From Golden Spirals to Constant Slope Surfaces

TL;DR

This paper classifies all constant slope surfaces in Euclidean 3-space, which are surfaces where the position vector maintains a constant angle with the surface normal, extending the concept of generalized helices.

Contribution

It provides a complete characterization and parametric equations for all constant slope surfaces in Euclidean 3-space.

Findings

Derived parametric equations for constant slope surfaces.

Identified these surfaces as bi-dimensional analogues of generalized helices.

Visual representations of the surfaces are included.

Abstract

In this paper, we find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point. These surfaces could be thought as the bi-dimensional analogue of the generalized helices. Some pictures are drawn by using the parametric equations we found.