THC Science

The Tobler hyperelliptical projection is a family of equal-area pseudocylindrical projections that may be used for world maps. Waldo R. Tobler introduced the construction in 1973 as the hyperelliptical projection, now usually known as the Tobler hyperelliptical projection.^[1]

Overview[edit]

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 $x^{k}+y^{k}=\gamma ^{k}$ , where $\gamma$ and $k$ are free parameters. Tobler's hyperelliptical projection is given as:

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

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