[Annotated] Rhumb line

In navigation, a rhumb line, rhumb, or loxodrome is an arc^[각주:1] crossing all meridians of^[각주:2] longitude at the same angle, that is, a path with constant bearing^[각주:3] as measured relative to true or magnetic north^[각주:4].

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^[각주:5] Defense of the Marine Chart, with further mathematical development by Thomas Harriot in the 1590s.

A rhumb line can be contrasted with^[각주:6] a great circle^[각주:7], which is the path of shortest distance between two points on the surface of a sphere^[각주:8]. On a great circle, the bearing to the destination point^[각주:9] does not remain constant^[각주:10]. If one were to drive a car along a great circle one would hold the steering wheel^[각주:11] 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^[각주:12] curvature^[각주:13], whereas a rhumb line has non-zero geodesic curvature.

Meridians of longitude and parallels of latitude^[각주:14] provide special cases of the rhumb line, where their angles of intersection^[각주:15] are respectively^[각주:16] 0˚ and 90˚. On a north-south passage the rhumb line course coincides with^[각주:17] a great circle, as it does on an east-west passage along the equator^[각주:18].

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^[각주:19] a loxodrome can extend beyond^[각주:20] the right edge of the map, where it then continues at the left edge with the same slope^[각주:21] (assuming that the map covers exactly 360 degrees of longitude).

Rhumb lines which cut meridians at oblique angles^[각주:22] are loxodromic^[각주:23] curves which spiral towards^[각주:24] the poles. 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^[각주:25] the two edges. On a stereographic^[각주:26] projection map, a loxodrome is an equiangular spiral^[각주:27] whose center is the north or south pole. 

All loxodromes spiral one pole to the other. Near the poles, they are close to being logarithmic spiral^[각주:28] (on a stereographic projection they are exactly), so they wind around^[각주:29] each pole an infinite number of times^[각주:30] 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 divided by the course of the bearing away from true north. Loxodromes are not defined at the poles.

Etymology^[각주:31] and historical description

The word loxodrome comes from Ancient Greek λοξός loxós: "oblique" + δρόμος drómos: "running" (from δραμεῖν drameîn: "to run"). The word rhumb may come from Spanish or Portuguese rumbo/rumo ("course" or "direction") and Greek ῥόμβος rhómbos, from rhémbein.

The 1878 edition of The Globe Encyclopaedia of Universal Information describes a loxodrome line as:

Loxodrom'ic Line is a curve which cuts every member of a system of lines of curvature of a given surface^[각주:32] at the same angel. A ship sailing towards^[각주:33] 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.^[각주:34]) the Loxodromic lines are evidently^[각주:35] straight.

A misunderstanding could arise^[각주:36] because the term "rhumb" had no precise meaning when it came into use. It applied equally well to the windrose lines^[각주:37] 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^[각주:38] that that implies. Therefore "rhumb" was applicable to^[각주:39] the straight lines on Mercator charts^[각주:40]. For short distances portolan "rhumbs" do not meaningfully differ from Mercator rhumbs, but these days "rhumb" is synonymous with^[각주:41] the mathematically precise "loxodrome" because it has been made synonymous retrospectively^[각주:42]. 

As Leo Bagrow states: "...the word ("Rhumbline") is wrongly applied to the sea-charts of^[각주:43] this period, since a loxodrome gives an accurate course only when the chart is drawn on a suitable projection. Cartometric investigation^[각주:44] has revealed that no projection was used in the early charts, for which we therefore retain^[각주:45] the name 'portolan'."

Application

Its use in navigation is directly linked to the style, or projection of certain navigational maps. A rhumb line appears as^[각주:46] a straight line on Mercator projection map. 

The name is derived from Old French or Spanish respectively: "rumb" or "rumbo", a line on the chart which intersects^[각주:47] all meridians at the same angle. On a plane surface^[각주:48] 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^[각주:49] a vehicle, aircraft or ship. Over longer distances and/or at higher latitudes the great circle route^[각주:50] 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.

The point can be illustrated with^[각주:51] 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 5,400 nautical miles (10,000 km). At 20 degrees north the great circle distance is 4,997 miles (8,042 km) while the rhumb line distance is 5,074 miles (8,166 km), about 1 1/2 percent further. But at 60 degrees north the great circle distance is 2,485 miles (3,999 km) while the rhumb line is 2,700 miles (4,300 km), a difference of 8 1/2 percent. A more extreme case^[각주:52] is the air route between^[각주:53] New York City and Hong Kong, for which the rhumb line path is 9,700 nautical miles (18,000 km). The great circle route over the North Pole is 7,000 nautical miles (13,000 km), or 5 1/2 hours less flying time^[각주:54] at a typical cruising speed^[각주:55]. 

