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

  • List of map projections

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.

Media files used on this page

Tobler hyperelliptical projection SW.jpg
Author/Creator: Strebe, Licence: CC BY-SA 3.0
The world on Tobler hyperelliptical projection. 15° graticule; α = 0, γ = 1.18314; k = 2.5. Imagery is a derivative of NASA’s Blue Marble summer month composite with oceans lightened to enhance legibility and contrast. Image created with the Geocart map projection software.
Tobler Hyperelliptical with Tissot's Indicatrices of Distortion.svg
Author/Creator: Justin Kunimune, Licence: CC BY-SA 4.0
The world on a Tobler hyperelliptical projection, with 10° graticule and Tissot's indicatrices overlaid. α=0; k=3. Each red circle is 1,000 km in diameter. Coastline data from www.naturalearthdata.com. Colors inspired by Eric Gaba. Projection generated with my own code, at github.com/jkunimune15/Map-Projections.