If terrestrial points are projected geometrically

Its principal use is in navigational astronomy because it is useful for illustrating and solving the navigational triangle. It is also useful for illustrating celestial coordinates. If the plane is tangent at a point on the equator, the parallels (including the equator) appear as straight lines. The meridians would appear as ellipses, except that the meridian through the point of tangency would appear as a straight line and the one 90° away would appear as a circle (Figure 2).

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 nofollowing (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.