Stereographic projection and uneven scaling

[Tahmeed]

Lets assume we are mapping one face of earth. we place a plane touching the Earth at 0 lattitude and 0 longitude. Now we take the plane of projection. suppose that we expand the projection unevenly. The small projectional area of a certain lattitude and longitude is expanded by a factor which is the function of it's lattitude abd longitude.

Evidently we won't get a circular projection. But how do i find the shape/equation of the projection if i know the expansion factor as a function of lattitude and longitude?

 

[.Scott]

The first thing you need to know is your map projection: https://en.wikipedia.org/wiki/Map_projection
The projection you are describing is Azimuthal, but there are variants.
From the wiki article:

Some azimuthal projections are true perspective projections; that is, they can be constructed mechanically, projecting the surface of the Earth by extending lines from a https://en.wikipedia.org/w/index.php?title=Point_of_perspective&action=edit&redlink=1 (along an infinite line through the tangent point and the tangent point's antipode) onto the plane:

• The gnomonic projection displays great circles as straight lines. Can be constructed by using a point of perspective at the center of the Earth. r(d) = c tan d/R; so that even just a hemisphere is already infinite in extent.[24][25]
• The General Perspective projection can be constructed by using a point of perspective outside the earth. Photographs of Earth (such as those from the International Space Station) give this perspective.
• The orthographic projection maps each point on the Earth to the closest point on the plane. Can be constructed from a point of perspective an infinite distance from the tangent point; r(d) = c sin d/R.[26] Can display up to a hemisphere on a finite circle. Photographs of Earth from far enough away, such as the Moon, approximate this perspective.
• The stereographic projection, which is conformal, can be constructed by using the tangent point's antipode as the point of perspective. r(d) = c tan d/2R; the scale is c/(2R cos2 d/2R).[27] Can display nearly the entire sphere's surface on a finite circle. The sphere's full surface requires an infinite map.

Other azimuthal projections are not true perspective projections:

• Azimuthal equidistant: r(d) = cd; it is used by amateur radio operators to know the direction to point their antennas toward a point and see the distance to it. Distance from the tangent point on the map is proportional to surface distance on the Earth (;[28] for the case where the tangent point is the North Pole, see the flag of the United Nations)
• Lambert azimuthal equal-area. Distance from the tangent point on the map is proportional to straight-line distance through the earth: r(d) = c sin d/2R[29]
• https://en.wikipedia.org/w/index.php?title=Logarithmic_azimuthal_projection&action=edit&redlink=1 is constructed so that each point's distance from the center of the map is the logarithm of its distance from the tangent point on the Earth. r(d) = c ln d/d0); locations closer than at a distance equal to the constant d0 are not shown.[30][31]