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.
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
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
and using Viète's method, the roots are, for :
all three roots are real and .
An inspection of the graph of shows that the desired minimum positive root is the intermediary one, .
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: