On the Lambert conformal conical projection and the general map of the Russian Empire

TL;DR

This work analyzes the Lambert conformal conical projection as a principled alternative for mapping a spherical annulus to the plane, with a focus on conformality and distortion control for large regions such as the Russian Empire. It constructs a normalized conformal representation of the sphere onto a circular cone via a pair of maps, derives explicit infinitesimal Lipschitz constants $L(\rho,\alpha,\rho_0)$, and identifies an optimal apex angle $\alpha_0$ by solving $\Lambda(\rho_2,\alpha_0,\rho_1)=1$ (equivalently $F(\rho_2,a_0,\rho_1)=1$ with $a_0=\sin\alpha_0$). The authors provide a detailed comparison with other conical projections (notably Delisle–Euler) for a region representing the Russian Empire, showing that the Lambert projection achieves distortion values comparable to the best known maps and offers a near-conformal performance linked to the Teichmüller map. They also use the modulus of annuli to quantify distortion and demonstrate that the Lambert projection can be tuned to minimize bi-Lipschitz distortion across the region. Overall, the paper establishes Lambert’s projection as a mathematically rigorous, practically competitive option for large-scale cartography, connecting classical geometry to modern distortion theory.

Abstract

The problem of drawing geographical maps is the one of mapping a subset of the sphere, representing a country or some other region on the surface of the Earth, into the Euclidean plane, minimising certain distortion properties that are specified in advance. It is known that from the purely mathematical point of view, this is an extremely difficult problem. One of Leonhard Euler's duties during his first stay at the Imperial Academy of Sciences of Saint Petersburg (1727-1741) was to help establishing maps of the Russian Empire. He worked on this project under the direction of the famous French geographer Joseph-Nicolas Delisle, who was the head of the astronomy and geography departments of the Academy. The general map of the Russian Empire, together with several maps of its particular regions were published under Euler's direction in the so-called Russian Atlas in 1745. In his later memoir ``De proiectione geographica De Lisliana in mappa generali imperii russici usitata'', written in 1777, Euler developed the mathematical theory of the method used by Delisle on a heuristic basis, which he himself used for drawing the general map of the Russian Empire. This method usually carries now the name Delisle--Euler map. In a previous paper, the first two authors of the present paper compared the Delisle--Euler map with several other maps of the conical type, with respect to various mathematical distorsion properties. They showed that this map is the best one from all the points of view considered, when it is applied to the drawing of the Russian Empire. In the present paper, we compare the Euler--Delisle map with a map which was not considered in the paper mentioned, namely, the so-called Lambert conformal conical projection, applied to the same region of the Earth. We show that the latter is better in several respects than all the other maps considered in the previous paper, including the Delisle--Euler map.

Figures (11)

• Figure 1: An example of a projection map: Euler's map of the Russian Empire, from the Atlas Russicusatlas-Russicus, published in Saint Petersburg in 1745. Source: Gallica, Bibliothèque nationale de France.
• Figure 2: Another example of a conical projection: A rendition of the map Universalior Cogniti Orbis Tabula ex Recentibus Confecta Observationibus by Johann Ruysch. Publisher Bernardinus Venetus de Vitalibus. From the William C. Wonders Map Collection, the University of Alberta Library.
• Figure 3: The locations of the cone $C(\alpha,\rho_0)$ and the annulus $A(\rho_1,\rho_2)$. The cone $C(\alpha,\rho_0)$ has apex angle $2\alpha$ and the lower circle in the intersection $\mathbb{S}^2\cap C(\alpha,\rho_0)$ is the parallel at $z=\rho_0$.
• Figure 4: The domain and the range of a Lambert conical projection
• Figure 5: A Lambert conical map of Europe, extracted from his memoir Beyträge zum Gebrauche der Mathematik und deren AnwendungLamG.

Theorems & Definitions (6)

• Proposition 1.1
• Proposition 6.1
• Theorem 7.1
• Proposition 7.2
• proof
• proof : Proof of \ref{['optimal is isometric']}