Map Projections: Deducing the Sanson-Flamsteed Projection

Map Projections

Deducing the Sanson-Flamsteed Projection

Given the Earth radius R, suppose the equatorial aspect of an equal-area projection with the following properties:

Parallels map into equally spaced parallel straight lines
All parallels are standard lines
The central meridian is a standard line

Such a map would be appropriate for certain meteorological presentations since linear parallel spacing make easy comparing latitude effects; constant scale on parallels help measuring main wind speeds. Finally, areas are preserved so comparing choropleths and other color-coded datasets makes sense.

Imagine at first an Earth-sized map; since −π/ 2 <= φ <= π/ 2, and parallels are uniformly spaced, y-coordinates are proportional to latitude only; −πR <= y <= πR, thus y = φR.
We want an area-preserving map, so the circumference of any parallel equals the Earth circumference at that latitude. The radius of a spherical cap at angle φ is R cos φ. Therefore, the corresponding projected parallel has length 2kπ R cos φ. At the Equator φ = 0, parallel length is 2πR, thus k = 0.5.
Since horizontal scale is constant and −π <= λ <= π, x / π R cos φ = λ / π.

The resulting transformation

x = Rλ cos φ
y = Rφ

yields the so-called Sanson-Flamsteed projection, also known as the sinusoidal for obvious reasons. Sanson-Flamsteed grid

Mathematically one of the simplest projections, it has fairly satisfactory results except perhaps at higher latitudes. One could use oblique Sanson-Flamsteed maps for a clearer view of polar areas (at the cost of losing the parallel spacing property), or interrupted versions avoiding high-latitude shearing.