Ruled surfaces and hyper-dual tangent sphere bundle

TL;DR

This paper explores the properties of ruled surfaces and hyper-dual tangent sphere bundles, establishing isomorphisms and developability conditions within hyper-dual vector spaces and their geometric interpretations.

Contribution

It introduces the hyper-dual sphere in hyper-dual vectors, proves an isomorphism with tangent bundles, and relates ruled surfaces in hyper-dual spaces to those in Euclidean space.

Findings

Hyper-dual sphere $S_{ ext{D}_2}$ defined and characterized.

Isomorphism between $S_{ ext{D}_2}^2$ and tangent bundle $TS_{ ext{D}}^2$ established.

Developability condition for ruled surfaces in hyper-dual space derived.

Abstract

In this study, we define the unit hyper-dual sphere $S_{\mathbb{D} _{2}}$ in hyper-dual vectors $\mathbb{D}_{2}$ and we give E-Study map version in $\mathbb{D}_{2}$ which prove that $S_{\mathbb{D} _{2}}^{2} $ is isomorphism to the tangent bundle $TS_{\mathbb{D} }^{2}.$ Next, we define ruled surfaces in $\mathbb{D}$, we give its developability condition and a geometric interpretation in $\mathbb{R}^{3}$ of any curves in $\mathbb{D}_{2}$. Finally, we present a relationship between a ruled surfaces set in $\mathbb{R}^{3}$ and curves in hyper dual vectors $\mathbb{D}_{2}$. We close each study with examples.