arXiv subjects

Rimvydas Krasauskas

Publications and source records attributed to Rimvydas Krasauskas.

4 recordsLinked to original sources

arXiv2025

Formula for Dupin Cyclidic Cube and Miquel Point

Dupin cyclides are surfaces conformally equivalent to a torus, a circular cone, or a cylinder. Their patches admit rational bilinear quaternionic Bézier parametrizations and are used in geometric design and architecture. Dupin cyclidic cubes are a natural trivariate generalization of Dupin cyclide patches. In this article, we derive explicit formulas for control points and weights of rational trilinear quaternionic Bézier parametrizations of Dupin cyclidic cubes and relate them with the classical construction of the Miquel point.

math.AG↗

arXiv2024

Classification of Dupin Cyclidic Cubes by Their Singularities

Triple orthogonal coordinate systems having coordinate lines as circles or straight lines are considered. Technically, they are represented by trilinear rational quaternionic maps and are called Dupin cyclidic cubes, naturally generalizing the bilinear rational quaternionic parametrizations of principal patches of Dupin cyclides. Dupin cyclidic cubes and their singularities are studied and classified up to Möbius equivalency in Euclidean space.

math.AG↗

arXiv2015

Surfaces containing two circles through each point and Pythagorean 6-tuples

We study analytic surfaces in 3-dimensional Euclidean space containing two circular arcs through each point. The problem of finding such surfaces traces back to the works of Darboux from XIXth century. We reduce finding all such surfaces to the algebraic problem of finding all Pythagorean 6-tuples of polynomials. The reduction is based on the Schicho parametrization of surfaces containing two conics through each point and a new approach using quaternionic rational parametrization.

math.AG↗

arXiv2003

Universal Rational Parametrizations and Toric Varieties

This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal rational parametrizations of certain projective varieties. We give numerous examples and then discuss what happens in the singular case. We also describe rational maps to smooth toric varieties.

math.AG↗