Trilinear mappings appear naturally when performing spatial isogeometric discretizations of degree . Among them, birational mappings are characterized by the property that both the mapping and the associated inverse mapping are rational and thus easy to evaluate. These mappings have recently been analyzed by Busé et al. (2023) . Among other results, the authors provide a classification of these mappings over the field of complex numbers. The parameter lines of trilinear mappings form three two-parameter systems of straight lines, and thus it is promising to analyze these mappings with the tools provided by the field of line geometry, which is a classical branch of higher geometry (Pottmann and Wallner, 2001) . Indeed, in the birational case, the three systems of lines form space-filling line congruences associated with rational mappings that can be used to parameterize certain algebraic surfaces (Jüttler and Rittenschober, 2003) . Moreover, the three systems are closely related, and based on these observations we will present a geometric discussion of the results of Busé et al. (2023) together with a more detailed analysis of the classification over the field of real numbers.

Abstract For every graph that is minimally rigid in the plane, its Galois group is defined as the Galois group generated by the coordinates of its planar realizations, assuming that the edge lengths are transcendental and algebraically independent. In this paper we compute the Galois group of all minimally rigid graphs that can be constructed from a single edge by repeated Henneberg 1-steps. It turns out that any such group is totally imprimitive, i.e., it is determined by all the partitions it preserves.

We describe a new de Casteljau–type algorithm for complex rational Bézier curves. After proving that these curves exhibit the maximal possible circularity, we construct their points via a de Casteljau–type algorithm over complex numbers. Consequently, the line segments that correspond to convex linear combinations in affine spaces are replaced by circular arcs. In difference to the algorithm of Sánchez-Reyes (2009), the construction of all the points is governed by (generically complex) roots of the denominator, using one of them for each level. Moreover, one of the bi-polar coordinates is fixed at each level, independently of the parameter value. A rational curve of the complex degree n admits generically n! distinct de Casteljau–type algorithms, corresponding to the different orderings of the denominator's roots.

We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation of realizations allows us to provide a combinatorial characterization of these graphs in terms of the existence of particular colorings of the edges. Moreover, we determine necessary relations for flexibility between the spherical lengths of the edges. We conclude by classifying all possible motions on the sphere of the complete bipartite graph with 3+3 vertices where no two vertices coincide or are antipodal.

Answering connectivity queries in semi-algebraic sets is a longstanding and challenging computational issue with applications in robotics, in particular for the analysis of kinematic singularities. One task there is to compute the number of connected components of the complementary of the singularities of the kinematic map. Another task is to design a continuous path joining two given points lying in the same connected component of such a set. In this paper, we push forward the current capabilities of computer algebra to obtain computer-aided proofs of the analysis of the kinematic singularities of various robots used in industry.

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least $$n^2/4 - O(n)$$ ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most $$n^3/24 - O(n^2)$$ circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most $$Kn^2$$ ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

We give an algorithm for deciding whether a given Del Pezzo surface of degree 6 is rational over the ground field, and for computing a proper parametrisation in the affirmative case. This problem is reduced to the simultanuous solution of two norm equations.

Given a rational parametrization of an algebraic surface, we try to reduce the degree by a suitable reparametrization. We give an algorithm that produces a parametrization with a degree that is at most twice the minimal degree. The problem is closely related to the simplification of linear systems of plane curves by Cremona transformations.

This paper contains several improvements of Villamayor's algorithm for the problem of resolution of the singularities of a hypersurface. The first improves the management of the charts which represent the blown up variety. The second improves the way how new resolution problems are created in the recursion, based on Hironaka's theory of idealistic exponents. The remaining two improve the way how discrete information is used, based on the adaption of Encinas and Villamayor of Abhyankar's theory of good points.

A canal surface S, generated by a parametrized curve m(t), in R3 is the envelope of the set of spheres with radius r(t) centered at m(t). This concept generalizes the classical offsets (for r(t) = const) of plane curves. In this paper we develop elementary symbolic methods for generating a rational parametrization of canal surfaces generated by rational curves m(t) with rational radius variation r(t). In a pipe surface r(t) is constant.

We give an algorithm for the following problem: given a surface defined over Q, and a rational pencil on it, compute a proper parametrization with coefficients in Q.

The parameterization problem asks for a real parameterization of an implicitly given real algebraic surface, in terms of rational functions in two v ariables.We give an algorithm for the parameterization of tubular surfaces.Also, it is shown that many instances of the parameterization problem can be reduced to the tubular case.