Masatomo Takahashi

Bio: Masatomo Takahashi is an academic researcher from Muroran Institute of Technology. The author has contributed to research in topics: Euclidean space & Legendre polynomials. The author has an hindex of 16, co-authored 74 publications receiving 859 citations. Previous affiliations of Masatomo Takahashi include Hokkaido University.

Existence and uniqueness for Legendre curves

Abstract: We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.

Horospherical flat surfaces in Hyperbolic 3-space

Abstract: Recently we discovered a new geometry on submanifolds in hyperbolic $n$-space which is called {\it horospherical geometry} Unfortunately this geometry is not invariant under the hyperbolic motions (it is invariant under the canonical action of $SO(n)$), but it has quite interesting features For example, the flatness in this geometry is a hyperbolic invariant and the total curvatures are topological invariants In this paper, we investigate the {\it horospherical flat surfaces} (flat surfaces in the sense of horospherical geometry) in hyperbolic $3$-space Especially, we give a generic classification of singularities of such surfaces As a consequence, we can say that such a class of surfaces has quite a rich geometric structure

Framed curves in the Euclidean space

Abstract: Abstract A framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We define smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.

Horospherical flat surfaces in Hyperbolic 3-space

Abstract: Recently we discovered a new geometry on submanifolds in hyperbolic n-space which is called horospherical geometry. Unfortunately this geometry is not invariant under the hyperbolic motions (it is invariant under the canonical action of SO(n)), but it has quite interesting features. For example, the flatness in this geometry is a hyperbolic invariant and the total curvatures are topological invariants. In this paper, we investigate the horospherical flat surfaces (flat surfaces in the sense of horospherical geometry) in hyperbolic 3-space. Especially, we give a generic classification of singularities of such surfaces. As a consequence, we can say that such a class of surfaces has quite a rich geometric structure.

The geometry of fronts

Abstract: We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges Moreover, it is closely related to the behavior of the Gaussian curvature of a surface near cuspidal edges and swallowtails

Singularities of maximal surfaces

Abstract: We show that the singularities of spacelike maximal surfaces in Lorentz–Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap.