Editing Great-circle distance (section)

=== From chord length ===
A line through three-dimensional space between points of interest on a [[spherical Earth]] is the [[Chord (geometry)|chord]] of the great circle between the points. The [[central angle]] between the two points can be determined from the chord length. The great circle distance is proportional to the central angle.

The great circle chord length, <math>\Delta\sigma_\text{c}\,\!</math>, may be calculated as follows for the corresponding unit sphere, by means of [[Cartesian coordinate system|Cartesian subtraction]]:
:<math>\begin{align}
  \Delta{X} &= \cos\phi_2\cos\lambda_2 - \cos\phi_1\cos\lambda_1;\\
  \Delta{Y} &= \cos\phi_2\sin\lambda_2 - \cos\phi_1\sin\lambda_1;\\
  \Delta{Z} &= \sin\phi_2 - \sin\phi_1;\\
          \Delta\sigma_\text{c} &= \sqrt{(\Delta{X})^2 + (\Delta{Y})^2 + (\Delta{Z})^2}.
\end{align}</math>

Substituting <math>\lambda_1 = -\tfrac12\Delta \lambda </math> and <math>\lambda_2 = \tfrac12 \Delta \lambda</math> this formula can be algebraically manipulated to the form shown above in {{slink||Computational formulae}}.