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,
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
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 :
all three roots are real and .
An inspection of the graph of shows that the desired minimum positive root is the intermediary one, .
|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: