Tobler hyperelliptical projection

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https://en.wikipedia.org/wiki/Tobler_hyperelliptical_projection
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]

Overview

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 with meridians of longitude that follow a particular kind of curve known as superellipses [3] or Lamé curves or sometimes as hyperellipses. The curve is described by xk + yk = γk. The relative weight of the cylindrical equal-area projection is given as α, ranging from all cylindrical equal-area with α = 1 to all hyperellipses with α = 0.

When α = 0 and k = 1 the projection degenerates to the Collignon projection; when α = 0, k = 2, and γ ≈ 1.2731 the projection becomes the Mollweide projection. [4] Tobler favored the parameterization shown with the top illustration; that is, α = 0, k = 2.5, and γ = 1.183136.

See also

References

  1. ^ Snyder, John P. (1993). Flattening the Earth: 2000 Years of Map Projections. Chicago: University of Chicago Press. p. 220.
  2. ^ The Tobler Hyperelliptical Projection on the Center for Spatially Integrated Social Science's site
  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  10.1.1.495.6424. doi: 10.1029/JB078i011p01753.