Foucaut Sinusoidal¶

The y-axis is based upon a weighted mean of the cylindrical equal-area and the sinusoidal projections. Parameter $$n=n$$ is the weighting factor where $$0 <= n <= 1$$.

\begin{align}\begin{aligned}\begin{array}\\x &= \lambda \cos \phi / (n + (1 - n) \ cos \phi)\\y &= n \phi + (1 - n) \sin \phi\\\end {array}\end{aligned}\end{align}

For the inverse, the Newton-Raphson method can be used to determine $$\phi$$ from the equation for $$y$$ above. As $$n \rightarrow 0$$ and $$\phi \rightarrow \pi/2$$, convergence is slow but for $$n = 0$$, $$\phi = \sin^1y$$