Editing Great-circle distance (section)

== Relation between central angle and chord length ==
The central angle <math>\Delta\sigma</math> is related with the [[Chord (geometry)|chord]] length of unit sphere <math>\Delta\sigma_\text{c}\,\!</math>:
:<math>\begin{align}
  \Delta\sigma &= 2\arcsin \frac{\Delta\sigma_\text{c}}{2} ,\\
  \Delta\sigma_\text{c} &= 2\sin \frac{\Delta\sigma}{2} .
\end{align}</math>

For short-distance approximation (<math>|\Delta\sigma_\text{c}| \ll 1</math>),
:<math>\Delta\sigma = \Delta\sigma_\text{c} \left(1 + \frac{1}{24} \left(\Delta\sigma_\text{c}\right)^2 + \cdots \right).</math>

===Computational formulae===
On computer systems with low [[floating point]] precision, the spherical law of cosines formula can have large [[rounding error]]s if the distance is small (if the two points are a kilometer apart on the surface of the Earth, the cosine of the central angle is near 0.99999999). For modern [[IEEE 754|64-bit floating-point numbers]], the spherical law of cosines formula, given above, does not have serious rounding errors for distances larger than a few meters on the surface of the Earth.<ref>{{cite web |url=http://www.movable-type.co.uk/scripts/latlong.html |title=Calculate distance, bearing and more between Latitude/Longitude points |access-date=10 Aug 2013}}</ref> The [[haversine formula]] is [[condition number|numerically better-conditioned]] for small distances by using the chord-length relation:<ref>{{cite journal |last1=Sinnott |first1=Roger W. |title=Virtues of the Haversine |journal=Sky and Telescope |date=August 1984 |volume=68 |issue=2 |page=159}}</ref>

:<math>\begin{align}
  \Delta\sigma &= \operatorname{archav}\left( \operatorname{hav}\left(\Delta\phi\right) + \left(1 - \operatorname{hav}(\Delta\phi) - \operatorname{hav}(\phi_1 + \phi_2)\right) \operatorname{hav}\left(\Delta\lambda\right)\right) .
\end{align}</math>

Historically, the use of this formula was simplified by the availability of tables for the [[haversine]] function: <math>\operatorname{hav} \theta = \sin^2 \frac{\theta}{2}</math> and <math>\operatorname{archav} x = 2 \arcsin \sqrt{x}</math>.

The following shows the equivalent formula expressing the chord length explicitly:
:<math>\begin{align}\Delta\sigma_\text{c}&=2\sqrt{\sin^2\left(\frac{\Delta\phi}{2}\right)+\cos{\phi_1}\cdot\cos{\phi_2}\cdot\sin^2\left(\frac{\Delta\lambda}{2}\right)} \ , \\
&=2\sqrt{\left(\sin \frac{\Delta \lambda}{2} \cos\phi_\textrm{m} \right)^2 + \left(\cos \frac{\Delta \lambda}{2} \sin \frac{\Delta \phi}{2} \right)^2} \ ,
\end{align}</math>
where <math>\phi_\text{m}=\tfrac12(\phi_1+\phi_2)</math>.

Although this formula is accurate for most distances on a sphere, it too suffers from rounding errors for the special (and somewhat unusual) case of antipodal points. A formula that is accurate for all distances is the following special case of the [[Vincenty's formulae|Vincenty formula]] for an ellipsoid with equal major and minor axes:<ref>{{cite journal |last = Vincenty |first = Thaddeus |author-link = Thaddeus Vincenty |title = Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations |journal = Survey Review |volume = 23 |issue = 176 |pages = 88–93 |publisher = [[Directorate of Overseas Surveys]] |location = Kingston Road, Tolworth, Surrey |date = 1975-04-01 |url =http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf |access-date = 2008-07-21 |doi = 10.1179/sre.1975.23.176.88|bibcode = 1975SurRv..23...88V }}</ref>

:<math>\begin{align}
\Delta\sigma = {\operatorname{atan2}} \Bigl( &\sqrt{\left(
      \cos\phi_2\sin\Delta\lambda\right)^2 +
      \left(\cos\phi_1\sin\phi_2 - \sin\phi_1\cos\phi_2\cos\Delta\lambda
    \right)^2}, \\
    &\quad {\sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos\Delta\lambda} \Bigr),
\end{align}</math>

where {{tmath|\operatorname{atan2}(y, x)}} is the [[atan2|two-argument arctangent]]. Using atan2 ensures that the correct quadrant is chosen.

=== Vector version ===
Another representation of similar formulas, but using [[n-vector|normal vectors]] instead of latitude and longitude to describe the positions, is found by means of 3D [[Vector calculus#Vector operations|vector algebra]], using the [[dot product]], [[cross product]], or a combination:<ref>{{cite journal |last1= Gade |first1= Kenneth |year= 2010 |title= A non-singular horizontal position representation |journal= The Journal of Navigation |publisher= Cambridge University Press |volume= 63 |issue= 3 |pages=395–417 |url=http://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf |doi= 10.1017/S0373463309990415 |bibcode= 2010JNav...63..395G }}</ref>

:<math>\begin{align}
  \Delta\sigma &= \arccos \left(\mathbf n_1 \cdot \mathbf n_2\right) \\
               &= \arcsin \left| \mathbf n_1 \times \mathbf n_2 \right| \\
               &= \arctan \frac{\left| \mathbf n_1 \times \mathbf n_2 \right|}{\mathbf n_1 \cdot \mathbf n_2} \\
\end{align}</math>

where <math>\mathbf n_1</math> and <math>\mathbf n_2</math> are the normals to the sphere at the two positions 1 and 2. Similarly to the equations above based on latitude and longitude, the expression based on arctan is the only one that is [[Inverse trigonometric functions#Numerical accuracy|well-conditioned for all angles]]. The expression based on arctan requires the magnitude of the cross product over the dot product.

=== 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}}.