Editing Rhumb line

{{short description|Arc crossing all meridians of longitude at the same angle}}
{{Distinguish|Rhumbline network}}
{{for multi|the album|The Rhumb Line|the board game|Rhumb Line (board game)}}
{{More citations needed|date=August 2017}}
{{Use dmy dates|date=October 2019}}
[[File:Loxodrome.png|thumb|right|220px|Image of a loxodrome, or rhumb line, spiraling towards the [[North Pole]]]]

In [[navigation]], a '''rhumb line''', '''rhumb''' ({{IPAc-en|r|ʌ|m}}), or '''loxodrome''' is an [[arc (geometry)|arc]] crossing all [[meridian (geography)|meridians]] of [[longitude]] at the same [[angle]], that is, a path with constant [[azimuth]] ([[bearing (navigation)|bearing]] as measured relative to [[true north]]).
Navigation on a fixed [[course (navigation)|course]] (i.e., [[steering]] the vessel to follow a constant [[cardinal direction]]) would result in a rhumb-line [[track (navigation)|track]].

==Introduction==
{{further|Bearing (navigation)#Arcs}}

The effect of following a rhumb line course on the surface of a globe was first discussed by the [[Portuguese people|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 theoretically a loxodrome can extend beyond the right edge of the map, where it then continues at the left edge with the same slope (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.<ref name="EOS" /> On a Mercator projection the [[North Pole]] and [[South Pole]] occur at infinity and are therefore never shown. However the full loxodrome on an infinitely high map would consist of infinitely many line segments between the two edges. On a [[stereographic projection]] map, a loxodrome is an [[equiangular spiral]] whose center is the north or south pole.

All loxodromes spiral from one [[Geographical pole|pole]] to the other. Near the poles, they are close to being [[logarithmic spiral]]s (which they are exactly on a [[stereographic projection]], see below), so they wind around each pole an infinite number of times but reach the pole in a finite distance. The pole-to-pole length of a loxodrome (assuming a perfect [[sphere]]) is the length of the [[meridian (geography)|meridian]] divided by the [[cosine]] of the bearing away from true north. Loxodromes are not defined at the poles.

<gallery caption="Three views of a pole-to-pole loxodrome" widths="250px" heights="250px" perrow="3">
File:Loxodrome-1.gif
File:Loxodrome-2.gif
File:Loxodrome-3.gif
</gallery>

==Etymology and historical description==
The word ''loxodrome'' comes from [[Ancient Greek language|Ancient Greek]] λοξός ''loxós'': "oblique" + δρόμος ''drómos'': "running" (from δραμεῖν ''drameîn'': "to run"). The word ''rhumb'' may come from [[Spanish language|Spanish]] or [[Portuguese language|Portuguese]] ''rumbo/rumo'' ("course" or "direction") and Greek [[rhombus|ῥόμβος]] ''rhómbos'',<ref>''[http://www.thefreedictionary.com/rhumb Rhumb]'' at TheFreeDictionary</ref> from ''rhémbein''.

The 1878 edition of ''The Globe Encyclopaedia of Universal Information'' describes a ''loxodrome line'' as:<ref name="Globe"/>

<blockquote>'''Loxodrom′ic''' Line is a curve which cuts every member of a system of lines of curvature of a given surface at the same angle. A ship sailing towards the same point of the compass describes such a line which cuts all the meridians at the same angle. In Mercator's Projection (q.v.) the Loxodromic lines are evidently straight.<ref name="Globe">Ross, J.M. (editor) (1878). ''[https://archive.org/details/globeencyclopae01rossgoog The Globe Encyclopaedia of Universal Information]'', Vol. IV, Edinburgh-Scotland, Thomas C. Jack, Grange Publishing Works, retrieved from [[Google Books]] 2009-03-18;</ref></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 line]]s 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 (navigation)|bearing]], with all the imprecision that that implies. Therefore, "rhumb" was applicable to the straight lines on [[portolan]]s 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:<ref name="Bagrow2010">{{cite book|author=Leo Bagrow|title=History of Cartography|url=https://books.google.com/books?id=OBeB4tDmJv8C&pg=PA65|year=2010|publisher=Transaction Publishers|isbn=978-1-4128-2518-4|page=65}}</ref>
<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 {{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]].

