Orthographic projection (cartography)

Orthographic projection of the hemisphere 30W–150E

Orthographic projection is a map projection of cartography. Like the Stereographic projection, Orthographic projection is a perspective projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the Orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from deep space. The shapes and areas are distorted, particularly near the edges, but distances are preserved along parallels.

History

The orthographic projection was called the "analemma" by the Greeks. The name was changed to "orthographic" in 1613 by François d'Aiguillon of Antwerp. Albrecht Dürer (1471 – 1528) prepared the first known polar and equatorial orthographic maps of the Earth. Photographs of the Earth and other planets has inspired renewed interest in the Orthographic projection in astronomy and planetary science.

Mathematics

The formulas for the Orthographic projection are in terms of longitude ($\lambda\,$) and latitude ($\phi\,$) on the sphere. Defining the radius of the sphere R and the center point and origin of the projection as ($\lambda_0,\,\phi_2$), the equations for the Orthographic projection onto the ($x,\,y$) plane reduce to the following:

$x = R\,\cos(\phi)\sin(\lambda - \lambda_{0})$

$y = R\,[\cos(\phi_1)\sin(\phi) - \sin(\phi_1)\cos(\phi)\cos(\lambda-\lambda_0)]$

Latitudes beyond the range of the map should be clipped by calculating the distance $c$ form the center of the projection. This ensures that points on the opposite hemisphere are not plotted:

$\cos(c) = \sin(\phi_1)\sin(\phi) + \cos(\phi_1)\cos(\phi)\cos(\lambda - \lambda_0)\,$

The point should be clipped from the map if $\cos(c)$ is negative.

For the inverse formulas for the sphere, to find $(\lambda,\,\phi)$ given $R,\,\phi_1,\,\lambda_0,\,x,\;and\;y$:

$\phi = \arcsin \left[\cos(c)\sin(\phi_1) + \frac{y\sin(c)\cos(\phi_1)}{\rho}\right]\,$

$\lambda = \lambda_0 + \arctan \left[\frac{x\sin(c)}{\rho\cos(\phi_1)\cos(c)-y\sin(\phi_1)\sin(c)}\right]\,$

where

$\rho = \sqrt{x^2 + y^2}\,$

$c = \arcsin \left(\frac{\rho}{R}\right)\,$

References

• Weisstein, Eric W., Orthographic Projection From MathWorld--A Wolfram Web Resource. Accessed 17 August 2010.

• USGS Map Projections: A Working Manual, USGS Professional Paper 1395 (1987). A freely downloadable book with details on most projections, including formulas and sample calculations.