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# The Maurer Equal-Area Polyconic

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.

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