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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.

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 angles on the map are not distorted.
At present, this method of Earth's 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 a 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.

Even the ancient sailors noticed that the rhumb line is not always the shortest way between the two points, and it's self-evident 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 transatlantic crossing distance 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)

PLANETCALC, Calculation of constant azimuth and rhumb line length

Calculation of constant azimuth and rhumb line length

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Digits after the decimal point: 2
Azimuth
 
Distance in kilometers
 
Distance in nautical miles
 

For the calculation of the course angle the following formulas are used:
\alpha = \arctan \left(\frac{{\Delta}\lambda}{{\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
where
\Delta}\lambda = \begin{cases}\lambda_2-\lambda_1 &{\text{if }} |\lambda_2-\lambda_1|\leq180\textdegree\\360\textdegree+\lambda_2-\lambda_1 &{\text{if }} \lambda_2-\lambda_1{<}-180\textdegree\\\lambda_2-\lambda_1-360\textdegree &{\text{if }} \lambda_2-\lambda_1{>}180\textdegree\end{cases} 2
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]3

, 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|\lambda_2-\lambda_1|\cdot\cos\left(\varphi\right)

  1. V.S. Mikhailov, Navigation and Pilot book]] ↩

  2. Noè Murr comment ↩

  3. Miljenko Petrović DIFFERENTIAL EQUATION OF A LOXODROME ON THE SPHEROID ↩

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PLANETCALC, Course angle and the distance between the two points on loxodrome (rhumb line).
Anton2021-02-16 13:42:05

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