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Orthographic projection (cartography)

Orthographic projection (equatorial aspect) of the hemisphere 30W–150E

An orthographic projection is a map projection of cartography. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective (or azimuthal) 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 outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges, but distances are preserved along parallels (Snyder 1987, 1993).

Contents

History

Orthographic projection was called "analemma" by the Greeks (see Ptolemy). The current name was adopted in 1613 by François d'Aiguillon of Antwerp, though it is thought to originate from the Roman writer and architect Vitruvius. Albrecht Dürer prepared the first known polar and equatorial Orthographic maps of the Earth. Photographs of the Earth and other planets from spacecraft have inspired renewed interest in the Orthographic projection in astronomy and planetary science.

Mathematics

The formulas for the Orthographic projection are derived using trigonometry. They are written in terms of longitude (λ) and latitude (φ) on the sphere. Define the radius of the sphere R and the center point (and origin) of the projection (\lambda_0,\,\phi_1). The equations for the Orthographic projection onto the (x,y) tangent plane reduce to the following (Snyder 1987):

\begin{align}
x &= R\,\cos(\phi) \sin\left(\lambda - \lambda_0\right) \\
y &= R\big[\cos(\phi_1) \sin(\phi) - \sin(\phi_1) \cos(\phi) \cos\left(\lambda - \lambda_0\right)\big]
\end{align}

Latitudes beyond the range of the map should be clipped by calculating the distance c from the center of the Orthographic 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\left(\lambda - \lambda_0\right)\,.

The point should be clipped from the map if cosc is negative.

The inverse formulas are given by:

\begin{align}
\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]
\end{align}

where

\begin{align}
\rho &= \sqrt{x^2 + y^2} \\
   c &= \arcsin\left(\frac{\rho}{R}\right)
\end{align}

For computation of the inverse formulas (e.g., using C/C++, Fortran, or other programming language), the use of the two-argument atan2 form of the inverse tangent function (as opposed to atan) is recommended. This ensures that the sign of the Orthographic projection as written is correct in all quadrants.

The inverse formulas are particularly useful when trying to project a variable defined on a (λ,φ) grid onto a rectilinear grid in (x,y). Direct application of the Orthographic projection yields scattered points in (x,y), which creates problems for plotting and numerical integration. One solution is to start from the (x,y) projection plane and construct the image from the values defined in (λ,φ) by using the inverse formulas of the Orthographic projection.

Orthographic projections over cylinders

In a wide sense, it is considered as orthographic all the projections with the point of perspective at infinity (and therefore parallel proyecting lines), regardless of the surface over which we project. These kind of projections distort angles and areas close to the poles.

A good example of these orthographic projections is Behrmann projection.

References

  • Snyder, J. P. (1987). Map Projections—A Working Manual (US Geologic Survey Professional Paper 1395). Washington, D.C.: US Government Printing Office. pp. 145–153. 
  • Snyder, J. P. (1993). Flattening the Earth: two thousand years of map projections. The University of Chicago Press. ISBN 0-226-76747-7 (paperback). 

Related

  • Orthographic projection

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