trigonometric interpolation of vertices of the given polyline and consequent computation of the symmetry group of the obtained trigonometric curve. The algorithm exploits the fact that the introduced unique assigning of the trigonometric curve to each closed discrete curve commutes with isometries. For understandable reasons, an essential...

We present a new approach using differential invariants to detect projective equivalences and symmetries between two rational parametric $3D$ curves properly parametrized. In order to do this, we introduce two differential invariants that commute with M\"obius transformations, which are the transformations in the parameter space associated with the projective equivalences between the curves. The M\"obius transformations are found by first computing the gcd of two polynomials ...

Given two rational, properly parametrized space curves ${\mathcal C}_1$ and ${\mathcal C}_2$, where $\CCC_2$ is contained in some plane $\Pi$, we provide an algorithm to check whether or not there exist perspective or parallel projections mapping $\CCC_1$ onto $\CCC_2$, i.e. to recognize $\CCC_2$ as the projection of $\CCC_1$. In the affirmative case, the algorithm provides the eye point(s) of the perspective transformation(s), or the direction(s) of the parallel projection(s...

We make use of the complex implicit representation in order to provide a deterministic algorithm for checking whether or not two implicit algebraic curves are related by a similarity, a central question in Pattern Recognition and Computer Vision. The algorithm has been implemented in the computer algebra system Maple 2015. Examples and evidence of the good practical performance of the algorithm are given.

In this paper a new intrinsic geometric characterization of the symmetric square of a curve and of the ordinary product of two curves is given. More precisely it is shown that the existence on a surface of general type S of irregularity q of an effective divisor D having self-intersection D^2>0 and arithmetic genus q implies that S is either birational to a product of curves or to the second symmetric product of a curve.

An expository description of smooth cubic curves in the real or complex projective plane.

We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau--Ginzburg models for Fano varieties; how to apply them to classification problems; and how to compute invariants of Fano varieties via Landau--Ginzburg models.

We present a method for computing all the symmetries of a rational ruled surface defined by a rational parametrization which works directly in parametric rational form, i.e. without computing or making use of the implicit equation of the surface. The method proceeds by translating the problem into the parameter space, and relies on polynomial system solving. If we want all the symmetries of the surface, including rotational symmetries, we need to deal with polynomial systems ...

The revised version has two additional references and a shorter proof of Proposition 5.7. This version also makes numerous small changes and has an appendix containing a proof of the degree formula for a parametrized surface.

Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we review several applications and discuss implementation and computational aspects.