Related papers: Translation Covers Among Triangular Billiard Surfa…
A translational surface is a tensor product surface constructed from two space curves by translating one along the other. These surfaces are common within geometric modeling and, since their description is parametric, it is desirable to…
A translation surface is a surface formed by identifying edges of a collection of polygons in the complex plane that are parallel and of equal length using only translations. We determined that the same circle packing can be realized on…
Consider a collection of finitely many polygons in $\mathbb C$, such that for each side of each polygon, there exists another side of some polygon in the collection (possibly the same) that is parallel and of equal length. A translation…
Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat $3$-manifolds which we call translation prisms. Using ideas of Furstenberg and Veech, we connect results about weak mixing properties of…
A nice trick for studying the billiard flow in a rational polygon is to unfold the polygon along the trajectories. This gives rise to a translation or half-translation surface tiled by the original polygon, or equivalently an Abelian or…
In this paper we will classify those translation surfaces in E3 involving polynomials which are Weingarten surfaces. We analyze Weingarten translation surfaces satisfying 2aH + bK = 0. We study also other types of translation surfaces,…
The algebraic translational surface is a typical modeling surface in computer aided design and architecture industry. In this paper, we give a necessary and sufficient condition for that algebraic surface having a standard parametric…
In the present paper we deal with non-constant curvature Thurston geometries \cite{M97}, \cite{S}, \cite{Sz22-3},\cite{W06}. We define and determine the generalized trans\-lation-like Apollonius surfaces and thus also bisector surfaces as a…
Rational polygonal billiards are one of the key models among the larger class of pseudo-integrable billiards. Their billiard flow may be lifted to the geodesic flow on a translation surface. Whereas such classical billiards have been much…
An oblivious point on a translation surface is a point with no closed geodesic passing through it. Nguyen, Pan, and Su (2017) showed that there are at most finitely many oblivious points on any given translation surface and constructed a…
A planar polygonal billiard $\P$ is said to have the finite blocking property if for every pair $(O,A)$ of points in $\P$ there exists a finite number of ``blocking'' points $B_1, ..., B_n$ such that every billiard trajectory from $O$ to…
A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with…
In this paper, we show that billiard orbits in rational polygons and geodesics on translation surfaces exhibit super-fast spreading, an optimal time-quantitative majority property about the corresponding linear flow that implies uniformity…
Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces…
There are only a few invariants one classically associates with precompact translation surfaces, among them certain numberfields, i.e. fields which are finite extensions of the field Q of rational numbers. These fields are closely related…
In this paper we study the Nil geometry that is one of the eight homogeneous Thurston 3-geomet\-ri\-es. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a…
We give a relatively simple proof that a translation surface in Euclidean space that satisfies a relation of type $aH+bK=c$, for some real numbers $a,b,c$, where $H$ and $K$ are the mean curvature and the Gauss curvature of the surface,…
We prove an effective version of a celebrated result of Eskin and Masur: for any affine invariant manifold of translation surfaces, almost every translation surface has quadratic growth for the saddle connection holonomy vectors, with an…
We investigated singular points of translation surfaces under the linearly independent condition. In this paper, as completion, we investigate singular points of translation surfaces under the linearly dependent condition, using the…
Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting…