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]
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, 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 [math]\displaystyle{ x^k + y^k = \gamma^k }[/math], where [math]\displaystyle{ \gamma }[/math] and [math]\displaystyle{ k }[/math] are free parameters. Tobler's hyperelliptical projection is given as:
- [math]\displaystyle{ \begin{align} &x = \lambda [\alpha + (1 - \alpha) \frac{(\gamma^k - y^k)^{1/k}}{\gamma}] \\ \alpha &y = \sin \varphi + \frac{\alpha - 1}{\gamma} \int_0^y (\gamma^k - z^k)^{1/k} dz \end{align} }[/math]
where [math]\displaystyle{ \lambda }[/math] is the longitude, [math]\displaystyle{ \varphi }[/math] is the latitude, and [math]\displaystyle{ \alpha }[/math] is the relative weight given to the cylindrical equal-area projection. For a purely cylindrical equal-area, [math]\displaystyle{ \alpha = 1 }[/math]; for a projection with pure hyperellipses for meridians, [math]\displaystyle{ \alpha = 0 }[/math]; and for weighted combinations, [math]\displaystyle{ 0 \lt \alpha \lt 1 }[/math].
When [math]\displaystyle{ \alpha = 0 }[/math] and [math]\displaystyle{ k = 1 }[/math] the projection degenerates to the Collignon projection; when [math]\displaystyle{ \alpha = 0 }[/math], [math]\displaystyle{ k = 2 }[/math], and [math]\displaystyle{ \gamma = 4 / \pi }[/math] the projection becomes the Mollweide projection.[4] Tobler favored the parameterization shown with the top illustration; that is, [math]\displaystyle{ \alpha = 0 }[/math], [math]\displaystyle{ k = 2.5 }[/math], and [math]\displaystyle{ \gamma \approx 1.183136 }[/math].
See also
References
- ↑ Snyder, John P. (1993). Flattening the Earth: 2000 Years of Map Projections. Chicago: University of Chicago Press. p. 220.
- ↑ Mapthematics directory of map projections
- ↑ "Superellipse" in MathWorld encyclopedia
- ↑ Tobler, Waldo (1973). "The hyperelliptical and other new pseudocylindrical equal area map projections". Journal of Geophysical Research 78 (11): 1753–1759. doi:10.1029/JB078i011p01753. Bibcode: 1973JGR....78.1753T.
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