As we saw in the discussion of the McCaw Modified Rectangular Polyconic projection, the scale transverse to a parallel is equal to:

1 1 ( --------------- + 1 ) cos(phi) + --------------- 2 2 ( sin(lat) ) ( sin(lat) )

To make the projection conformal for one parallel, we integrated the reciprocal of the scale along the parallel. To make it equal-area, we must instead integrate the scale as a function of phi to obtain the longitude.

The integral of the cosine is the sine, making the integral trivial, so we have:

1 1 long = ( --------------- + 1 ) sin(phi) + --------------- phi 2 2 ( sin(lat) ) ( sin(lat) )

and this function has no analytic inverse, so we must deal with it as we had dealt with latitude as a function of y for the Mollweide and Eckert IV projections, determining phi by an iterative process.

On the Equator, the scale of 1 + (x ^ 2)/2 gives, when integrated:

long = x + (x ^ 3)/6

which *is* invertible, being a cubic equation.

[Next] [Up] [Previous]