Map Projections

### Deducing Craster's Parabolic Projection

Suppose a pointed-polar equal-area pseudocylindrical projection in the equatorial aspect whose boundary meridians are parabolic arcs with axes on the Equator, which is twice as long as the central meridian.

Schematic development of Craster's parabolic projection

Since the projection is pseudocylindrical,

in other words,

Due to symmetry around the Equator and central meridian, with no loss of generality let us consider only the northeastern quadrant where .

For , the following constraints hold:

For , therefore and the boundary meridian is defined by:

As seen in the development of the sinusoidal projection, half the area on Earth between the Equator and the parallel is . This time, instead of “discovering” that a projection is equal-area, we will use that property as a constraint to calculate the ordinate . Half the area on the map between the Equator and any given ordinate is

For , and

Therefore and

thus

Solving the cubic equation ,

and because there are three real roots; however, since

the root terms involve complex expressions. With the substitutions for the depressed cubic

Graph of S(y).

and using Viète's method, the roots are, for :

Because

all three roots are real and .

An inspection of the graph of shows that the desired minimum positive root is the intermediary one, .

For

Or, since ,

Those are the forward equations of the parabolic equal-area projection, the best known of all projections presented by J.E.E.Craster.

J.E.E.Craster's parabolic projection

Compared deformation patterns of Craster's parabolic (top left and bottom right quadrants) and the sinusoidal projections

The overall shapes of the sinusoidal and parabolic projections are easily confused, but there are differences:

• at identical scales, the sinusoidal is both taller and broader, while the parabolic's boundary meridian is more convex
• the parabolic's central meridian is not a standard line, with a shorter region of low angular deformation along this line
• conversely, at higher latitudes nearer the map's boundary, the region of higher deformation is somewhat smaller in the parabolic

www.progonos.com/furuti/MapProj/Normal/CartHow/HowCPar/howCPar.html — August  5, 2018