As with any pseudocylindrical projection, in the projection’s normal aspect, 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 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{align} &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{align} 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 . 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. Tobler favored the parameterization shown with the top illustration; that is, \alpha = 0, k = 2.5, and \gamma \approx 1.183136. ==See also==