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 parallelsoflatitude 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]orLamé curves or sometimes as hyperellipses. A hyperellipse is described by , where
and
are free parameters. Tobler's hyperelliptical projection is given as:
where is the longitude,
is the latitude, and
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,
.
When and
the projection degenerates to the Collignon projection; when
,
, and
the projection becomes the Mollweide projection.[4] Tobler favored the parameterization shown with the top illustration; that is,
,
, and
.