==Application==
Its use in navigation is directly linked to the style, or ''[[map projection|projection]]'' of certain navigational maps. A rhumb line appears as a straight line on a [[Mercator projection]] map.<ref name="EOS">Oxford University Press [http://www.encyclopedia.com/doc/1O225-rhumbline.html Rhumb Line]. The Oxford Companion to Ships and the Sea, Oxford University Press, 2006. Retrieved from Encyclopedia.com 18 July 2009.</ref>

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.<ref name="EOS" /> On a plane surface this would be the shortest distance between two points. Over the Earth's surface at low latitudes or over short distances it can be used for plotting the course of a vehicle, aircraft or ship.<ref name="EOS" /> Over longer distances and/or at higher latitudes the [[great circle]] route is significantly shorter than the rhumb line between the same two points. However the inconvenience of having to continuously change bearings while travelling a great circle route makes ''rhumb line navigation'' appealing in certain instances.<ref name="EOS" />

The point can be illustrated with an east–west passage over [[90 degrees]] of longitude along the [[equator]], for which the great circle and rhumb line distances are the same, at {{convert|5400|nmi|km|abbr=off|order=flip}}. At 20 degrees north the great circle distance is {{convert|4997|nmi|km|abbr=on|order=flip}} while the rhumb line distance is {{convert|5074|nmi|km|abbr=on|order=flip}}, about 1.5% further. But at 60 degrees north the great circle distance is {{convert|2485|nmi|km|abbr=on|order=flip}} while the rhumb line is {{convert|2700|nmi|km|abbr=on|order=flip}}, a difference of 8.5%. A more extreme case is the air route between [[New York City]] and [[Hong Kong]], for which the rhumb line path is {{convert|9700|nmi|km|abbr=on|order=flip}}. The great circle route over the North Pole is {{convert|7000|nmi|km|abbr=on|order=flip}}, or {{frac|5|1|2}} hours less flying time at a typical [[Cruise (flight)|cruising speed]].

Some old maps in the Mercator projection have grids composed of lines of [[latitude]] and [[longitude]] but also show rhumb lines which are oriented directly towards north, at a right angle from the north, or at some angle from the north which is some simple rational fraction of a right angle. These rhumb lines would be drawn so that they would converge at certain points of the map: lines going in every direction would converge at each of these points. See [[compass rose]]. Such maps would necessarily have been in the Mercator projection therefore not all old maps would have been capable of showing rhumb line markings.

The radial lines on a compass rose are also called ''rhumbs''. The expression ''"sailing on a rhumb"'' was used in the 16th–19th centuries to indicate a particular compass heading.<ref name="EOS" />

Early navigators in the time before the invention of the [[marine chronometer]] used rhumb line courses on long ocean passages, because the ship's latitude could be established accurately by sightings of the Sun or stars but there was no accurate way to determine the longitude. The ship would sail north or south until the latitude of the destination was reached, and the ship would then sail east or west along the rhumb line (actually a [[Circle of latitude|parallel]], which is a special case of the rhumb line), maintaining a constant latitude and recording regular estimates of the distance sailed until evidence of land was sighted.<ref>A Brief History of British Seapower, David Howarth, pub. Constable & Robinson, London, 2003, chapter 8.</ref>

==Generalizations==

===On the Riemann sphere===
{{main|Möbius transformation}}
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 transformation]]s.

