A new characterization of canal surfaces with parallel transport frame in Euclidean space $\mathbb{E}^{4}$

TL;DR

This paper investigates the geometric properties of canal surfaces in four-dimensional Euclidean space using the parallel transport frame, providing curvature analysis, examples, and visualizations.

Contribution

It introduces a new characterization of canal surfaces in $ extbf{E}^4$ using the parallel transport frame, including curvature properties and specific surface classifications.

Findings

Curvature properties depend on principal curvature functions $k_1$, $k_2$, $k_3$.

If the spine curve is straight, the surface is a Weingarten canal surface.

Visualization of projected canal surfaces in $ extbf{E}^3$ is provided.

Abstract

In this study, we consider canal surfaces according to parallel transport frame in Euclidean space $\mathbb{E}^{4}$. The curvature properties of these surfaces are investigated with respect to $k_{1}$, $k_{2}$ and $k_{3}$ which are principal curvature functions according to parallel transport frame. We also give an example of canal surfaces in $\mathbb{E}^{4}.$ Further, we point out that if spine curve $\gamma $ is a straight line, then $M$ is a Weingarten canal surface and also $M$ is a linear Weingarten tube surface. Finally, the visualization of the projections of canal surfaces in $\mathbb{E}^{3}$ are shown.