Some old maps in the Mercator projection have grids^[각주:56] composed of^[각주:57] lines of latitude and longitude but also show rhumb lines which are oriented directly towards^[각주:58] north, at a right angle from the north, or at some angle from the north which is some simple rational fraction of^[각주:59] a right angle^[각주:60]. These rhumb lines would be drawn so that they would converge at^[각주:61] certain points of the map: lines going in every direction^[각주:62] would converge at each of these points. See compass rose^[각주:63]. Such maps would necessarily have been in the Mercator projection therefore not all old maps would have been capable of showing rhumb line markings^[각주:64]. 

The radial lines on^[각주:65] 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^[각주:66].

Early navigators in the time before the invention of the marine chronometer^[각주:67] used rhumb line courses on long ocean passages, because the ship's latitude could be established accurately by sightings of^[각주:68] 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 parallel, which is a special case of the rhumb line), maintaining a constant latitude and recording regular estimates of^[각주:69] the distance sailed until evidence of land was sighted^[각주:70]. 

1. arc ; 1. (기하) 호, 원호 2. 둥근[활] 모양 [본문으로]
2. meridian ; [명사] 자오선 ;; US·UK [mə|rɪdiən] [본문으로]
3. bearing ; 3. [C] (전문 용어) (일정한 지점에서 나침반으로 측정한) 방향[방위] [본문으로]
4. magnetic north ; [명사] 자북(磁北: 자기 나침반에 나타나는 북쪽) ;; 참고 ; true north [본문으로]
5. treatise ; [명사] ~ (on sth) 논문 ;; US.UK [|tri:tɪs ; |tri:tɪz] [본문으로]
6. be contrasted with ; …와 대조를 이루다[대비되다]. [본문으로]
7. great circle ; (구면(球面)의) 대원(大圓)(cf. SMALL CIRCLE); (지구의) 대권(大圈) [본문으로]
8. sphere ; 1. (기하) 구(球) 2. 구체 [본문으로]
9. destination point ; 목적지점, 목표지점 [본문으로]
10. remain constant ; 일정하게 유지되다 [본문으로]
11. steering wheel ; [명사] (자동차의) 핸들 [본문으로]
12. geodesic ; [형용사] (전문 용어) 측지(선)의, 지름길의 [본문으로]
13. curvature ; [U] (전문 용어) 굽음, 만곡; 굽음률, 곡률 ;; geodesic curvature ; 측지곡률 [본문으로]
14. parallel of latitude ; 1. (수학) 위선(緯線) 2. (천문학) 위도권(緯度圏) 3. 위도선(緯度線) [본문으로]
15. angle of intersection ; (과학용어) 교각(交角) [본문으로]
16. respectively ; [부사] 각자, 각각, 제각기 [본문으로]
17. coincide with ; ~와 동시에 일어나다, 일치하다 [본문으로]
18. the equator ; 적도 [본문으로]
19. theoretically ; [부사] 이론(상)으로; 공론으로. [본문으로]
20. extend beyond ; …너머까지 미치다. [본문으로]
21. slope ; 3. [sing., U] 경사(도), 기울기 [본문으로]
22. oblique angle ; (수학) 사각(斜角) ;; 빗각(-角)의 옛 용어. 예각(銳角)이나 둔각(鈍角)과 같이 직각(直角)이나 평각(平角)이 아닌 각(角) ;; 90도, 270도 같은 직각이나 180도, 360도 같은 평각이 아닌 모든 각을 '빗각(oblique angle)'이라고 합니다. '빗각'이란 비스듬한 각이라는 뜻입니다. '사각(斜角)'은 '빗각'에 해당하는 한자어이고 비탈을 '경사'라고 부르듯이 '斜'는 '비낄 사'입니다. [본문으로]
23. loxodromic ; [형용사] [항해] 등사(等斜) 항법의; 등사 곡선의 ;; US [lɑ̀ksədrɑ́mik] UK [lɔ́ksədrɔ́m-] [본문으로]
24. spiral ; (-ll- , 美 주로 -l-), [v , + adv. / prep.] 1. 나선형으로 움직이다, 나선형을 그리다 [본문으로]
25. line segment ; (수학) 선분(線分) [본문으로]
26. stereographic ; [형용사] 입체[실체] 화법의. [본문으로]
27. equiangular spiral ; [명사] (수학) 등각나사선 [본문으로]
28. logarithmic spiral ; [명사] (수학) 로그나사선 ;; US [lɔ́:ɡəríðmik(əl),-ríθ-,lɑ̀ɡ-] UK [lɔ́ɡ-] [본문으로]
29. wind around ; …에 감다, 휘말다, 굽이돌다, 감아 말다, 휘감다 [본문으로]
30. infinite number of times ; 무한번 [본문으로]
31. etymology ; (pl. -ies) 1. [U] 어원학, 어원 연구 2. [C] (특정 낱말의) 어원 US [|etɪ|mɑ:lədƷi] UK [|etɪ|mɒlədƷi] [본문으로]
