Vincenty — Fórmula literal única

$$ \begin{aligned} s &= 6356752.314245179 \cdot \Bigg( 1 + \frac{\cos^2\alpha \frac{(6378137^2-6356752.314245179^2)}{6356752.314245179^2}}{16384} \big( 4096 + \cos^2\alpha \frac{(6378137^2-6356752.314245179^2)}{6356752.314245179^2} (-768 + (\cos^2\alpha \frac{(6378137^2-6356752.314245179^2)}{6356752.314245179^2}) (320-175 (\cos^2\alpha \frac{(6378137^2-6356752.314245179^2)}{6356752.314245179^2})) ) \big) \Bigg) \\ &\quad \cdot \Bigg[ \sigma - \frac{\cos^2\alpha \frac{(6378137^2-6356752.314245179^2)}{6356752.314245179^2}}{1024} (256 + \cos^2\alpha \frac{(6378137^2-6356752.314245179^2)}{6356752.314245179^2} (-128 + \cos^2\alpha \frac{(6378137^2-6356752.314245179^2)}{6356752.314245179^2} (74-47 \cos^2\alpha \frac{(6378137^2-6356752.314245179^2)}{6356752.314245179^2}))) ) \\ &\quad \quad \cdot \sin\sigma \Big( \cos 2\sigma_m + \frac{f}{16}\cos^2\alpha(4+f(4-3\cos^2\alpha)) \cdot \sin\sigma (\cos 2\sigma_m + \frac{f}{16}\cos^2\alpha(4+f(4-3\cos^2\alpha)) \cdot \cos\sigma(-1+2\cos^2 2\sigma_m)) \Big) \Bigg] \end{aligned} $$

On \(\sigma, \alpha, \cos2\sigma_m\) i \(\lambda\) es defineixen implícitament per les expressions de Vincenty amb:

$$ \begin{aligned} U_1 &= \arctan((1-1/298.257223563)\tan 0.8114900154100151), \quad U_2 = \arctan((1-1/298.257223563)\tan 0.2836571932194256),\\ \lambda &= -1.0381432891827342 + (1-C) \frac{1}{298.257223563} \sin\alpha (\sigma + C \sin\sigma (\cos 2\sigma_m + C \cos\sigma(-1+2\cos^2 2\sigma_m))),\\ \sin\sigma &= \sqrt{(\cos U_2 \sin \lambda)^2 + (\cos U_1 \sin U_2 - \sin U_1 \cos U_2 \cos \lambda)^2},\\ \cos\sigma &= \sin U_1 \sin U_2 + \cos U_1 \cos U_2 \cos \lambda,\\ \sigma &= \operatorname{atan2}(\sin\sigma, \cos\sigma),\\ \sin\alpha &= \frac{\cos U_1 \cos U_2 \sin\lambda}{\sin\sigma}, \quad \cos^2\alpha = 1 - \sin^2\alpha,\\ \cos 2\sigma_m &= \cos\sigma - \frac{2 \sin U_1 \sin U_2}{\cos^2\alpha},\\ C &= \frac{1/298.257223563}{16} \cos^2\alpha \left(4 + \frac{1}{298.257223563} (4-3 \cos^2\alpha)\right) \end{aligned} $$