===Spheroid===
The formulation above can be easily extended to a [[spheroid]].<ref>
{{Cite journal| doi = 10.1093/mnras/106.2.124| title = On a Problem in Navigation| journal = Monthly Notices of the Royal Astronomical Society| volume = 106| issue = 2| pages = 124&ndash;127| year = 1946| last1 = Smart | first1 = W. M.| bibcode = 1946MNRAS.106..124S| doi-access = free}}
</ref><ref>
{{Cite journal| doi = 10.1017/S0373463300045549| title = Loxodromic Distances on the Terrestrial Spheroid| journal = Journal of Navigation| volume = 3| issue = 2| pages = 133&ndash;140| year = 1950| last1 = Williams | first1 = J. E. D.| bibcode = 1950JNav....3..133W| s2cid = 128651304}}
</ref><ref>
{{Cite journal| doi = 10.1017/S0373463300010791| title = On Loxodromic Navigation| journal = Journal of Navigation| volume = 45| issue = 2| pages = 292&ndash;297| year = 1992| last1 = Carlton-Wippern | first1 = K. C. | bibcode = 1992JNav...45..292C| s2cid = 140735736}}
</ref><ref>
{{Cite journal| doi = 10.1017/S0373463300013151| title = Practical Rhumb Line Calculations on the Spheroid| journal = Journal of Navigation| volume = 49| pages = 112&ndash;119| year = 1996| last1 = Bennett | first1 = G. G.| issue = 1| bibcode = 1996JNav...49..112B| s2cid = 128764133}}
</ref><ref>
{{cite journal
|last1 = Botnev |first1 = V.A
|last2 = Ustinov |first2 = S.M.
|script-title=ru:Методы решения прямой и обратной геодезических задач с высокой точностью
|language = Russian
|trans-title = Methods for direct and inverse geodesic problems solving with high precision
|journal = St. Petersburg State Polytechnical University Journal
|volume = 3
|number = 198
|year = 2014
|pages= 49&ndash;58
|url = http://ntv.spbstu.ru/fulltext/T3.198.2014_05.PDF
}}
</ref><ref>{{ cite journal
| last = Karney
| first = C. F. F.
| year = 2024
| volume = 68
| title = The area of rhumb polygons
| journal = Studia Geophysica et Geodaetica
| issue = 3–4
| pages = 99–120
| doi = 10.1007/s11200-024-0709-z
| doi-access = free| arxiv = 2303.03219
| bibcode = 2024StGG...68...99K
}}</ref> The course of the rhumb line is found merely by using the ellipsoidal [[isometric latitude]]. In formulas above on this page, substitute the [[latitude#conformal latitude|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==
{{reflist|30em}}
'''''Note:''' this article incorporates text from the 1878 edition of '''The Globe Encyclopaedia of Universal Information''', a work in the public domain''

==Further reading==
{{Refbegin}}
*{{cite book|last1=Monmonier|first1=Mark|title=Rhumb lines and map wars. A social history of the Mercator projection|url=https://archive.org/details/rhumblinesmapwar00monm|url-access=registration|date=2004|publisher=University of Chicago Press|location=Chicago|isbn=9780226534329}}
{{Refend}}

==External links==
{{wiktionary|loxodrome|rhumb}}
* [http://www.mathpages.com/home/kmath502/kmath502.htm Constant Headings and Rhumb Lines] at MathPages.
* [https://geographiclib.sourceforge.io/C++/doc/RhumbSolve.1.html RhumbSolve(1)], a utility for ellipsoidal rhumb line calculations (a component of [https://geographiclib.sourceforge.io GeographicLib]).
* [https://geographiclib.sourceforge.io/cgi-bin/RhumbSolve An online version of RhumbSolve].
* [https://sites.google.com/site/navigationalalgorithms/papersnavigation Navigational Algorithms] {{Webarchive|url=https://web.archive.org/web/20181016042619/https://sites.google.com/site/navigationalalgorithms/papersnavigation |date=16 October 2018 }} Paper: The Sailings.
* [https://sites.google.com/site/navigationalalgorithms/ Chart Work - Navigational Algorithms] Chart Work free software: Rhumb line, Great Circle, Composite sailing, Meridional parts. Lines of position Piloting - currents and coastal fix.
* [http://mathworld.wolfram.com/Loxodrome.html Mathworld] Loxodrome.

{{DEFAULTSORT:Rhumb Line}}
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[[Category:Spirals]]
[[Category:Spherical curves]]
[[Category:Navigation]]
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