Nonlinear Projections - Transformations and Projections in Computer Graphics - page 207

meridian. Because of the aspect ratio chosen by Mollweide, the central meridian is half
as long as the equator. The two meridians 90 ◦ east and west of the central meridian
form a circle.
The mathematical expression of this projection starts with a point with longitude
θ and latitude φ on the sphere. The point is mapped by this projection to the point
x = 2 √ 2( θ y = √ 2sin α

− θ 0 )cos α

π

and

on the map, where θ 0 is the longitude at the center of the map and α is the solution to
the equation 2 α +sin(2 α )= π sin φ .
This projection is also called homalographic, homolographic (from the Greek homo ,
meaning “same”), elliptical, or Babinet. There is also an interrupted version of the
Mollweide projection. Mathematically, this projection is pseudocylindrical equal-area.
This projection is sometimes used in thematic world maps. It preserves scale up
to latitude 40 ◦ (north and south). North and south of this latitude, distortions become
more and more severe.

The sinusoidal projection (Figure 4.57), also known as the Sanson-Flamsteed pro-
jection and the Mercator equal-area projection, is the simplest pseudocylindrical equal-
area projection.

Figure 4.57: Sinusoidal Projection.

The width of a degree of longitude is proportional to the cosine of the latitude,
and the lines of latitude become straight segments placed uniformly on the map. This
combination preserves areas. Specifically, a point with longitude θ and latitude φ on
the sphere will be mapped by this projection to the point (( θ − θ 0 )cos φ, φ )onthemap

(where θ 0 is the longitude at the center of the map).