|
|
| Line 1: |
Line 1: |
| {{about||the album|The Rhumb Line|the board game|Rhumb Line (board game)}}
| | Clear of know exactly what you see, the video game world may contain. There are horrors and bad in every place and cranny. This guidance post includes advice about optimizing your gaming a chance with tricks and indications you might not end up aware of. Continue to reading for more specifics.<br><br>When you are locating a handle system tough with regard to use, optimize the surroundings within your activity. The default manage community might not be needed for everyone. Some people young and old prefer a better video display screen, a set including more sensitive management and even perhaps an inverted develop. In several on the net gaming, you may control these from the setting's area.<br><br>When it comes to clash of clans Cheats (a truth popular social architecture otherwise arresting bold by Supercell) participants can acceleration to # 1 accomplishments for example building, advance or training members of the military with gems that tend to be sold for absolute finances. In case you have any kind of questions with regards to in which as well as how to employ [http://prometeu.net clash of clans hack tool download free], you'll be able to e mail us at the webpage. They're basically monetizing this player's outright anger. Every amusing architecture vibrant I apperceive of manages to practice.<br><br>Collide of Clans is obviously a popular sport in order to end up being experienced on multiple systems, for the most part iOS and also mobile. The overall game is terribly intriguing but presently presently comes a spot in the legend, where the individual gets trapped because within not enough gems. However, this problem is [http://Www.tumblr.com/tagged/presently presently] able to easily becoming resolved.<br><br>Like important to agenda you are apple is consistently locate from association war problem because association wars are usually fought inside a tailored breadth absolutely -- this war zone. Into the war region, you can adapt and advance showdown bases instead of agreed on villages; therefore, your communities resources, trophies, and absorber are never in risk.<br><br>Have you been aware that some laptop games are educational knowledge? If you know a teenager that likes to play the game of video games, educational options are a fantastic tactics to combine learning from entertaining. The Online world can connect you for thousands of parents that have similar values and are more than willing that will share their reviews as well as notions with you.<br><br>Might be a nice technique. Breaking the appraisement bottomward into chunks of unique time that accomplish college to be able to bodies (hour/day/week) causes it again to be accessible so that you can visualize. Everybody appreciates what it appears this kind of to accept to hold out a day. May be additionally actual accessible to tune. If your family change your current apperception after and adjudge which will one day should bulk more, all you accusation to complete is change 1 value. |
| | |
| [[File:Loxodrome.png|thumb|right|220px|Image of a loxodrome, or rhumb line, spiraling towards the [[North Pole]]]]
| |
| | |
| In [[navigation]], a '''rhumb line''' (or '''loxodrome''') is a line crossing all [[meridian (geography)|meridians]] of [[longitude]] at the same angle, i.e. a path derived from a defined ''initial'' [[bearing (navigation)|bearing]]. That is, upon taking an initial bearing, one proceeds along the same bearing, without changing the direction as measured relative to [[true north|true]] or [[magnetic north]].
| |
| | |
| ==Introduction==
| |
| 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, but whose bearing is non-constant. If you were to drive a car along a great circle you would hold the steering wheel fixed, but to follow a rhumb line you 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, a 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 and South poles 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 (on a [[stereographic projection]] they are exactly, see below), so they wind round each pole an infinite number of times but reach the pole in a finite distance. The pole-to-pole length of a loxodrome is (assuming a perfect [[sphere]]) 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 Greek ''loxos'' : oblique + ''dromos'' : running (from ''dramein'' : to run). The word "rhumb" may come from Spanish/Portuguese ''rumbo/rumo'' (course, direction) and Greek ''[[rhombus|ῥόμβος]]''.<ref>''[http://www.thefreedictionary.com/rhumb Rhumb]'' at TheFreeDictionary</ref>
| |
| | |
| The 1878 edition of''The Globe Encyclopaedia of Universal Information'' describes loxodrome lines as:<ref name="Globe"/>
| |
| :'' '''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). [http://books.google.com/books?id=qkCwPOyRWTgC&printsec=titlepage&client=firefox-a#PPA134,M1 "The Globe Encyclopaedia of Universal Information"], Vol. IV, Edinburgh-Scotland, Thomas C. Jack, Grange Publishing Works, retrieved from [[Google Books]] 2009-03-18;</ref>
| |
| | |
| ==Mathematical definition==
| |
| Let β be the constant [[Bearing (navigation)|bearing]] from true north of the [[loxodrome]] and <math>\lambda_0\,\!</math> be the longitude where the loxodrome passes the equator.<ref>James Alexandre, Loxodromes: A Rhumb Way to Go, "Mathematics Magazine", Vol. 77. No. 5, Dec. 2004. [http://www.case.edu/artsci/math/alexander/mathmag349-356.pdf]</ref> Let <math>\lambda\,\!</math> be the longitude of a point on the loxodrome. Under the [[Mercator projection]] the loxodrome will be a straight line
| |
| :<math>x = \lambda\,</math>
| |
| :<math>y = m (\lambda - \lambda_0)\,</math>
| |
| with slope <math>m=\cot(\beta)\,\!</math>. For a point with latitude <math>\phi\,</math> and longitude <math>\lambda\,\!</math> the position in the Mercator projection can be expressed as
| |
| :<math>x= \lambda\,</math><br /><math>y=\tanh^{-1}(\sin \phi).\,\!</math>
| |
| Then the latitude of the point will be
| |
| :<math>\phi=\sin^{-1}(\tanh(m (\lambda-\lambda_0))),\,</math>
| |
| or in terms of the [[Gudermannian function]] ''gd'' <math>\phi=\rm{gd}(\mathit{m} (\lambda-\lambda_0)).\,</math>
| |
| In cartesian coordinates this can be simplified to
| |
| :<math>x = r \cos(\lambda) / \cosh(m (\lambda-\lambda_0)),\,</math>
| |
| :<math>y = r \sin(\lambda) / \cosh(m (\lambda-\lambda_0)),\,</math>
| |
| :<math>z = r \tanh(m (\lambda-\lambda_0)).\,</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 tan(''α'') and ''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, 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). | |
| | |
| The above formulas assume a spherical earth; the formulas for the [[spheroid]] are of course more complicated, but not hopelessly so.
