Orthographic projection (cartography)
From Wiki.GIS.com
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.
[edit] 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.[1] Photographs of the Earth and other planets has inspired renewed interest in the Orthographic projection in astronomy and planetary science.
[edit] Mathematics
The formulas for the Orthographic projection are in terms of longitude (
) and latitude (
) on the sphere. Defining the radius of the sphere R and the center point and origin of the projection as (
), the equations for the Orthographic projection onto the (
) plane reduce to the following:
![y = R\,[\cos(\phi_1)\sin(\phi) - \sin(\phi_1)\cos(\phi)\cos(\lambda-\lambda_0)]](/wiki/images/math/2/3/7/237d848ef8875fbaf1cf9caed639be6c.png)
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:
The point should be clipped from the map if cos(c) is negative.
For the inverse formulas for the sphere, to find 
given 
:
![\phi = \arcsin \left[\cos(c)\sin(\phi_1) + \frac{y\sin(c)\cos(\phi_1)}{\rho}\right]\,](/wiki/images/math/3/2/a/32a9efa6e9c350f0737a4c3e7d8b6190.png)
![\lambda = \lambda_0 + \arctan \left[\frac{x\sin(c)}{\rho\cos(\phi_1)\cos(c)-y\sin(\phi_1)\sin(c)}\right]\,](/wiki/images/math/9/6/9/9691bb719b98a184b8e44310f869e8ba.png)
where
[edit] References
- ↑ Keuning, Johannes. The History of Geographical Map projections until 1600: Imago Mundi, v.12, p.1-24
- 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.
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