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The general case of azimuthal projections is mainly defined by a function that transforms distances from the center of projection T. |
The mathematical development of the azimuthal orthographic projection is purely geometric. Even though several azimuthal projections - like the two explained here - do not follow such a perspective process, all can be reduced to a general pattern.
On a sphere, two angles determine the distance and position of any point
relative to the center of projection
. For north polar aspects,
equals the longitude
, while
equals the colatitude
, or
.
On the map, due to the azimuthal property,
The fundamental characteristic of an azimuthal projection is a function which transforms distances from the center of the map, and therefore determines the spacing of parallels in the polar aspects:
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The defining distance function for some azimuthal projections |
And
The austral aspect is just as easy, with
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Let a thin spherical zone be the element, given the
colatitude The corresponding element on a polar azimuthal map is a ring with area: |
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For any given colatitude ,
we want
such that the
spherical cap bounded by
and
the disc bounded by
on the map
have identical areas. This is enough for areal preservation because
the scales along circumferences on both the zone and
ring, although different, are constant - again, due to the azimuthal property.
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A similar sign change applies if the south polar aspect is intended.
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Combined north polar azimuthal equidistant (top) and equal-area (bottom) maps. |
Calculating other aspects for azimuthal maps is possible by introducing coordinate transformations and rotations in space. However, the important equatorial aspect can be obtained in a more direct way, using two properties of triangles on a spherical surface.
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A triangle on the sphere is defined by the center of projection T, the point to be projected P and an equatorial point with the same longitude. |
On the equatorial
aspect, the center of projection lies on
the intersection
of the Equator and an arbitrary central meridian. It is one
vertex of a spherical triangle, the second is
projected point
, and the third one lies
on the Equator at the same longitude as
.
The respective angles at the center of the sphere are
,
and
, and we want the angle at
,
which corresponds to
in the first diagram.
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If ,
but
Now using two trigonometric identities,
the common factor can be expanded:
Finally,
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Combined azimuthal equidistant (top) and equal-area (bottom) equatorial maps. |