Map Projections: Two Arbitrary Azimuthal Projections

Map Projections

Two Aspects for Two Arbitrary Azimuthal Projections

General Polar Azimuthal Projections

The mathematical development of the azimuthal orthographic projection is purely geometric. Even though some azimuthal projections do not follow such a perspective process, all can be reduced to a general pattern, given the tangency point T:

Outlined projection plane tangent to sphere at point T

θ = θ' due to the azimuthal property
ρ = f (r)
x = ρ cos θ
y = ρ sin θ

For the North polar aspect, θ = λ.

The Azimuthal Equidistant Projection

For the azimuthal equidistant, an important projection for navigational applications, distance ρ from the center of the map is directly proportional to radial distance from the tangent point. In the North polar aspect:

ρ = (π / 2 − φ)R

The austral aspect is just as easy, with

ρ = (π / 2 + φ)R and θ = −λ

Lambert's Azimuthal Equal-area Projection

In the only projection both azimuthal and equal-area, created by Lambert and suitable for world maps, distance from the center of the map is progressively reduced in order to keep areal equivalence. Formulas follow from basic integral calculus.

Area element on sphere, given colatitude Φ = π / 2 − φ: ds = 2πR sin Φ R dΦ = 2πR² sin Φ dΦ
Corresponding element on map: ds = 2πrdr

For a given Φ₁, we want ρ: Radial distance for Lambert map

A similar sign change applies if the south polar aspect is intended.

Azimuthal equidistant and equal-area polar maps

Combined azimuthal equidistant (top) and equal-area (bottom) polar maps

The combined map shows both projections virtually identical at latitudes above 70°N. Beyond that, parallels get closer and closer together in Lambert's half, while remaining equally spaced in the azimuthal equidistant portion. The resulting areal difference is clearly visible in Antarctica.

General Equatorial Aspect for Azimuthal Maps

Calculating other aspects for azimuthal maps is possible applying 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.

	Given A, B, C angles on a spherical triangle's vertices, and α, β, γ the corresponding angles between edges connecting triangle vertices and center O of sphere,
	Law of sines:	sin A / α = sin B / β = sin C / γ
	Law of cosines:	cos γ = cos α cos β + sin α sin β sin C

Spherical triangle on Earth On the equatorial aspect, tangent point T lies on the intersection of Equator and an arbitrary central meridian. The projected point P marks a shaded triangle, whose vertices define central angle α, latitude φ and longitude λ.
Formulas for equatorial aspect

The Equatorial Azimuthal Equidistant

Just substituting results for cos θ and sin θ,

ρ = r = αR
x = αR cos φ sin λ / sin α
y = αR sin φ / sin α

If λ = φ = 0, sin α = 0, x = y = 0.

The Equatorial Azimuthal Equal-Area

In equations for Lambert's equal-area azimuthal projection, substitute α for Φ₁:

x = ρ cos θ = ρ cos φ / sin λ sin α
y = ρ sin θ = ρ sin φ / sin α

sin (a + b) = sin a cos b + cos a sin b, therefore sin 2a = 2 sin a cos a
cos (a + b) = cos a cos b − sin a sin b; cos 2a = cos²a − sin²a = 2 cos²a − 1, therefore (cos 2a + 1) / 2 = cos²a

Formulas for equatorial aspect of Lambert's projection

Combined azimuthal equidistant (top) and equal-area (bottom) equatorial maps

Again, both projections are very similar near the tangent point: the northern and southern portions of Africa join almost seamlessly.