Tobler hyperelliptical projection: Difference between revisions
y is correct, and it doesn’t cause “infinite recursion”; it just doesn’t have a closed-form solution. Undid revision 1185098460 by Atavoidturk (talk) |
|||
(6 intermediate revisions by 4 users not shown) | |||
Line 19: | Line 19: | ||
:<math>\begin{align} |
:<math>\begin{align} |
||
&x = \lambda [\alpha + (1 - \alpha) \frac{(\gamma^k - y^k)^{1/k}}{\gamma}] \\ |
&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 - |
\alpha &y = \sin \varphi + \frac{\alpha - 1}{\gamma} \int_0^y (\gamma^k - z^k)^{1/k} dz |
||
\end{align}</math> |
\end{align}</math> |
||
where <math>\lambda</math> is the longitude, <math>\varphi</math> is the latitude, and <math>\alpha</math> is the relative weight given to the cylindrical equal-area projection. For a purely cylindrical equal-area, <math>\alpha = 1</math>; for a projection with pure hyperellipses for meridians, <math>\alpha = 0</math>; and for weighted combinations, <math>0 < \alpha < 1</math>. |
where <math>\lambda</math> is the longitude, <math>\varphi</math> is the latitude, and <math>\alpha</math> is the relative weight given to the cylindrical equal-area projection. For a purely cylindrical equal-area, <math>\alpha = 1</math>; for a projection with pure hyperellipses for meridians, <math>\alpha = 0</math>; and for weighted combinations, <math>0 < \alpha < 1</math>. |
||
When <math>\alpha = 0</math> and <math>k = 1</math> the projection [[Degeneracy (mathematics)|degenerates]] to the [[Collignon projection]]; when <math>\alpha = 0</math>, <math>k = 2</math>, and <math>\gamma |
When <math>\alpha = 0</math> and <math>k = 1</math> the projection [[Degeneracy (mathematics)|degenerates]] to the [[Collignon projection]]; when <math>\alpha = 0</math>, <math>k = 2</math>, and <math>\gamma = 4 / \pi</math> the projection becomes the [[Mollweide projection]].<ref>{{cite journal| |
||
last=Tobler| |
last=Tobler| |
||
first=Waldo| |
first=Waldo| |
||
Line 46: | Line 46: | ||
{{reflist}} |
{{reflist}} |
||
{{Map |
{{Map projections}} |
||
[[Category:Map projections]] |
[[Category:Map projections]] |
Latest revision as of 15:56, 14 November 2023
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[edit]
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]
References[edit]
- ^ 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.