Map Projections: Deducing the Orthographic Projection

Map Projections

Deducing the Azimuthal Orthographic Projection

The purely geometrical azimuthal orthographic projection can be entirely visualized as a physical model. The easiest case, a polar aspect, is presented here for the Northern hemisphere.


First construction method	Second method

Suppose the Earth lying over a plane parallel to the Equator. Light rays emanating from a point infinitely far away on the north-south polar axis pierce a semitransparent northern hemisphere and "draw" its features onto the plane. The southern hemisphere is considered completely transparent.
Alternatively, one can imagine an observer infinitely far away on that axis. Parallel light rays emanating from Earth's surface hit an intervening plane (or disc) perpendicular to the rays, where the image is developed. Since all rays are parallel ("cylindric perspective"), the plane may or may not be tangent to the sphere without affecting the result.
Anyway, only one hemisphere can be seen at any time.

Geometrically, the orthographic projection can be imagined as converting polar coordinates to 3-D cartesian, then flattening it (ignoring one coordinate). The projection formulae are easily derived in polar coordinates:

r = R cos φ, 0 <= φ <= π/2, θ = λ, ρ = r

On the left, Earth set for a polar azimuthal orthographic projection, resting on the projection plane; distance of projected point to center of map depends only on latitude. On the right, Earth and projection plane viewed from above; angle of point around center of map equals longitude

Only points with φ >= 0 (for the north polar case) or φ <= 0 (south polar) are visible.
Converting to Cartesian coordinates,

x = ρ cos θ = R cos φ cos λ

y = ρ sin θ = R cos φ sin λ

Conversion of the equations to inverse mapping is straightforward.

North polar aspect of azimuthal orthographic map; parallel spacing manifests the distortion pattern, especially near the rim

A more general aspect, either equatorial or oblique, can be obtained by first rotating Earth coordinates in 3D space, then applying the previous equations. Early cartographers designed general orthographic maps by first plotting the polar aspect, then locating key grid points by a set of parallel lines drawn from the polar map.

Two other important azimuthal projections are created just by changing the light source's position.

For a practical presentation, a cartographer could conceivably paint coastlines and other geographical features onto a glass globe or bowl and, using reflected sunlight or a strong flashlight at a convenient distance, project the globe shadows on the wall, thus creating a variety of azimuthal projections like the orthographic and stereographic.