Catenary and Mercator projection

TL;DR

The paper addresses the geometric connection between the Mercator projection and a central cylindrical projection by showing that bending a dense, soft central cylindrical map into a horizontal catenary profile under uniform gravity yields the Mercator projection. It formalizes the relationship between latitude $\varphi$ and Mercator height $\psi$ via $\psi = \int_0^{\varphi} \frac{dt}{\cos t} = \ln\left(\tan\left(\frac{\varphi}{2}+\frac{\pi}{4}\right)\right)$ and uses the Gudermannian $\varphi = \mathrm{gd}(\psi)$ to connect circular and hyperbolic measures, with $L = \tan\varphi = \sinh\psi$ and $L_{catenary}(\psi) = \sinh(\psi)$ showing equality of lengths between the central projection and the catenary profile. It derives a bending parameter from height $H$ and latitude $\alpha$, giving $D_{unit\;sphere} = 2\,\mathrm{arsinh}(\tan\alpha)$ and, for radius $R = \frac{H}{2\tan\alpha}$, $D = \frac{H}{\tan\alpha}\mathrm{arsinh}(\tan\alpha)$, enabling construction of the Mercator map from the central cylindrical form. The work provides a physical-geometric interpretation of Mercator projection construction and elucidates how loxodromes appear as straight lines when the bent map is viewed appropriately, offering a conceptual bridge between classical map projections and catenary physics. The results have potential implications for visualization and hand-crafted map design in educational or workshop settings.

Abstract

The Mercator projection is sometimes confused with another mapping technique, specifically the central cylindrical projection, which projects the Earth's surface onto a cylinder tangent to the equator, as if a light source is at the Earth's center. Accidentally, this misconception is rather close to a truth. The only operation that the map needs is a free bending in a uniform gravitational field if the map's material is dense and soft enough to produce a catenary profile. The north and south edges of the map should be parallel and placed in the same plane at the appropriate distance. In this case, the bent map been projected onto this plane gives the Mercator projection. This property is rather curious, since it allows to make such a sophisticated one-to-one mapping as the Mercator projection using simple tools available in the workroom.

Figures (3)

• Figure 1: To calculate the Mercator projection of a given point $N$ with latitude $\varphi$, we first project the point $N$ to the point $B$ on the vertical line $AB$, then move it vertically downward to the point $Q$. The exact vertical coordinate of the point $Q$ is $\psi$, which is expressed by the equation (\ref{['eq:psi_ln']}).
• Figure 2: The hyperbolic amplitude $\varphi$ is constructed for an arbitrary point $M$ belonging to the unit hyperbola (blue curve), as shown here. Also $\varphi$ is the latitude of the point $N$ figured out in the Mercator projection (see figure \ref{['fig:mercator']} for comparison).
• Figure 3: After introduction of the hyperbolic amplitude $\varphi$ and disclose its relation to the hyperbolic measure $\psi$ via the Gudermannian function, the length of the catenary (solid magenta curve $AC$) introduced implicitly via the area hyperbolic cosine as described by equation (\ref{['eq:psi_ln_arcosh']}) is equal to the length $L$ (solid green line $AB$) that is vertical component of the central cylindrical map projection.