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Light pillars over rippled water: why their sides appear parallel
[Submitted on 16 Jun 2026] · 2026-06-18 · via math updates on arXiv.org

Mathematics > Differential Geometry

arXiv:2606.18547 (math)

[Submitted on 16 Jun 2026]

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Abstract:Streetlights shining across rippled water often produce tall, narrow reflections with strikingly parallel sides. These light pillars appear almost architectural, yet the water surface is neither vertical nor smooth. We develop a geometric optics model that explains the phenomenon using the specular reflection rule, projection geometry, and the physics and statistics of surface slopes. A rippled water surface can be viewed as an ensemble of small facets (tangent planes on waves) acting as tiny mirrors. As one looks farther across the water (coordinate x), the physical width of the region whose facets reflect the light into the eye (or camera/pinhole) increases, but the pinhole projection onto the image plane compresses this widening by a factor proportional to 1/x. The two effects cancel, producing a reflection whose image width remains constant. This paper appears to be the first to formally document this. (Deviations from the strict cancellation at both ends of the image are also explained.) The analysis clarifies how earlier qualitative treatments failed through a lack of identifying the role of projection geometry. The model also qualitatively explains the brightness variation along the pillar, and why extended objects such as buildings do not produce elongated reflections. The treatment is intended for scholars of introductory optics and for those interested in the physics and mathematics of everyday visual phenomena. We conclude by outlining how the same framework extends naturally to more complex viewing situations, such as the setting sun viewed from a cliff top over the sea.

Submission history

From: Philip Kuchel Professor [view email]
[v1] Tue, 16 Jun 2026 23:55:13 UTC (513 KB)

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