Models from the 19th Century used for Visualizing Optical Phenomena and Line Geometry

TL;DR

This paper explores 19th-century models of quartic surfaces, particularly complex surfaces, used for visualizing optical phenomena and line geometry, highlighting their historical and mathematical significance.

Contribution

It analyzes the historical development and mathematical properties of complex quartic surfaces introduced by Plücker and their relation to optical caustics and wave surfaces.

Findings

Complex surfaces relate to caustic surfaces in optics

Kummer surfaces generalize Fresnel's wave surface

Historical models aid understanding of optical phenomena

Abstract

The main focus of this paper is on models of quartic surfaces, especially so-called complex surfaces. These are special fourth-degree surfaces that Julius Pl\"ucker introduced in the 1860s for visualizing the local structure of a quadratic line complex. Pl\"ucker's complex surfaces turned out to be closely related to Kummer surfaces and both of these types of quartics are examples of caustic surfaces, which arise in geometrical optics. Indeed, Kummer surfaces represent a natural generalization of the wave surface, first introduced by Augustin Fresnel to explain double refraction in biaxial crystals.