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]
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
xk+yk=\gammak
\gamma
k
\begin{align} &x=λ[\alpha+(1-\alpha)
| (\gammak-yk)1/k | |
| \gamma |
]\\ \alpha&y=\sin\varphi+
| \alpha-1 | |
| \gamma |
| y | |
| \int | |
| 0 |
(\gammak-zk)1/kdz \end{align}
where
λ
\varphi
\alpha
\alpha=1
\alpha=0
0<\alpha<1
When
\alpha=0
k=1
\alpha=0
k=2
\gamma=4/\pi
\alpha=0
k=2.5
\gamma ≈ 1.183136
. Flattening the Earth: 2000 Years of Map Projections . John P. Snyder . 1993 . . Chicago . 220 .