Suppose a pointed-polar equal-areapseudocylindrical 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,

With no loss of generality due to symmetry, 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

J.E.E.Craster's parabolic projection

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

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.

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