Nicolas-Auguste Tissot: A link between cartography and quasiconformal theory

TL;DR

This paper explores Nicolas-Auguste Tissot's pioneering work on map distortion and its deep connections to the mathematical theory of quasiconformal mappings, highlighting his overlooked contributions to the field.

Contribution

It uncovers and contextualizes Tissot's early mathematical insights on map distortion, linking them to modern quasiconformal mapping theory.

Findings

Tissot introduced the indicatrix for map distortion visualization.

His work predates and influences quasiconformal mapping theory.

Tissot's contributions have been historically underrecognized.

Abstract

Nicolas-Auguste Tissot (1824--1897) published a series of papers on cartography in which he introduced a tool which became known later on, among geographers, under the name of the "Tissot indicatrix." This tool was broadly used during the twentieth century in the theory and in the practical aspects of the drawing of geographical maps. The Tissot indicatrix is a graphical representation of a field of ellipses on a map that describes its distortion. Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane that are used in drawing geographical maps, and more generally he developed a theory for the distorsion of mappings between general surfaces. His ideas are at the heart of the work on quasiconformal mappings that was developed several decades after him by Gr{\"o}tzsch, Lavrentieff, Ahlfors and Teichm{\"u}ller. Gr{\"o}tzsch mentions the work of Tissot and he uses the terminology related to his name (in particular, Gr{\"o}tzsch uses the Tissot indicatrix). Teichm{\"u}ller mentions the name of Tissot in a historical section in one of his fundamental papers where he claims that quasiconformal mappings were used by geographers, but without giving any hint about the nature of Tissot's work. The name of Tissot is also missing from all the historical surveys on quasiconformal mappings. In the present paper, we report on this work of Tissot. We shall also mention some related works on cartography, on the differential geometry of surfaces, and on the theory of quasiconformal mappings. This will place Tissot's work in its proper context. The final version of this paper will appear in the journal Arch. Hist. Exact Sciences.