Editing Orthographic map projection

{{Short description|Azimuthal perspective map projection}}
[[File:Orthographic projection SW.jpg|thumb|upright=1.25|Orthographic projection (equatorial aspect) of eastern hemisphere 30W&ndash;150E]]
[[File:Orthographic with Tissot's Indicatrices of Distortion.svg|upright=1.25|thumb|The orthographic projection with [[Tissot's indicatrix]] of deformation.]]

'''Orthographic projection in cartography''' has been used since antiquity. Like the [[Stereographic map projection|stereographic projection]] and [[gnomonic projection]], [[orthographic projection]] is a [[General Perspective projection|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 [[infinity|infinite]] distance.  It depicts a [[Sphere|hemisphere]] of the [[globe]] as it appears from [[outer space]], where the [[horizon]] is a [[great circle]].  The shapes and areas are [[Distortion#Map projections|distorted]], particularly near the edges.<ref name="SnyderWorkingManual">{{Cite book | author=Snyder, J. P.| title=Map Projections—A Working Manual (US Geologic Survey Professional Paper 1395) | publisher=US Government Printing Office | location=Washington, D.C.| year=1987 | pages=[https://archive.org/details/Snyder1987MapProjectionsAWorkingManual/page/n159 145]–153 | url=https://archive.org/details/Snyder1987MapProjectionsAWorkingManual}}</ref><ref name="Snyder16">Snyder, John P. (1993). ''Flattening the Earth: Two Thousand Years of Map Projections'' pp. 16–18. Chicago and London: The University of Chicago Press. {{ISBN|9780226767475}}.</ref>

==History==
The [[orthographic projection]] has been known since antiquity, with its cartographic uses being well documented. [[Hipparchus]] used the projection in the 2nd century BC to determine the places of star-rise and star-set. In about 14 BC, Roman engineer [[Vitruvius|Marcus Vitruvius Pollio]] used the projection to construct sundials and to compute sun positions.<ref name="Snyder16" />

Vitruvius also seems to have devised the term orthographic (from the Greek ''orthos'' (= “straight”) and graphē (= “drawing”)) for the projection. However, the name ''[[analemma]]'', which also meant a sundial showing latitude and longitude, was the common name until [[François d'Aguilon]] of [[Antwerp]] promoted its present name in 1613.<ref name="Snyder16"/>

The earliest surviving maps on the projection appear as crude woodcut drawings of terrestrial globes of 1509 (anonymous), 1533 and 1551 (Johannes Schöner), and 1524 and 1551 (Apian). A highly-refined map, designed by Renaissance [[polymath]] [[Albrecht Dürer]] and executed by [[Johannes Stabius]], appeared in 1515.<ref name="Snyder16"/>

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 spherical orthographic projection are derived using [[trigonometry]].  They are written in terms of [[longitude]] (''&lambda;'') and [[latitude]] (''&phi;'') on the [[sphere]]. Define the [[radius]] of the [[sphere]] ''R'' and the ''center'' [[Point (geometry)|point]] (and [[Origin (mathematics)|origin]]) of the projection (''&lambda;''<sub>0</sub>, ''&phi;''<sub>0</sub>).  The [[equations]] for the orthographic projection onto the (''x'', ''y'') tangent plane reduce to the following:<ref name="SnyderWorkingManual" />

:<math>\begin{align}
x &= R\,\cos\varphi \sin\left(\lambda - \lambda_0\right) \\
y &= R\big(\cos\varphi_0 \sin\varphi - \sin\varphi_0 \cos\varphi \cos\left(\lambda - \lambda_0\right)\big)
\end{align}</math>

Latitudes beyond the range of the map should be clipped by calculating the [[angular distance]] ''c'' from the ''center'' of the orthographic projection.  This ensures that points on the opposite hemisphere are not plotted:

:<math>\cos c = \sin\varphi_0 \sin\varphi + \cos\varphi_0 \cos\varphi \cos\left(\lambda - \lambda_0\right)\,</math>.

The point should be clipped from the map if cos(''c'') is negative. That is, all points that are included in the mapping satisfy:
:<math>-\frac{\pi}{2} < c < \frac{\pi}{2}</math>.

The inverse formulas are given by:

:<math>\begin{align}
\varphi &= \arcsin\left(\cos c \sin\varphi_0 + \frac{y\sin c \cos\varphi_0}{\rho}\right) \\
\lambda &= \lambda_0 + \arctan\left(\frac{x\sin c}{\rho \cos c \cos\varphi_0 - y \sin c \sin\varphi_0}\right)
\end{align}</math>

where

:<math>\begin{align}
\rho &= \sqrt{x^2 + y^2} \\
   c &= \arcsin\frac{\rho}{R}
\end{align}</math>

For [[computation]] of the inverse formulas the use of the two-argument [[atan2]] form of the [[inverse tangent]] function (as opposed to [[Inverse trigonometric functions|atan]]) is recommended.  This ensures that the [[sign (mathematics)|sign]] of the orthographic projection as written is correct in all [[Cartesian coordinate system|quadrants]].

The inverse formulas are particularly useful when trying to project a variable defined on a (''&lambda;'', ''&phi;'') grid onto a rectilinear grid in (''x'', ''y'').  Direct application of the orthographic projection yields scattered points in (''x'', ''y''), which creates problems for [[graph of a function|plotting]] and [[numerical integration]].  One solution is to start from the (''x'', ''y'') projection plane and construct the image from the values defined in (''&lambda;'', ''&phi;'') by using the inverse formulas of the orthographic projection.

See References for an ellipsoidal version of the orthographic map projection.<ref>{{cite web |url=http://www.hydrometronics.com/downloads/Ellipsoidal%20Orthographic%20Projection.pdf |title=Ellipsoidal Orthographic Projection via ECEF and Topocentric (ENU)|author=Zinn, Noel |date=June 2011 |access-date=2011-11-11}}</ref>

{{comparison_azimuthal_projections.svg}}

==Orthographic projections onto cylinders==
In a wide sense, all projections with the point of perspective at infinity (and therefore parallel projecting lines) are considered as orthographic, regardless of the surface onto which they are projected. Such projections distort angles and areas close to the poles.{{Clarify|date=November 2014}}

An example of an orthographic projection onto a cylinder is the [[Lambert cylindrical equal-area projection]].

==See also==

* [[List of map projections]]
* [[Stereographic projection in cartography]]

==References==
{{Reflist}}

==External links==
{{Commons category|Orthographic projection (cartography)}}
*[http://mathworld.wolfram.com/OrthographicProjection.html Orthographic Projection—from MathWorld]

{{Map projections}}

[[Category:Map projections]]