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

Abstract

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.