The Natural 3D Spiral

Abstract

Logarithmic spirals are ubiquitous in nature. This paper presents a novel mathematical definition of a 3D logarithmic spiral, which provides a proper description of objects found in nature. To motivate our work, we scanned spiral-shaped objects and studied their geometric properties. We consider the extent to which the existing 3D definitions capture these properties. We identify a property that is shared by the objects we investigated and is not satisfied by the existing 3D definitions. This leads us to present our definition in which both the radius of curvature and the radius of torsion change linearly along the curve. We prove that our spiral satisfies several desirable properties, including invariance to similarity transformations, smoothness, symmetry, extensibility, and roundness. Finally, we demonstrate the utility of our curves in the modeling of several animal structures.