The inverse surfaces of tangent developables with respect to s_{c}(r)

TL;DR

This paper introduces the concept of inverse surfaces of tangent developables with respect to a sphere, deriving their geometric properties and conditions for flatness and minimality.

Contribution

It defines inverse surfaces of tangent developables relative to a sphere and computes their curvatures, Christoffel symbols, and shape operator, providing new geometric insights.

Findings

Derived the curvatures of inverse tangent developable surfaces.

Established conditions for the inverse surface to be flat.

Established conditions for the inverse surface to be minimal.

Abstract

In this paper, we define the inverse surface of a tangent developable surface with respect to the sphere S_{c}(r) with the center $c\in \mathbb{E}^{3}$ and the radius r in 3-dimensional Euclidean space $\mathbb{E}^{3}$. We obtain the curvatures, the Christoffel symbols and the shape operator of this inverse surface by the help of these of the tangent developable surface. Morever, we give some necessary and sufficient conditions regarding the inverse surface being flat and minimal.