README.md

Inverse Vincenty's method

Solving the inverse problem of geodesy on the WGS-84 Earth model. It should be said that I have not considered special cases such as two nearly antipodal points, etc.

Algorithm

Suppose that there are two points on the Earth's surface, $\overline{A}\left(\varphi_1, L_1\right)$ and $\overline{B}\left(\varphi_2, L_2\right)$, each of which is defined by certain latitude ($\varphi$) and longitude ($L$) coordinates, then the distance $D$ between these points can be determined using the inverse Vincenty's method.

Introduce the following notation:
$a$ is the equatorial radius of the Earth model,
$f$ is the flattening factor of the Earth model,
$b$ is the polar radius of the Earth model,
$\alpha_1,\alpha_2$ — direct azimuths in points,
$U_1 = \arctan{\left(\left(1 - f\right)\tan{\varphi_1}\right)}$,
$U_2 = \arctan{\left(\left(1 - f\right)\tan{\varphi_2}\right)}$,
$L = L_2 - L_1$,
$\varepsilon_\lambda$ — the permissible error.

Set the initial value of $\lambda_k = L, k = 0$, and calculate the following expressions:

$$ \sin \sigma ={\sqrt {\left(\cos U_{2}\sin \lambda_k \right)^{2}+\left(\cos U_{1}\sin U_{2}-\sin U_{1}\cos U_{2}\cos \lambda_k \right)^{2}}}, $$

$$ \cos \sigma =\sin U_{1}\sin U_{2}+\cos U_{1}\cos U_{2}\cos \lambda_k, $$

$$ \sigma =\text {arctan2} \left(\sin \sigma ,\cos \sigma \right), $$

$$ \sin \alpha ={\frac {\cos U_{1}\cos U_{2}\sin \lambda_k }{\sin \sigma }}, $$

$$ \cos \left(2\sigma _{\text{m}}\right)=\cos \sigma -{\frac {2\sin U_{1}\sin U_{2}}{1-\sin ^{2}\alpha }}, $$

$$ C={\frac {1}{16}}f\cos ^{2}\alpha \left[4+f\left(4-3\cos ^{2}\alpha \right)\right], $$

$$ \lambda_{k+1} =L+(1-C)f\sin \alpha \left\{\sigma +C\sin \sigma \left[\cos \left(2\sigma _{\text{m}}\right)+C\cos \sigma \left(-1+2\cos ^{2}\left(2\sigma _{\text{m}}\right)\right)\right]\right\}. $$

The iterative calculation continues until the end condition is met:

$$ \left|\lambda_{k+1}-\lambda_{k}\right|\leq\varepsilon_\lambda, $$

or until the limit on iterations is reached.

The geodetic distance and azimuths are then calculated:

$$ u^2 = \frac{a^2 - b^2}{b^2} \cos^2 \alpha, $$

$$ A = 1 + \frac{u^2}{16384} \left\{ 4096 + u^2 \left[ -768 +u^2 (320 - 175u^2) \right] \right\}, $$

$$ B = \frac{u^2}{1024} \left\{ 256 + u^2 \left[ -128 + u^2 (74-47 u^2) \right] \right\}, $$

$$ \Delta \sigma = B \sin \sigma \left\{\cos(2 \sigma_m) + \frac{1}{4}B \left[\cos \sigma \left(-1+2 \cos^2(2 \sigma_m)\right) - \frac{1}{6} B \cos(2 \sigma_m) (-3+4 \sin^2 \sigma) \left(-3+4 \cos^2 (2 \sigma_m)\right)\right]\right\}, $$

$$ D = b A\left(\sigma - \Delta \sigma\right), $$

$$ \alpha _{1}=\text {arctan2} \left(\cos U_{2}\sin \lambda_{k+1} ,\cos U_{1}\sin U_{2}-\sin U_{1}\cos U_{2}\cos \lambda_{k+1} \right), $$

$$ \alpha _{2}=\text {arctan2} \left(\cos U_{1}\sin \lambda_{k+1} ,\cos U_{1}\sin U_{2}\cos \lambda_{k+1} -\sin U_{1}\cos U_{2}\right). $$

Vincenty then proposed a more compact formula for $A$ ta $B$ using the Helmert parameter:

$$ F = \frac { \sqrt {(1 + u^2)} - 1}{ \sqrt {(1 + u^2)} + 1},$$

$$ A = \frac {1 + \frac {1}{4} F^2}{1 - F}, $$

$$ B = F\left(1 - \frac {3}{8}F^2\right).$$

References

1. T. Vincenty. DIRECT AND INVERSE SOLUTIONS OF GEODESICS ON THE ELLIPSOID WITH APPLICATION OF NESTED EQUATIONS. Survey Review, 23(176):88–93, 4 1975. https://doi.org/10.1179/sre.1975.23.176.88
2. T. Vincenty. GEODETIC INVERSE SOLUTION BETWEEN ANTIPODAL POINTS. DMAAC Geodetic Survey Squadron. 1975. https://doi:10.5281/zenodo.32999
3. International Civil Aviation Organization. World Geodetic System – 1984 (WGS-84) manual, 2002. https://www.icao.int/NACC/Documents/Meetings/2014/ECARAIM/REF08-Doc9674.pdf