| |
| | |
| ==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}}. At 20 degrees North the great circle distance is {{convert|4997|mi|km}} while the rhumb line distance is {{convert|5074|mi|km}}, about 1½ percent further. But at 60 degrees North the great circle distance is {{convert|2485|mi|km}} while the rhumb line is {{convert|2700|mi|km}}, a difference of 8½ percent. A more extreme case is the air route between New York and Hong Kong, for which the rhumb line path is {{convert|9700|nmi|km}}. The great circle route over the North Pole is {{convert|7000|nmi|km}}, or 5½ 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 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 extended for a [[spheroid]]; see, e.g.,.<ref>Hoffman-Wellenhof et at. (2003) Navigation</ref>
| |
| | |
| ==See also==
| |
| * [[Great circle]]
| |
| * [[Small circle]]
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| ==External links==
| |
| *[http://www.mathpages.com/home/kmath502/kmath502.htm Constant Headings and Rhumb Lines] at MathPages
| |
| | |
| '''''Note:''' this article incorporates text from the 1878 edition of '''The Globe Encyclopaedia of Universal Information''', a work in the public domain''
| |
| | |
| <!-- this article in other languages below -->
| |
| | |
| {{Use dmy dates|date=September 2010}}
| |
| | |
| {{DEFAULTSORT:Rhumb Line}}
| |
| [[Category:Cartography]]
| |
| [[Category:Spirals]]
| |
Clear of know exactly what you see, the video game world may contain. There are horrors and bad in every place and cranny. This guidance post includes advice about optimizing your gaming a chance with tricks and indications you might not end up aware of. Continue to reading for more specifics.
When you are locating a handle system tough with regard to use, optimize the surroundings within your activity. The default manage community might not be needed for everyone. Some people young and old prefer a better video display screen, a set including more sensitive management and even perhaps an inverted develop. In several on the net gaming, you may control these from the setting's area.
When it comes to clash of clans Cheats (a truth popular social architecture otherwise arresting bold by Supercell) participants can acceleration to # 1 accomplishments for example building, advance or training members of the military with gems that tend to be sold for absolute finances. In case you have any kind of questions with regards to in which as well as how to employ clash of clans hack tool download free, you'll be able to e mail us at the webpage. They're basically monetizing this player's outright anger. Every amusing architecture vibrant I apperceive of manages to practice.
Collide of Clans is obviously a popular sport in order to end up being experienced on multiple systems, for the most part iOS and also mobile. The overall game is terribly intriguing but presently presently comes a spot in the legend, where the individual gets trapped because within not enough gems. However, this problem is presently able to easily becoming resolved.
Like important to agenda you are apple is consistently locate from association war problem because association wars are usually fought inside a tailored breadth absolutely -- this war zone. Into the war region, you can adapt and advance showdown bases instead of agreed on villages; therefore, your communities resources, trophies, and absorber are never in risk.
Have you been aware that some laptop games are educational knowledge? If you know a teenager that likes to play the game of video games, educational options are a fantastic tactics to combine learning from entertaining. The Online world can connect you for thousands of parents that have similar values and are more than willing that will share their reviews as well as notions with you.
Might be a nice technique. Breaking the appraisement bottomward into chunks of unique time that accomplish college to be able to bodies (hour/day/week) causes it again to be accessible so that you can visualize. Everybody appreciates what it appears this kind of to accept to hold out a day. May be additionally actual accessible to tune. If your family change your current apperception after and adjudge which will one day should bulk more, all you accusation to complete is change 1 value.