arXiv subjects

Shun Kumagai

Publications and source records attributed to Shun Kumagai.

5 recordsLinked to original sources

arXiv2026

The Klein-Lie W-curves as a new class of aesthetic curves based on self-affinity

In this paper, we consider planar curves and present a family of planar curves characterized by a symmetry called the extendable self-affinity (ESA). The ESA has been recognized through the investigation of the symmetry of the log-aesthetic curve (LAC), which has been studied as a reference for designing aesthetic shapes in CAGD and regarded as an analog of Euler's elastica in similarity geometry. We investigate the ESA and show that it gives rise to affine geometry and the Klein-Lie W-curve. Based on the observations on logarithmic curvature graphs, we propose the Klein-Lie W-curve as a new family of "aesthetic curves" in affine geometry, to be considered together with the LAC.

math.DG↗

arXiv2024

Self-affinities of planar curves: towards unified description of aesthetic curves

In this paper, we consider the self-affinity of planar curves. It is regarded as an important property to characterize the log-aesthetic curves which have been studied as reference curves or guidelines for designing aesthetic shapes in CAD systems. We reformulate the two different self-affinities proposed in the development of log-aesthetic curves. We give rigorous proof that one self-affinity actually characterizes log-aesthetic curves, while another one characterizes parabolas. We then propose a new self-affinity which, in equiaffine geometry, characterizes the constant curvature curves (the quadratic curves). It integrates the two self-affinities, by which constant curvature curves in similarity and equiaffine geometries are characterized in a unified manner.

math.DG↗

arXiv2023

Calculation of Veech groups and Galois invariants of general origamis

Nontrivial examples of Teichmüller curves have been studied systematically with notions of combinatorics invariant under affine homeomorphisms. An origami (square-tiled surface) induces a Teichmüller curve for which the absolute Galois group acts on the embedded curve in the moduli space. In this paper, we study general origamis not admitting pure half-translation structure. Such a flat surface is given by a cut-and-paste construction from origami that is a translation surface. We present an algorithm for the simultaneous calculation of the Veech groups of origamis of given degree. We have calculated the equivalence classes, the $PSL(2,\mathbb{Z})$-orbits, and some Galois invariants for all the patterns of origamis of degree $d\leq 7$.

math.GT↗

arXiv2022

General origamis and Veech groups of flat surfaces

In this century, a square-tiled translation surface (an origami) is intensively studied as an object with special properties of its translation structure and its $SL(2,\mathbb{R})$-orbit embedded in the moduli space. We generalize this concept in the language of flat surfaces appearing naturally in the Teichmüller theory. We study the combinatorial structure of origamis and show that a certain system of linear equations realizes the flat surface in which rectangles of specified moduli replace squares of an origami. This construction gives a parametrization of the family of flat surfaces with two finite Jenkins-Strebel directions for each combinatorial structure of two-directional cylinder decomposition. Moreover, we obtain the inclusion of Veech groups of such flat surfaces under a covering relation with specific branching behavior.

math.GT↗

arXiv2020

A characterization of Veech groups in terms of origamis

Schmithüsen proved in 2004 that the Veech group of an origami is closely related to a subgroup of the automorphism group of the free group $F_2$. This result is significant in the sense that the framework of approachable Veech groups is greatly extended. In this paper, we continue the analysis and consider what kind of settings of flat surfaces allow Veech groups to be characterized combinatorially like origamis. We show that elements in the Veech group of a flat surface with two finite Jenkins-Strebel directions are characterized to allow a concurrence between two `origamis' defined by geodesics in the surface. In the proof we use an observation presented by Earle and Gardiner that a flat surface with two finite Jenkins-Strebel directions is decomposed into a finite number of parallelograms and is proved to be of finite analytic type. Using our results we can decide whether a matrix belongs to the Veech group for various kinds of flat surfaces of finite analytic type.

math.GT↗