Details

The transformation formulas used in this Demonstration are based on the Mathematica built-in function ToSphericalCoordinates and the conversion formulas in [1, 2].

For the equidistant projection, the following mapping is used to convert the Cartesian coordinates of a point on the unit sphere to the corresponding point on the cylinder wrapped around it: 
.

The equidistant projection to a plane uses the following mapping to convert the Cartesian coordinates of a point on the unit sphere to the corresponding point on the 2D plane: .

For the equal area projection, the following mapping is used to convert the Cartesian coordinates of a point on the unit sphere to the corresponding point on the cylinder wrapped around it: .

The equal-area projection to a plane uses the following mapping to convert the Cartesian coordinates of a point on the unit sphere to the corresponding point on the 2D plane: .

References

[1] E. W. Weisstein. "Cylindrical Equidistant Projection" from MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/CylindricalEquidistantProjection.html (Wolfram MathWorld).

[2] E. W. Weisstein. "Cylindrical Equal-Area Projection" from MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/CylindricalEqual-AreaProjection.html (Wolfram MathWorld).