User talk:Tim Zukas/nav

If a navigator is starting at latitude $$\scriptstyle\phi_1\,\!$$ and plans to travel the great circle to a point at latitude $$\scriptstyle\phi_2\,\!$$, with a longitude difference between the points of L (positive eastward), his initial course $$\alpha\,\!$$ is given by


 * $$\tan \alpha

= \frac{\sin L}{(\cos \phi_1)(\tan \phi_2)- (\sin\phi_1)(\cos L)}$$

If a navigator begins at latitude $$\scriptstyle\phi_s\,\!$$ (the "standpoint") and plans to travel the great circle to a point at latitude $$\scriptstyle\phi_s\,\!$$ (the "forepoint"), with a longitude difference between the points of $$\scriptstyle\Delta\lambda\,\!$$ (positive eastward), his initial course $$\alpha\,\!$$ is given by
 * $$\begin{align}S\!A&=\cos(\phi_f)\sin(\Delta\lambda);\\

S\!B&=\cos(\phi_s)\sin(\phi_f)-\sin(\phi_s)\cos(\phi_f)\cos(\Delta\lambda);{}_{\color{white}.}\\ \tan(\alpha_s)&=\frac{S\!A}{T\!B};{}_{\color{white}.}\end{align}\,\!$$

The central angle $$ \sigma $$ between the two points is given by


 * $$\cos \;\sigma = (\cos \phi_1)(\cos \phi_2)(\cos L) + (\sin \phi_1)(\sin \phi_2)$$

The central angle between the two points, $$ \Delta\sigma $$, is given by
 * $$\tan(\Delta\sigma)=\tan(\sigma_f-\sigma_s)=\frac{\sqrt{S\!A^2+T\!B^2}}{\sin(\phi_s)\sin(\phi_f)+\cos(\phi_s)\cos(\phi_f)\cos(\Delta\lambda)};{}_{\color{white}.}\,\!$$ Tim Zukas (talk) 02:30, 27 February 2011 (UTC)