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 $x^{k}+y^{k}=\gamma ^{k}$ , where $\gamma$ and $k$ are free parameters. Tobler's hyperelliptical projection is given as:

{\begin{aligned}&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{aligned}}

where $\lambda$ is the longitude, $\varphi$ is the latitude, and $\alpha$ is the relative weight given to the cylindrical equal-area projection. For a purely cylindrical equal-area, $\alpha =1$ ; for a projection with pure hyperellipses for meridians, $\alpha =0$ ; and for weighted combinations, $0<\alpha <1$ .

When $\alpha =0$ and $k=1$ the projection degenerates to the Collignon projection; when $\alpha =0$ , $k=2$ , and $\gamma =4/\pi$ the projection becomes the Mollweide projection.^[4] Tobler favored the parameterization shown with the top illustration; that is, $\alpha =0$ , $k=2.5$ , and $\gamma \approx 1.183136$ .

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. Bibcode:1973JGR....78.1753T. CiteSeerX 10.1.1.495.6424. doi:10.1029/JB078i011p01753.

Map projection

By surface

Cylindrical

Mercator-conformal	Gauss–Krüger Transverse Mercator Oblique Mercator
Equal-area	Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert Smyth equal-surface Trystan Edwards
Cassini Central Equirectangular Gall stereographic Gall isographic Miller Space-oblique Mercator Web Mercator

Pseudocylindrical

Equal-area	Collignon Eckert II Eckert IV Eckert VI Equal Earth Goode homolosine Mollweide Sinusoidal Tobler hyperelliptical
Kavrayskiy VII Wagner VI Winkel I and II

Conical

Pseudoconical

Azimuthal
(planar)

General perspective	Gnomonic Orthographic Stereographic
Equidistant Lambert equal-area

Bonne	Sinusoidal Werner
Bottomley	Sinusoidal Werner
Cylindrical	Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert cylindrical equal-area Smyth equal-surface Trystan Edwards
Tobler hyperelliptical	Collignon Mollweide
Albers Briesemeister Eckert II Eckert IV Eckert VI Equal Earth Goode homolosine Hammer Lambert azimuthal equal-area Quadrilateralized spherical cube Strebe 1995

Equidistant in
some aspect

Gnomonic

Gnomonic

Loxodromic

Retroazimuthal
(Mecca or Qibla)

Planar	Gnomonic Orthographic Stereographic
Central cylindrical

Polyhedral

See also
Interruption (map projection) Latitude Longitude Tissot's indicatrix Map projection of the tri-axial ellipsoid

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Tobler hyperelliptical projection

Overview

See also

References

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