thumb|right|220px|Image of a loxodrome, or rhumb line, spiraling towards the [[North Pole]]

In navigation, a rhumb line (also rhumb () or loxodrome) is an arc crossing all meridians of longitude at the same angle. It is a path of constant azimuth relative to true north, which can be steered by maintaining a course of fixed bearing. When drift is not a factor, accurate tracking of a rhumb line course is independent of speed.

In practical navigation, a distinction is made between this true rhumb line and a magnetic rhumb line, with the latter being a path of constant bearing relative to magnetic north. While a navigator could easily steer a magnetic rhumb line using a magnetic compass, this course would not be true because the magnetic declination—the angle between true and magnetic north—varies across the Earth's surface.

To follow a true rhumb line, using a magnetic compass, a navigator must continuously adjust magnetic heading to correct for the changing declination. This was a significant challenge during the Age of Sail, as the correct declination could only be determined if the vessel's longitude was accurately known, the central unsolved problem of pre-modern navigation.

Using a sextant, under a clear night sky, it is possible to steer relative to a visible celestial pole star. The magnetic poles are not fixed in location. In the northern hemisphere, Polaris has served as a close approximation to true north for much of recent history. In the southern hemisphere, there is no such star, and navigators have relied on more complex methods, such as inferring the location of the southern celestial pole by reference to the Crux constellation (also known as the Southern Cross).

Steering a true rhumb line by compass alone became practical with the invention of the modern gyrocompass, an instrument that determines true north not by magnetism, but by referencing a stable internal vector of its own angular momentum.

Introduction

The effect of following a rhumb line course on the surface of a globe was first discussed by the Portuguese mathematician Pedro Nunes in 1537, in his Treatise in Defense of the Marine Chart, with further mathematical development by Thomas Harriot in the 1590s.

A rhumb line can be contrasted with a great circle, which is the path of shortest distance between two points on the surface of a sphere. On a great circle, the bearing to the destination point does not remain constant. If one were to drive a car along a great circle one would hold the steering wheel fixed, but to follow a rhumb line one would have to turn the wheel, turning it more sharply as the poles are approached. In other words, a great circle is locally "straight" with zero geodesic curvature, whereas a rhumb line has non-zero geodesic curvature.

Meridians of longitude and parallels of latitude provide special cases of the rhumb line, where their angles of intersection are respectively 0° and 90°. On a north–south passage the rhumb line course coincides with a great circle, as it does on an east–west passage along the equator.

On a Mercator projection map, any rhumb line is a straight line; a rhumb line can be drawn on such a map between any two points on Earth without going off the edge of the map. But it can extend beyond a side edge of the map, where it then continues from the opposite edge at the same slope and latitude it departed at (assuming that the map covers exactly 360 degrees of longitude).

Rhumb lines which cut meridians at oblique angles are loxodromic curves which spiral towards the poles. from rhémbein.

The 1878 edition of The Globe Encyclopaedia of Universal Information describes a loxodrome line as:</blockquote>

A misunderstanding could arise because the term "rhumb" had no precise meaning when it came into use. It applied equally well to the windrose lines as it did to loxodromes because the term only applied "locally" and only meant whatever a sailor did in order to sail with constant bearing, with all the imprecision that that implies. Therefore, "rhumb" was applicable to the straight lines on portolans when portolans were in use, as well as always applicable to straight lines on Mercator charts. For short distances portolan "rhumbs" do not meaningfully differ from Mercator rhumbs, but these days "rhumb" is synonymous with the mathematically precise "loxodrome" because it has been made synonymous retrospectively.

As Leo Bagrow states:

<blockquote>the word ('Rhumbline') is wrongly applied to the sea-charts of this period, since a loxodrome gives an accurate course only when the chart is drawn on a suitable projection. Cartometric investigation has revealed that no projection was used in the early charts, for which we therefore retain the name 'portolan'.</blockquote>

Mathematical description

For a sphere of radius 1, the azimuthal angle , the polar angle (defined here to correspond to latitude), and Cartesian unit vectors , , and can be used to write the radius vector 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>

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 scalar products

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

for constant traces out a parallel of latitude, while for constant 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 with the unit vector for any and , 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 with all meridians of longitude, and therefore must be parallel to the unit vector . As a result, a differential length 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 sine.

With this relationship between and , 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 isometric latitude.

In the Rhumb line, as the latitude tends to the poles, , , the isometric latitude , and longitude increases without bound, circling the sphere ever so fast in a spiral towards the pole, while tending to a finite total arc length Δ given by

:<math>\Delta s = R \, \big|(\pm\pi/2 - \varphi_0) \cdot \sec \beta\big|</math>

Connection to the Mercator projection

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 be the longitude of a point on the sphere, and 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 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 and . 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 , measured along a loxodrome, is simply the absolute value of the 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 is one of the earth average radii.

Application

Its use in navigation is directly linked to the style, or projection of certain navigational maps. A rhumb line appears as a straight line on a Mercator projection map.

The name is derived from Old French or Spanish respectively: "rumb" or "rumbo", a line on the chart which intersects all meridians at the same angle.

Generalizations

On the Riemann sphere

The surface of the Earth can be understood mathematically as a Riemann sphere, that is, as a projection of the sphere to the complex plane. In this case, loxodromes can be understood as certain classes of Möbius transformations.

Spheroid

The formulation above can be easily extended to a spheroid. The course of the rhumb line is found merely by using the ellipsoidal isometric latitude. In formulas above on this page, substitute the conformal latitude on the ellipsoid for the latitude on the sphere. Similarly, distances are found by multiplying the ellipsoidal meridian arc length by the secant of the azimuth.

See also

  • Great circle
  • Geodesics on an ellipsoid
  • Great ellipse
  • Isoazimuthal
  • Rhumbline network
  • Seiffert's spiral
  • Small circle

References

Note: this article incorporates text from the 1878 edition of The Globe Encyclopaedia of Universal Information, a work in the public domain

Further reading

  • Constant Headings and Rhumb Lines at MathPages.
  • RhumbSolve(1), a utility for ellipsoidal rhumb line calculations (a component of GeographicLib).
  • An online version of RhumbSolve.
  • Navigational Algorithms Paper: The Sailings.
  • Chart Work - Navigational Algorithms Chart Work free software: Rhumb line, Great Circle, Composite sailing, Meridional parts. Lines of position Piloting - currents and coastal fix.
  • Mathworld Loxodrome.