Editing Rhumb line (section)

==Mathematical description==
For a sphere of radius 1, the azimuthal angle {{mvar|λ}}, the polar angle {{math|−{{sfrac|π|2}} ≤ ''φ'' ≤ {{sfrac|π|2}}}} (defined here to correspond to latitude), and [[Cartesian coordinate system#Representing a vector in the standard basis|Cartesian unit vectors]] {{math|'''i'''}}, {{math|'''j'''}}, and {{math|'''k'''}} can be used to write the radius vector {{math|'''r'''}} as

:<math>\mathbf{r}(\lambda,\varphi) = (\cos{\lambda} \cdot \cos{\varphi})  \mathbf{i} + (\sin{\lambda} \cdot \cos{\varphi})  \mathbf{j} + (\sin{\varphi}) \mathbf{k} \, .</math>

[[Orthogonality#Euclidean vector spaces|Orthogonal unit vectors]] in the azimuthal and polar directions of the sphere can be written

:<math>\begin{align}
\boldsymbol{\hat\lambda}(\lambda,\varphi) &= \sec{\varphi} \frac{\partial\mathbf{r}}{\partial\lambda} = (-\sin{\lambda}) \mathbf{i} + (\cos{\lambda}) \mathbf{j} \, , \\[8pt]
\boldsymbol{\hat\varphi}(\lambda,\varphi) &= \frac{\partial\mathbf{r}}{\partial\varphi} = (-\cos{\lambda} \cdot \sin{\varphi}) \mathbf{i} + (-\sin{\lambda} \cdot \sin{\varphi}) \mathbf{j} + (\cos{\varphi}) \mathbf{k} \, ,
\end{align}</math>

which have the [[Dot product#Geometric definition|scalar products]]

:<math>\boldsymbol{\hat\lambda} \cdot \boldsymbol{\hat\varphi} = \boldsymbol{\hat\lambda} \cdot \mathbf{r} = \boldsymbol{\hat\varphi} \cdot \mathbf{r} = 0 \, .</math>

{{math|'''λ̂'''}} for constant {{mvar|φ}} traces out a parallel of latitude, while {{math|'''φ̂'''}} for constant {{mvar|λ}} traces out a meridian of longitude, and together they generate a plane tangent to the sphere.

The unit vector
:<math>\mathbf{\boldsymbol{\hat\beta}}(\lambda,\varphi) = (\sin{\beta}) \boldsymbol{\hat\lambda} + (\cos{\beta}) \boldsymbol{\hat\varphi}</math>
has a constant angle {{mvar|β}} with the unit vector {{math|'''φ̂'''}} for any {{mvar|λ}} and {{mvar|φ}}, since their scalar product is

:<math>\boldsymbol{\hat\beta} \cdot \boldsymbol{\hat\varphi} = \cos{\beta} \, .</math>

A loxodrome is defined as a curve on the sphere that has a constant angle {{mvar|β}} with all meridians of longitude, and therefore must be parallel to the unit vector {{math|'''β̂'''}}. As a result, a differential length {{mvar|ds}} along the loxodrome will produce a differential displacement

:<math>\begin{align}
d\mathbf{r} &= \boldsymbol{\hat\beta} \, ds \\[8px]
\frac{\partial\mathbf{r}}{\partial\lambda} \, d\lambda + \frac{\partial\mathbf{r}}{\partial\varphi} \, d\varphi &= \bigl((\sin{\beta}) \, \boldsymbol{\hat\lambda} + (\cos{\beta}) \, \boldsymbol{\hat\varphi}\bigr) ds \\[8px]
(\cos{\varphi}) \, d\lambda \, \boldsymbol{\hat\lambda} + d\varphi \, \boldsymbol{\hat\varphi} &= (\sin{\beta}) \, ds \, \boldsymbol{\hat\lambda} + (\cos{\beta}) \, ds \, \boldsymbol{\hat\varphi} \\[8px]
ds &= \frac{\cos{\varphi} }{\sin{\beta}} \, d\lambda = \frac{d\varphi}{\cos{\beta}} \\[8px]
\frac{d\lambda}{d\varphi} &= \tan{\beta} \cdot \sec{\varphi} \\[8px]
\lambda(\varphi\,|\,\beta,\lambda_0,\varphi_0) &= \tan\beta \cdot \big( \operatorname{gd}^{-1}\varphi - \operatorname{gd}^{-1}\varphi_0 \big) + \lambda_0 \\[8px]
\varphi(\lambda\,|\,\beta,\lambda_0,\varphi_0) &= \operatorname{gd} \big((\lambda - \lambda_0) \cot\beta + \operatorname{gd}^{-1}\varphi_0\big)
\end{align}</math>

where <math>\operatorname{gd}</math> and <math>\operatorname{gd}^{-1}</math> are the [[Gudermannian function]] and its inverse, <math>\operatorname{gd}\psi = \arctan(\sinh\psi),</math> <math>\operatorname{gd}^{-1}\varphi = \operatorname{arsinh}(\tan\varphi),</math> and <math>\operatorname{arsinh}</math> is the [[inverse hyperbolic functions|inverse hyperbolic sine]].

With this relationship between {{mvar|λ}} and {{mvar|φ}}, the radius vector becomes a parametric function of one variable, tracing out the loxodrome on the sphere:

:<math>\mathbf{r}(\lambda\,|\,\beta,\lambda_0,\varphi_0) = \big(\cos{\lambda} \cdot \operatorname{sech} \psi \big) \mathbf{i} +
\big(\sin{\lambda} \cdot \operatorname{sech}\psi\big) \mathbf{j} + \big(\tanh\psi\big) \mathbf{k} \, ,</math>

where

:<math>\psi \equiv (\lambda - \lambda_0) \cot\beta + \operatorname{gd}^{-1}\varphi_0 = \operatorname{gd}^{-1}\varphi</math>

is the [[Latitude#Isometric latitude|isometric latitude]].<ref>James Alexander, Loxodromes: A Rhumb Way to Go, "Mathematics Magazine", Vol. 77. No. 5, Dec. 2004. [http://hans.fugal.net/src/lindbergh/mathmag349-356.pdf]</ref>

In the Rhumb line, as the latitude tends to the poles, {{math|''φ'' → ±{{sfrac|π|2}}}}, {{math|sin ''φ'' → ±1}}, the isometric latitude {{math|arsinh(tan ''φ'') → ± ∞}}, and longitude {{mvar|λ}} increases without bound, circling the sphere ever so fast in a spiral towards the pole, while tending to a finite total arc length Δ{{math|s}} given by
:<math>\Delta s = R \, \big|(\pm\pi/2 - \varphi_0) \cdot \sec \beta\big|</math>

===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]].