Orthographic projection (cartography)

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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.

[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 (\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]\,


\rho = \sqrt{x^2 + y^2}\,
c = \arcsin \left(\frac{\rho}{R}\right)\,

[edit] References

  1. Keuning, Johannes. The History of Geographical Map projections until 1600: Imago Mundi, v.12, p.1-24
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