A Vector-Algebraic Reconstruction of the Medieval Arab Formula for the Shadow of a Gnomon

Abstract

We present a modern reconstruction of the classical formula, first derived by medieval Arab astronomers, that describes the trajectory of the tip of a gnomon's shadow during the day as a function of latitude, solar declination, and gnomon height. Unlike the traditional derivations based on spherical trigonometry, our approach uses only elementary vector algebra and rotation matrices. The same vector framework naturally extends to other classical relations used in sundial construction.