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Course angle and the distance between the two points on loxodrome (rhumb line).

Calculation of a distance on loxodrome (rhumb line) and course angle (azimuth) between two points with a given geographical coordinates.
Anton2015-08-05 16:12:38

In the 16th century, Flemish geographer Gerhard Mercator made a navigation map of the world, depicting the earth's surface on a plane, so that corners are not distorted on the map.
At present this method of Earths image is known as Mercator conformal cylindrical projection. This map was very convenient for the sailors as to come from point A to point B on the Mercator's map it's enough to draw a straight line between these points, measure its angle to the meridian and constantly adhere to this direction, for example by using a sextant and a polar star as landmark or using a magnetic compass(actually it's not that simple with the compass as it's not always pointing to the true north).
Mercator projection is still widely used for navigational maps.

Although, even the ancient sailors started to notice that the rhumb line is not always the shortest way between the two points and it's especially obvious for the long distances. If you draw a line on the globe, crossing all meridians at the same angle, it becomes clear why this is happening. The straight line on the Mercator map turns on the globe into the endlessly spinning spiral to the poles. That line is called loxodrome which means "slanting run" in Greek.
The following calculator calculates the course angle and the distance of transatlantic crossing from Las Palmas (Spain) to Bridgetown (Barbados) on the loxodrome. The resulting distance is different by tens of kilometers of the shortest path (see Distance calculator)

Calculation of constant azimuth and rhumb line lengthCreative Commons Attribution/Share-Alike License 3.0 (Unported)
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For the calculation of the course angle the following formulas are used:
\alpha = \arctan \left(\frac{\lambda_2-\lambda_1}{{\ln\left(tan(\frac{\pi}{4}+\frac{\varphi_2}{2})\cdot\left[\frac{1-e\cdot \sin{\varphi_2}}{1+e\cdot \sin{\varphi_2}}\right]^{\frac{e}{2}}\right)}-{\ln\left(tan(\frac{\pi}{4}+\frac{\varphi_1}{2})\cdot\left[\frac{1-e\cdot \sin{\varphi_1}}{1+e\cdot \sin{\varphi_1}}\right]^{\frac{e}{2}}\right)}}\right)[1]

Loxodrome length is calculated by the following formula:
S=a\cdot\sec\alpha\left[\left(1-\frac{1}{4}e^2\right)\Delta\varphi-\frac{3}{8}e^2(\sin{2\varphi_2}-\sin{2\varphi_1})\right][2]

, where \varphi_1,\lambda_1 - latitude and longitude of the first point
\varphi_2,\lambda_2 - latitude and longitude of the second point
e=\sqrt{1-\frac{b^2}{a^2}} -the eccentricity of the spheroid (a - the length of the major semiaxis, b - the length of the minor semiaxis)

At angles of 90 ° or 270 °, for the calculation of the arc length the following formula was used
S=a\cdot \sec\alpha \left(\lambda_2-\lambda_1\right)\cdot\cos\left(\frac{\varphi_2-\varphi_1}{2}\right)

Sources:
[1]В.С. Михайлов и др. Навигация и Лоция

[2]Miljenko Petrović DIFFERENTIAL EQUATION OF A LOXODROME ON THE SPHEROID

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