Related papers: Degree-optimal moving frames for rational curves
We develop a theory and an algorithm for constructing minimal-degree polynomial moving frames for polynomial curves in an affine space. The algorithm is equivariant under volume-preserving affine transformations of the ambient space and the…
This paper investigates the construction of rational motions of a minimal quaternionic degree that generate a prescribed plane trajectory (a ``rational torse''). Using the algebraic framework of dual quaternions, we formulate the problem as…
Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically,…
The trajectories of a rational motion given by a polynomial of degree n in the dual quaternion model of rigid body displacements are generically of degree 2n. In this article we study those exceptional motions whose trajectory degree is…
The main purpose of this paper is to define dynamical degrees for rational maps over an algebraic closed field of characteristic zero and prove some basic properties (such as log-concavity) and give some applications. We also define…
In this paper, we study how to find rational motions that move a line along a given rational ruled surface. Our goal is to find motions with the lowest possible degree using dual quaternions. While similar problems for point trajectories…
We give a constructive proof for the existence of a unique rational motion of minimal degree in the dual quaternion model of Euclidean displacements with a given rational parametric curve as trajectory. The minimal motion degree equals the…
In this paper, we construct rotating frames for curves, including plane curves, space curves and curves on surfaces. Hence, the behaviour of an arbitrary moving point on a curve can be seen as the composite of linear motion and rotation.…
We present a method for constructing all bounded rational motions that frame a space curve $\mathbf{r}(t)$. This means that the motion guides an orthogonal frame along the curve such that one frame axis is in direction of the curve tangent.…
We study arithmetic degree of a dominant rational self-map on a smooth projective variety over a function field of characteristic zero. We see that the notion of arithmetic degree and some related problems over function fields are…
Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on…
We identify a common scheme in several existing algorithms addressing computational problems on linear differential equations with polynomial coefficients. These algorithms reduce to computing a linear relation between vectors obtained as…
The article presents a new method of linear programming, called the surface movement method. This method constructs an optimal objective path on the surface of the feasible polytope from the initial boundary point to the point at which the…
A (fully) dynamic graph algorithm is a data structure that supports edge insertions, edge deletions, and answers certain queries that are specific to the problem under consideration. There has been a lot of research on dynamic algorithms…
We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination…
In this work, we solve a discrete optimal transport problem in a nonuniform environment. To solve the optimal transport problem, we build the cost matrix and then use classical solvers for discrete optimal transport. The challenge is to…
We deal with the following closely related problems: (i) For a germ of a reduced plane analytic curve, what is the minimal degree of an algebraic curve with a singular point analytically equivalent (isomorphic) to the given one? (ii) For a…
This paper presents a framework for fast and robust motion planning designed to facilitate automated driving. The framework allows for real-time computation even for horizons of several hundred meters and thus enabling automated driving in…
We provide criteria 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. These criteria reduce the projection problem to a certain…
Molecular-orbital-based machine learning (MOB-ML) enables the prediction of accurate correlation energies at the cost of obtaining molecular orbitals. Here, we present the derivation, implementation, and numerical demonstration of MOB-ML…