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

Fabrizio Patuzzo

TL;DR

The paper reframes the medieval Arab formula for the shadow of a vertical gnomon using elementary vector algebra, replacing traditional spherical-trigonometric derivations with simple rotations of the Sun direction, the gnomon, and the dial plane. It shows that the shadow-tip trajectory is the intersection of a sun-ray cone with the local horizontal plane, yielding a conic section whose type is governed by $\Delta \propto \cos^{2}\lambda - \sin^{2}\delta$. Beyond reproducing the Arab result, the vector approach derives related sundial relations such as sunrise/sunset times, solar altitude, and hour angles, and provides a pedagogically transparent framework for general sundial surfaces. The work thus clarifies the underlying geometry of gnomonics and offers a compact, extensible method for sundial construction and analysis.

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.

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

TL;DR

The paper reframes the medieval Arab formula for the shadow of a vertical gnomon using elementary vector algebra, replacing traditional spherical-trigonometric derivations with simple rotations of the Sun direction, the gnomon, and the dial plane. It shows that the shadow-tip trajectory is the intersection of a sun-ray cone with the local horizontal plane, yielding a conic section whose type is governed by . Beyond reproducing the Arab result, the vector approach derives related sundial relations such as sunrise/sunset times, solar altitude, and hour angles, and provides a pedagogically transparent framework for general sundial surfaces. The work thus clarifies the underlying geometry of gnomonics and offers a compact, extensible method for sundial construction and analysis.

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.
Paper Structure (15 sections, 18 equations)

This paper contains 15 sections, 18 equations.

Table of Contents

  1. Introduction
  2. The Arab formula
  3. A Simple Case: the North Pole at the Equinox
  4. The Sun’s Direction Vector
  5. The Arab Formula when $\lambda=90^{\circ}$
  6. The Gnomon and Local Horizontal Plane
  7. The Arab Formula via Vector Algebra
  8. Interpretation and Conic Classification
  9. From the vector equation to the conic form
  10. Classification of the shadow path
  11. Relation to Other Sundial Formulas
  12. Sunrise and sunset times
  13. Solar altitude
  14. Hour angle
  15. Conclusion