Tobler hyperelliptical projection

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.

are free parameters.

Tobler's hyperelliptical projection is given as: where

is the relative weight given to the cylindrical equal-area projection.

For a purely cylindrical equal-area,

; for a projection with pure hyperellipses for meridians,

; and for weighted combinations,

the projection degenerates to the Collignon projection; when

the projection becomes the Mollweide projection.

[4] Tobler favored the parameterization shown with the top illustration; that is,

Tobler hyperelliptical projection of the world; α = 0, γ = 1.18314, k = 2.5
The Tobler hyperelliptical projection with Tissot's indicatrix of deformation; α = 0, k = 3