![Tobler hyperelliptical projection SW](https://upload.wikimedia.org/wikipedia/commons/3/34/Tobler_hyperelliptical_projection_SW.jpg)
The Tobler hyperelliptical projection is a family of
equal-area pseudocylindrical
In cartography, map projection is the term used to describe a broad set of Transformation (function) , transformations employed to represent the two-dimensional curved Surface (mathematics), surface of a globe on a Plane (mathematics), plane. In ...
projections that may be used for
world map
A world map is a map of most or all of the surface of Earth. World maps, because of their scale, must deal with the problem of map projection, projection. Maps rendered in two dimensions by necessity distort the display of the three-dimensiona ...
s.
Waldo R. Tobler
Waldo Rudolph Tobler (November 16, 1930 – February 20, 2018) was an American- Swiss geographer and cartographer. Tobler's idea that "Everything is related to everything else, but near things are more related than distant things" is referred ...
introduced the construction in 1973 as the ''hyperelliptical'' projection, now usually known as the Tobler hyperelliptical projection.
Overview
As with any pseudocylindrical projection, in the projection’s normal aspect, the
parallels of
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
are parallel,
straight lines
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
. Their spacing is calculated to provide the equal-area property. The projection blends the
cylindrical equal-area projection
In cartography, the normal cylindrical equal-area projection is a family of normal cylindrical, equal-area map projections.
History
The invention of the Lambert cylindrical equal-area projection is attributed to the Swiss mathematician Johan ...
, which has straight, vertical
meridians, with meridians that follow a particular kind of curve known as ''
superellipse
A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape.
In the ...
s'' 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
Degenerates is a musical group which originated in Grosse Pointe Park, Michigan in 1979, during the formative years of the Detroit hardcore scene. The group predated the Process of Elimination EP, which some reviewers view as the beginning of the ...
to the
Collignon projection; when
,
, and
the projection becomes the
Mollweide projection.
[
] Tobler favored the parameterization shown with the top illustration; that is,
,
, and
.
See also
*
List of map projections
This is a summary of map projections that have articles of their own on Wikipedia or that are otherwise notable
Notability is the property
of being worthy of notice, having fame, or being considered to be of a high degree of interest, signif ...
References
{{Map projections
Map projections
Equal-area projections