Cylindrical equal-area projection

[1] Variations of it appeared over the years by inventors who stretched the height of the Lambert and compressed the width commensurately in various ratios.

By the geometry of their construction, cylindrical projections stretch distances east-west.

The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude is given by φ): The only normal cylindrical projections that preserve area have a north-south compression precisely the reciprocal of east-west stretching (cos φ).

This divides north-south distances by a factor equal to the secant of the latitude, preserving area but distorting shapes.

The formulae presume a spherical model and use these definitions:[3] Relationship between