Geometry and Photometry of the Penumbra: From Monge’s Fortifications to Whitney Singularities
Seminars of the Institut de Physique de Nice,
Abstract:
The penumbra is usually regarded as a simple blurred transition between shadow and light. Yet, as early as the eleventh century, Ibn al-Haytham (Alhazen) had already developed a quantitative theory of penumbral regions, later revisited by Johannes Kepler in his analysis of the camera obscura. In the eighteenth century, Leonhard Euler observed that shadow boundaries generated by extended sources are supported by singular ruled surfaces that are neither cones nor cylinders. Inspired by Euler, Gaspard Monge further developed a geometric theory of shadows and penumbras for convex sources and obstacles.
In this seminar, we investigate experimentally and theoretically the penumbras produced by filamentary light sources partially occulted by wire-like obstacles. Simple experiments using commercial filament bulbs reveal remarkably rich structures, including bright penumbral lines with cusp singularities, strongly reminiscent of optical caustics.
The problem is modeled through the geometry of rays emitted by the source and blocked by the obstacle before reaching the screen. We show that the observed patterns admit two complementary descriptions: one based on contact bifurcations between planar curves, and another based on the singularity theory of mappings from the plane to the plane, leading naturally to the stable singularities classified by Hassler Whitney.
