arXiv subjects

Takeo Noda

Publications and source records attributed to Takeo Noda.

5 recordsLinked to original sources

arXiv2023

Billiards in a circle with trajectories circumscribing a triangle

Dogru and Tabachnikov in 2003 explored the polygonal outer billiard map in the hyperbolic plane and introduced a class of convex polygons called 'large'. They particularly sought conditions for a triangle to be classified as large. For a large triangle, there exist two triangles that are circumscribed around it and inscribed within the unit circle. In the Klein-Beltrami model of hyperbolic geometry, we reformulate the conditions for a triangle to be classified as 'large' in a more Euclidean geometric manner. A proposed measure of its Euclidean geometric size when the triangle is considered 'large' is introduced, and an evaluation of this measure is conducted. We also provide an explicit formula for an isosceles triangle.

math.DS↗

arXiv2023

Star-shaped trajectories of certain billiards around a triangle

We explore the triangle outer billiards map in points at infinity in the hyperbolic plane, focusing on the rotation number. Building on Dogru and Tabachnikov's work, which established the conditions for triangles where the rotation number of the billiard map is $1/3$, we examine cases where the rotation number is $2/5$. We provide a sufficient condition for this rotation number and show its necessity for large isosceles triangles. The results are framed within the context of the Beltrami-Klein model. We concludes with a conjecture based on the findings.

math.DS↗

arXiv2020

A remark on Chapple-Euler Theorem in non Euclidean geometry

In non-Euclidean geometry, there are several known correspondings to Chapple-Euler Theorem. This remark shows that those results yield expressions corredponding to the well-known formula $d=\sqrt{R(R-2r)}$.

math.MG↗

arXiv2019

On $3$-dimensional foliated dynamical systems and Hilbert type reciprocity law

We show some fundamental results concerning $3$-dimensional foliated dynamical systems (FDS$^3$ for short) introduced by Deninger. Firstly, we give a decomposition theorem for an FDS$^3$, which yields a classification of FDS$^3$'s. Secondly, for each type of the classification, we construct concrete examples of FDS$^3$'s. Finally, by using the integration theory for smooth Deligne cohomology, we introduce geometric analogues of local symbols and show a Hilbert type reciprocity law for an FDS$^3$. Our results answer the question posed by Deninger.

math.DS↗

arXiv2009

Homotopy classes of total foliations and bi-contact structures on three-manifolds

On every compact and orientable three-manifold, we construct total foliations (three codimension 1 foliations that are transverse at every point). This construction can be performed on any homotopy class of plane fields with vanishing Euler class. As a corollary we obtain similar results on bi-contact structures.

math.GT↗