Related papers: Euclidean and Affine Curve Reconstruction
We study the structure of Jacobians of geometrically reduced curves over arbitrary (i. e., not necessarily perfect) fields. We show that, while such a group scheme cannot in general be decomposed into an affine and an Abelian part as over…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf…
Planning trajectories for nonholonomic systems is difficult and computationally expensive. When facing unexpected events, it may therefore be preferable to deform in some way the initially planned trajectory rather than to re-plan entirely…
We explicitly construct the algebraic model of affine Jacobian of a generic algebraic curve of high genus and use it to compute the Euler characteristic of the Jacobian and investigate its structure.
In this paper we discuss the variety of planar spiral segments and their applications in objects in both the real and artificial world. The discussed curves with monotonic curvature function are well-known in geometric modelling and…
In recent years, neural implicit representations have made remarkable progress in modeling of 3D shapes with arbitrary topology. In this work, we address two key limitations of such representations, in failing to capture local 3D geometric…
The renormalization-group improved effective potential ---to leading-log and in the linear curvature approximation--- is constructed for ``finite'' theories in curved spacetime. It is not trivial and displays a quite interesting,…
We discuss some research problems on affine monomial curves, from the perspective of computation.
We introduce and analyze parallelizable algorithms to compress and accurately reconstruct finite simplicial complexes that have non-trivial automorphisms. The compressed data -- called a complex of groups -- amounts to a functor from (the…
We compute the presentations of fundamental groups of the complements of a class of rational cuspidal projective plane curves classified by Flenner, Zaidenberg, Fenske and Saito. We use the Zariski-Van Kampen algorithm and exploit the…
We are interested in unsupervised reconstruction of complex near-capillary vasculature with thousands of bifurcations where supervision and learning are infeasible. Unsupervised methods can use many structural constraints, e.g. topology,…
We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. The motivation comes from the problem of establishing a…
A general scheme for determining and studying integrable deformations of algebraic curves is presented. The method is illustrated with the analysis of the hyperelliptic case. An associated multi-Hamiltonian hierarchy of systems of…
We report on results about a study of algebraic graph invariants, based on computer exploration, and motivated by graph-isomorphism and reconstruction problems.
Several classes of *-algebras associated to the action of an affine transformation are considered, and an investigation of the interplay between the different classes of algebras is initiated. Connections are established that relate…
It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance $\epsilon>0$ and an $\epsilon$-irreducible algebraic affine plane curve $\mathcal C$ of…
The Euclidean distance degree of an algebraic variety is a well-studied topic in applied algebra and geometry. It has direct applications in geometric modeling, computer vision, and statistics. We use non-proper Morse theory to give a…
We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The…
We revisit the well-known Curve Shortening Flow for immersed curves in the $d$-dimensional Euclidean space. We exploit a fundamental structure of the problem to derive a new global construction of a solution, that is, a construction that is…