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Pseudocylindrical equal-area map 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
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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
Lame
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
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References
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