Focal Representation of k-slant Helices in E^{m+1}

TL;DR

This paper explores the relationship between k-slant helices and their focal representations in Euclidean space, revealing that the focal representation of a k-slant helix is an (m-k+2)-slant helix, thus connecting geometric properties.

Contribution

It establishes a new geometric link between k-slant helices and their focal representations in Euclidean space.

Findings

Focal representation of a k-slant helix is an (m-k+2)-slant helix.

Provides a characterization of focal representations of k-slant helices.

Enhances understanding of the geometric structure of helices in higher dimensions.

Abstract

A focal representation of a generic regular curve {\gamma} in E^{m+1} consists of the centers of the osculating hyperplanes. A k-slant helix {\gamma} in E^{m+1} is a (generic) regular curve whose unit normal vector V_{k} makes a constant angle with a fixed direction U in E^{m+1}. In the present paper we proved that if {\gamma} is a k-slant helix in E^{m+1}, then the focal representation C_{{\gamma}} of {\gamma} in E^{m+1} is a (m-k+2)-slant helix in E^{m+1}.