Orthographic map projection

Mathematics

The formulas for the spherical 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 (λ₀, φ₀). The equations for the orthographic projection onto the (x, y) tangent plane reduce to the following: $${\displaystyle \begin{array}{cc}x& =R\cos \phi \sin \left(\lambda -{\lambda }_0\right)\\ y& =R(\cos {\phi }_0\sin \phi -\sin {\phi }_0\cos \phi \cos \left(\lambda -{\lambda }_0\right))\end{array}}$$

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: $${\displaystyle \cos c=\sin {\phi }_0\sin \phi +\cos {\phi }_0\cos \phi \cos \left(\lambda -{\lambda }_0\right).}$$ The point should be clipped from the map if cos(c) is negative. That is, all points that are included in the mapping satisfy: $${\displaystyle -\frac{\pi }{2}<c<\frac{\pi }{2}.}$$

The inverse formulas are given by: $${\displaystyle \begin{array}{cc}\phi & =\arcsin \left(\cos c\sin {\phi }_0+\frac{y\sin c\cos {\phi }_0}{\rho }\right)\\ \lambda & ={\lambda }_0+\arctan \left(\frac{x\sin c}{\rho \cos c\cos {\phi }_0-y\sin c\sin {\phi }_0}\right)\end{array}}$$ where $${\displaystyle \begin{array}{cc}\rho & =\sqrt{x^2+y^2}\\ c& =\arcsin \frac{\rho }{R}\end{array}}$$

For computation of the inverse formulas 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.

See References for an ellipsoidal version of the orthographic map projection.

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.

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