Editing Great-circle distance (section)

=== 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.