Equiangular surfaces, self-similar surfaces and the geometry of sea shells

TL;DR

This paper explores special 3D surfaces where the normal vector maintains a constant angle with the radius vector, extending the concept of equiangular spirals into three dimensions and analyzing their geometric properties.

Contribution

It introduces a new class of surfaces characterized by a constant angle between the normal and radius vectors, generalizing equiangular spirals to three-dimensional geometry.

Findings

Characterization of equiangular surfaces in 3D

Extension of planar spirals to spatial surfaces

Geometric properties and potential applications

Abstract

We investigate three-dimensional surfaces where the normal vector forms a constant angle with the radius vector. These surfaces naturally extend equiangular (logarithmic) spirals in the plane.