Tobler hyperelliptical projection

From Wikipedia, the free encyclopedia
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

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 , 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 .

See also[edit]


  1. ^ Snyder, John P. (1993). Flattening the Earth: 2000 Years of Map Projections. Chicago: University of Chicago Press. p. 220.
  2. ^ Mapthematics directory of map projections
  3. ^ "Superellipse" in MathWorld encyclopedia
  4. ^ Tobler, Waldo (1973). "The hyperelliptical and other new pseudocylindrical equal area map projections". Journal of Geophysical Research. 78 (11): 1753–1759. Bibcode:1973JGR....78.1753T. CiteSeerX doi:10.1029/JB078i011p01753.