Orthographic Projection (cartography) - Mathematics

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:

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

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

.

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

The inverse formulas are given by:

\begin{align} \varphi &= \arcsin\left \\ \lambda &= \lambda_0 + \arctan\left \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.

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

Read more about this topic:  Orthographic Projection (cartography)

Famous quotes containing the word mathematics:

    “It is a monstrous thing to force a child to learn Latin or Greek or mathematics on the ground that they are an indispensable gymnastic for the mental powers. It would be monstrous even if it were true.”
    —George Bernard Shaw (1856–1950)

    “In mathematics he was greater
    Than Tycho Brahe, or Erra Pater:
    For he, by geometric scale,
    Could take the size of pots of ale;
    Resolve, by sines and tangents straight,
    If bread and butter wanted weight;
    And wisely tell what hour o’ th’ day
    The clock doth strike, by algebra.”
    —Samuel Butler (1612–1680)

    “Why does man freeze to death trying to reach the North Pole? Why does man drive himself to suffer the steam and heat of the Amazon? Why does he stagger his mind with the mathematics of the sky? Once the question mark has arisen in the human brain the answer must be found, if it takes a hundred years. A thousand years.”
    —Walter Reisch (1903–1963)