Tobler hyperelliptical projection

As with any pseudocylindrical projection, in the projection’s normal aspect,^[2] the parallels of latitude are parallel, straight lines. Their spacing is calculated to provide the equal-area property. The projection blends the cylindrical equal-area projection, which has straight, vertical meridians, with meridians that follow a particular kind of curve known as superellipses^[3] or Lamé curves or sometimes as hyperellipses. A hyperellipse is described by ${\displaystyle x^{k}+y^{k}=\gamma ^{k}}$, where ${\displaystyle \gamma }$ and ${\displaystyle k}$ are free parameters. Tobler's hyperelliptical projection is given as:

${\displaystyle {\begin{aligned}&x=\lambda [\alpha +(1-\alpha ){\frac {(\gamma ^{k}-y^{k})^{1/k}}{\gamma }}]\\\alpha &y=\sin \varphi +{\frac {\alpha -1}{\gamma }}\int _{0}^{y}(\gamma ^{k}-z^{k})^{1/k}dz\end{aligned}}}$

where ${\displaystyle \lambda }$ is the longitude, ${\displaystyle \varphi }$ is the latitude, and ${\displaystyle \alpha }$ is the relative weight given to the cylindrical equal-area projection. For a purely cylindrical equal-area, ${\displaystyle \alpha =1}$; for a projection with pure hyperellipses for meridians, ${\displaystyle \alpha =0}$; and for weighted combinations, ${\displaystyle 0<\alpha <1}$.

When ${\displaystyle \alpha =0}$ and ${\displaystyle k=1}$ the projection degenerates to the Collignon projection; when ${\displaystyle \alpha =0}$, ${\displaystyle k=2}$, and ${\displaystyle \gamma =4/\pi }$ the projection becomes the Mollweide projection.^[4] Tobler favored the parameterization shown with the top illustration; that is, ${\displaystyle \alpha =0}$, ${\displaystyle k=2.5}$, and ${\displaystyle \gamma \approx 1.183136}$.

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