arXiv subjects

Atsuki Hiramatsu

Publications and source records attributed to Atsuki Hiramatsu.

2 recordsLinked to original sources

arXiv2026

Parallel Surfaces of Cuspidal Cross Caps

This paper investigates the geometry and singularities of parallel surfaces of cuspidal cross caps, the fundamental non-front frontal singularities. We establish a criterion for the degeneracy of the distance squared function in terms of known geometric invariants and describe the resulting configuration of singularities. Our main result demonstrates that while the parallel surface is generically $\mathcal{A}$-equivalent to a cuspidal cross cap, it degenerates into a degenerated cuspidal $S_1$ singularity at specific distances characterized by the equation $C_2(\varepsilon)=0$. These distances act as a novel analogue of the principal radii of curvature. Indeed, although the Gaussian and mean curvatures diverge at the singularity, their asymptotic expansions reveal that their constant terms correspond to the product and average of the reciprocals of these distances, respectively.

math.DG↗

arXiv2023

Versality of Rotation Unfolding of Folding Maps for Surfaces in $\mathbb{R}^3$

We introduce the rotation unfolding of the folding map of a surface in $\mathbb{R}^3$, and investigate its $\mathcal{A}$-vesality. The rotation unfolding is a 2-parameter unfolding and can be considered as a subfamily of the folding family, which is introduced by Bruce and Wilkinson. They revealed relationships between a bifurcation set of this family and the focal/symmetry set of a surface in $\mathbb{R}^3$. We state the criteria of singularities of the folding map up to codimension 2 and prove when our rotation unfolding is versal. The conditions to be versal are stated in terms of geometry. As a by-product, we show the diffeomorphic type of the locus of the tangent planes of the focal set of regular surfaces, which passes through the origin.

math.DG↗