Editing Rhumb line (section)

===Connection to the Mercator projection===
[[File:Rhumb line vs great-circle arc.png|thumb|upright=1.3|A rhumb line (blue) compared to a great-circle arc (red) between Lisbon, Portugal and Havana, Cuba. Top: orthographic projection. Bottom: Mercator projection.]]
Let {{mvar|λ}} be the longitude of a point on the sphere, and {{mvar|φ}} its latitude. Then, if we define the map coordinates of the [[Mercator projection]] as
:<math>\begin{align}
x &= \lambda - \lambda_0 \, , \\
y &= \operatorname{gd}^{-1}\varphi = \operatorname{arsinh}(\tan\varphi)\, ,
\end{align}</math>
a loxodrome with constant [[Bearing (navigation)|bearing]] {{mvar|β}} from true north will be a straight line, since (using the expression in the previous section)
:<math>y = m x</math>
with a slope
:<math>m=\cot\beta\,.</math>

Finding the loxodromes between two given points can be done graphically on a Mercator map, or by solving a nonlinear system of two equations in the two unknowns {{math|1=''m'' = cot ''β''}} and {{math|''λ''<sub>0</sub>}}. There are infinitely many solutions; the shortest one is that which covers the actual longitude difference, i.e. does not make extra revolutions, and does not go "the wrong way around".

The distance between two points {{math|Δ''s''}}, measured along a loxodrome, is simply the absolute value of the [[secant (trigonometry)|secant]] of the bearing (azimuth) times the north–south distance (except for [[circles of latitude]] for which the distance becomes infinite):

:<math>\Delta s = R \, \big|(\varphi - \varphi_0)\cdot \sec \beta \big|</math>

where {{math|R}} is one of the [[Earth radius#Global average radii|earth average radii]].