Tobler hyperelliptical projection: Difference between revisions
m +wikilink Gabriel Lamé |
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) |
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{{Short description|Pseudocylindrical equal-area map projection}} |
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[[File:Tobler hyperelliptical projection SW.jpg| |
[[File:Tobler hyperelliptical projection SW.jpg|400px|thumb|Tobler hyperelliptical projection of the world; α = 0, γ = 1.18314, k = 2.5]] |
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⚫ | The '''Tobler hyperelliptical projection''' is a family of [[ |
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[[File:Tobler Hyperelliptical with Tissot's Indicatrices of Distortion.svg|400px|thumb|The Tobler hyperelliptical projection with [[Tissot's indicatrix]] of deformation; α = 0, k = 3]] |
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⚫ | The '''Tobler hyperelliptical projection''' is a family of [[Map projection#Equal-area|equal-area]] [[Map projection#Pseudocylindrical|pseudocylindrical]] projections that may be used for [[world map]]s. [[Waldo R. Tobler]] introduced the construction in 1973 as the ''hyperelliptical'' projection, now usually known as the Tobler hyperelliptical projection.<ref name="Snyder93">{{cite book |
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| title = Flattening the Earth: 2000 Years of Map Projections |
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| last = Snyder |
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| first = John P. |
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| authorlink = John P. Snyder |
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| year = 1993 |
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| publisher = [[University of Chicago Press]] |
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| location = Chicago |
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| page = 220 |
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⚫ | |||
==Overview== |
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As with any pseudocylindrical projection, in the projection’s normal aspect,<ref>[https://mapthematics.com/ProjectionsList.php?Projection=105#hyperelliptical Mapthematics directory of map projections]</ref> the [[circle of latitude|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 [[Meridian (geography)|meridians]], with meridians that follow a particular kind of curve known as ''[[superellipse]]s''<ref>[http://mathworld.wolfram.com/Superellipse.html "Superellipse" in MathWorld encyclopedia]</ref> or ''[[Gabriel Lamé|Lamé]] curves'' or sometimes as ''hyperellipses''. A hyperellipse is described by <math>x^k + y^k = \gamma^k</math>, where <math>\gamma</math> and <math>k</math> are free parameters. Tobler's hyperelliptical projection is given as: |
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:<math>\begin{align} |
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&x = \lambda [\alpha + (1 - \alpha) \frac{(\gamma^k - y^k)^{1/k}}{\gamma}] \\ |
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\alpha &y = \sin \varphi + \frac{\alpha - 1}{\gamma} \int_0^y (\gamma^k - z^k)^{1/k} dz |
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\end{align}</math> |
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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>. |
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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| |
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last=Tobler| |
last=Tobler| |
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first=Waldo| |
first=Waldo| |
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volume=78| |
volume=78| |
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issue=11| |
issue=11| |
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pages= |
pages=1753–1759| |
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year=1973| |
year=1973| |
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doi=10.1029/JB078i011p01753 |
doi=10.1029/JB078i011p01753 |
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|bibcode=1973JGR....78.1753T |
|bibcode=1973JGR....78.1753T |
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}} |
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citeseerx=10.1.1.495.6424}} |
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⚫ | |||
</ref> Tobler favored the parameterization shown with the top illustration; that is, <math>\alpha = 0</math>, <math>k = 2.5</math>, and <math>\gamma \approx 1.183136</math>. |
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==See also== |
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* [[List of map projections]] |
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In the projection’s normal aspect,<ref>[http://www.csiss.org/map-projections/Pseudocylindrical/Hyperelliptical.pdf The Tobler Hyperelliptical Projection on the Center for Spatially Integrated Social Science's site]</ref> the [[circle of latitude|parallels]] of [[latitude]] are parallel [[straight line]]s whose spacing is calculated to provide the equal-area property; the [[Meridian (geography)|meridian]]s of [[longitude]] (except for the central meridian, which is a straight line perpendicular to the lines representing parallels) are curves of the form ''a''|''x''|<sup>γ</sup> + ''b''|''y''|<sup>γ</sup> = 1 (with ''a'' dependent on longitude and ''b'' constant for a given map), known as ''[[superellipse]]s''<ref>[http://mathworld.wolfram.com/Superellipse.html "Superellipse" in MathWorld encyclopedia]</ref> or [[Gabriel Lamé|Lamé]] curves. When γ=1 it becomes the [[Collignon projection]]; when γ=2 the projection becomes the [[Mollweide projection]]; the [[limiting case]] as γ→∞ is the [[Cylindrical equal-area projection]] ([[Lambert cylindrical equal-area projection|Lambert cylindrical equal-area]], [[Gall–Peters projection|Gall–Peters]], or [[Behrmann projection]]). Values of γ that are favored by Tobler and others are generally greater than 2. |
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==References== |
==References== |
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{{reflist}} |
{{reflist}} |
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{{Map |
{{Map projections}} |
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[[Category: |
[[Category:Map projections]] |
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[[Category:Equal-area projections]] |
[[Category:Equal-area 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.