Karl Brandan Mollweide (1774-1825) designed an equal-area map projection, mapping the surface of the earth to an ellipse with an aspect ratio of 2:1. If you were looking at the earth from a distance, the face you’re looking at would correspond to a circle in the middle of the Mollweide map. The part of the earth that you can’t see is mapped to the rest of the ellipse.
The lines of latitude are not quite evenly spaced; some distortion is required to achieve equal area. Instead, latitude φ corresponds to θ on the map according to the equation
There is no closed-form solution for θ as a function of φ and so the equation must be solved numerically. For each φ we have to find the root of
Here’s a plot of θ as a function of φ.
Newton’s method
Newton’s method works efficiently (converges quadratically) except when φ is close to π/2. The reason Newton’s method slows down for high latitudes is that f and its derivative are both at π/2, i.e. f has a double root there.
Newton’s method slows down from quadratic to linear convergence near a double root, but there is a minor modification that maintains quadratic convergence near a double root.
When m = 1 this is the usual form of Newton’s method. Setting m = 2 tunes the method for double roots.
The modified version of Newton’s method with m = 2 works when the root you’re trying to is exactly a double root. However, if you’re trying to find a root near a double root, setting m = 2 can cause the method to diverge, so you have to be very careful with changing m.
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