Related papers: A minimum-energy quadratic curve through three poi…
We characterise, in terms of Dixmier-Ohno invariants, the types of singularities that a plane quartic curve can have. We then use these results to obtain new criteria for determining the stable reduction types of non-hyperelliptic curves of…
We study two minimization questions: the nature of curves $\gamma \subset \mathbb{H}$ which minimize the Loewner energy among all curves from 0 to a fixed $z_0 \in \mathbb{H}$, and the nature of $\gamma$ which minimize the Loewner energy…
We introduce a novel energy method that reinterprets ``curve shortening'' as ``tangent aligning''. This conceptual shift enables the variational study of infinite-length curves evolving by the curve shortening flow, as well as higher order…
In this paper, we address the problem of computing optimal paths through three consecutive points for the curvature-constrained forward moving Dubins vehicle. Given initial and final configurations of the Dubins vehicle, and a midpoint with…
We investigate minimizers and critical points for scale-invariant tangent-point energies ${\rm TP}^{p,q}$ of closed curves. We show that a) minimizing sequences in ambient isotopy classes converge to locally critical embeddings in all but…
We introduce a variational approach for extracting curves between a list of possible endpoints, based on the discretization of an energy and Smirnov's decomposition theorem for vector fields. It is used to design a bi-level minimization…
We provide explicit expressions for quadrature rules on the space of $C^1$ cubic splines with non-uniform, symmetrically stretched knot sequences. The quadrature nodes and weights are derived via an explicit recursion that avoids an…
We consider the approximation of minimal geodesics between two closed sets in $\mathbb{R}^D$ endowed with a smooth Riemannian metric. The continuous problem is formulated as the minimization of the energy functional over piecewise smooth…
This paper proposes a simple technique of curve and surface construction with B-splines. Given a control polygon or a control mesh together with node ordinates corresponding to all control points, a rational curve or surface is obtained by…
We study points of moderately low degree on a curve $C$ over a number field, which is embedded on a nice toric surface $S$. Recently, Smith and Vogt related the linear equivalence classes of such points to intersections of $C$ with curves…
We introduce a new concept for generating optimal quadrature rules for splines. Given a target spline space where we aim to generate an optimal quadrature rule, we build an associated source space with known optimal quadrature and transfer…
We use a tangent method approach to obtain the arctic curve in a model of non-intersecting lattice paths within the first quadrant, including a q-dependent weight associated with the area delimited by the paths. Our model is characterized…
Knot and link energies can be computed from sets of closed curves in three dimensional space, and each type of knot or link has a minimum energy associated with it. Here, we consider embeddings of links that locally or globally minimize the…
This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot. The most interesting phenomena found in these experiments is the dependence of the…
It is known that the Frenet-Serret apparatus of a space curve in three-dimensional Euclidean space determines the local geometry of curves. In particular, the Frenet-Serret apparatus specifies important geometric invariants, including the…
Let $\mathbb{K}$ be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree $n+1>3$ over $\mathbb{K}$, containing three points, $P_1,P_2,$ and $P_3$, such that $nP_1+P_2$, $nP_2+P_3$, and…
In this thesis, we consider the knot energy "integral Menger curvature" which is the triple integral over the inverse of the classic circumradius of three distinct points on the given knot to the power $p\in [2,\infty)$. We prove the…
In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles' Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter's construction. In addition,…
To construct a curve with a monotonic curvature (spiral), and given tangents and curvatures at the ends, the author proposed the following method. From given boundary conditions, the values of two inverse invariants are determined. Then, on…
We use a recent characterization of minimal value set polynomials and $q$- Frobenius nonclassical curves to construct curves that generalize the Hermitian curve. The genus $g$ and the number $N$ of $\mathbb{F}_q$-rational points of the…