Editing Orthographic map projection (section)

==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}}