Geodesics on Surfaces with Helical Symmetry: Cavatappi Geometry

TL;DR

This paper explores geodesics on helical tubular surfaces, generalizing surfaces of revolution, and illustrates their properties through the example of cavatappi pasta, highlighting the mathematical and physical aspects of such geometries.

Contribution

It introduces a family of helical surfaces with symmetry, providing a pedagogical framework for studying geodesics beyond simple revolution surfaces.

Findings

Characterization of geodesics on helical surfaces

Effective potential approach for geodesic analysis

Application to the geometry of cavatappi pasta

Abstract

A 3-parameter family of helical tubular surfaces obtained by screw revolving a circle provides a useful pedagogical example of how to study geodesics on a surface that admits a 1-parameter symmetry group, but is not as simple as a surface of revolution like the torus which it contains as a special case. It serves as a simple example of helically symmetric surfaces which are the generalizations of surfaces of revolution in which an initial plane curve is screw-revolved around an axis in its plane. The physics description of geodesic motion on these surfaces requires a slightly more involved effective potential approach than the torus case due to the nonorthogonal coordinate grid necessary to describe this problem. Amazingly this discussion allows one to very nicely describe the geodesics of the surface of the more complicated ridged cavatappi pasta.