Sinusoidal projection

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Sinusoidal projection of the Earth.

The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson-Flamsteed or the Mercator equal-area projection. It is defined by:

$x = \left(\lambda - \lambda_0\right) \cos \phi$

$y = \phi\,$

where $\phi\,$ is the latitude, $\lambda\,$ is the longitude, and $\lambda_0\,$ is the central meridian^[1].

The north-south scale is the same everywhere at the central meridian, and the east-west scale is throughout the map the same as that; correspondingly, on the map, as in reality, the length of each parallel is proportional to the cosine of the latitude; thus the shape of the map for the whole earth is the area between two symmetric rotated cosine curves. The true distance between two points on the same meridian corresponds to the distance on the map between the two parallels, which is smaller than the distance between the two points on the map. There is no distortion on the central meridian or the equator.

A sinusoidal projection shows relative sizes accurately, but distorts shapes and directions. Distortion can be reduced by "interrupting" the map.

Similar projections which wrap the east and west parts of the sinusoidal projection around the north pole are the Werner and the intermediate Bonne and Bottomley projections.