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$$\mathds{R}$$

The equations to convert a point on the sphere $$(\theta, \phi)$$ to a point on 2D plane $$(x, y)$$ are:

\begin{equation} x=R\frac{\beta\theta}{\sqrt{\alpha\beta}}\cos{(\alpha\phi)} \end{equation} \begin{eqnarray} y & = & R\frac{\alpha\phi}{\sqrt{\alpha\beta}} \\ \alpha & = & \frac{2\arccos{c}}{\pi} \\ \beta & = & \frac{\alpha}{2p} \end{eqnarray}

where $$\theta$$ is the longitude and $$\phi$$ the latitude. R is the radius of the sphere, which is 1 in our program. $$c=0.5$$ and $$p=0.5$$ are two constants controlling the shape of the projection.