32. of 가 네번 연달아 연결되어서 사용된것을 확인 [본문으로]
33. sail towards ; ~를 향해 항해하다 [본문으로]
34. q.v. ; [약어] … 참조, …를 보라(quod vide: 책에서 참조할 부분을 가리키는 표현) [본문으로]
35. evidently ; 1. 분명히, 눈에 띄게 [본문으로]
36. arise ; (arose / ə'rəUz ; 美 ə'roUz / , arisen / ə'rIzn /), [v] 1. (비교적 격식) (특히 문제나 곤란한 상황이) 생기다, 발생하다 [본문으로]
37. windrose ; (기상) 풍배도(風配圖), 바람장미 ;; 1. 어떤 관측지점의 어느 기간에 대하여 각 방위별 풍향 출현 빈도를 방사 모양의 그래프에 나타낸 것. 일반적으로 출현 빈도의 백분율(%)을 각각의 풍향에 대응하는 방위판 위에 방위선의 길이로 나타내거나, 그 바깥 끝을 연결한 선으로 나타낸다. 고요(calm)의 출현 빈도는 별도의 그래프의 중심이나 옆 자리에 표시한다. [본문으로]
38. imprecision ; 1. [U] 부정확, 불명확, 비정밀 2. 부정확[불명확]한 것, 애매한 것 [본문으로]
39. applicable ; [형용사] (대개 명사 앞에는 안 씀) ~ (to sb/sth) 해당[적용]되는 ;; US·UK [ə|plɪkəbl;|ӕplɪkəbl] [본문으로]
40. Mercator chart ; [명사] (지도) 메르카토르 해도[지도](海圖[地圖]). [본문으로]
41. be synonymous with ; ~와 비슷하다, 동의어이다, 밀접하다, 맥락을 같이 하다 [본문으로]
42. retrospectively ; [부사] 회고적으로, 추억에 잠겨, 과거로 거슬러 올라가 [본문으로]
43. sea chart ; 해도(海圖) [본문으로]
44. Cartometry ; the division of cartography in which methods of measurement of various geographic objects on maps to find their area, length, volume, and other quantitative characteristics are studied. Direct measurement of the length and area of actual objects using geodetic methods can only be done when the objects have very small dimensions. Cartometry is concerned with the computation of larger quantities, such as the areas of states and oceans, the lengths of coastlines and rivers, and the areas of river drainage basins. It also includes methods for the computation of various quantitative characteristics of terrain (average elevation, average angle of slope, volume, and density of the river network) by measurement on maps. [본문으로]
45. retain ; [vn], (비교적 격식), (참고: retention , retentive) 1. (계속) 유지[보유]하다 [본문으로]
46. appear as ; …으로 출연하다. [본문으로]
47. intersect ; 1. (선・도로 등이) 교차하다[만나다] [본문으로]
48. plane surface ; 평면 [본문으로]
49. plot ; 2. [타동사][VN] ~ sth (on sth) (위치・항로 등을 지도에) 표시하다 [본문으로]
50. Great Circle Route ; 대권 항로 : 대권에 있는 두 지점간에 가장 짧은 거리를 따르는 항로. [본문으로]
51. illustrate with ; …로 설명하다. [본문으로]
52. extreme case ; 극단적인 경우, 예 [본문으로]
53. air route ; 항공로 [본문으로]
54. flying time ; [명사] 비행시간 [본문으로]
55. cruising speed ; 순항 속도 ((경제 속도)), 항주력 [본문으로]
56. grid ; 3. (지도에서 위치를 나타내기 위한) 기준선망 [본문으로]
57. be composed of ; ~로 구성되어 있다 [본문으로]
58. orient ; [vn], (英 또한 orien・tate), (참고: disorientate) 1. [주로 수동태로] ~ sb/sth (to/towards sb/sth) (…을) 지향하게 하다; (특정 목적에) 맞추다 [본문으로]
59. rational fraction ; [명사] (수학) 유리분수(有理分數) [본문으로]
60. right angle ; [명사] 직각 ;; 참고 ; acute angle, obtuse angle, reflex angle [본문으로]
61. converge ; 2. (두 개 이상의 선이나 도로 등이) 만나다 [본문으로]
62. go in ~ direction ; ~방향으로 가다 [본문으로]
63. compass rose ; [항해] 나침도 ((해도상의 원형 방위도)); 방사선[방위]도 [본문으로]
64. marking ; 2. [C] [주로 복수로] (도로・차량 등에 그려진) 표시 [본문으로]
65. radial line ; [명사] (임학) 사선(射線) ;; . 화살표와 같이 방향을 나타내는 직선. [본문으로]
66. compass heading ; (항공) 나침반의 북을 기준으로 한 비행 방향 측정 [본문으로]
67. marine chronometer ; 항해용 시계 ;; A timepiece used as a portable time standard at sea so as to enable the determining of longitude by means of celestial navigation. [본문으로]
68. sighting ; [명사] 목격(특히 특이한 것・잠깐밖에 볼 수 없는 것에 대해 씀) [본문으로]
69. estimate ; 1. 추정(치), 추산 2. 견적서 [본문으로]
70. sight ; [타동사][VN] (격식) (특히 찾고 있던 것을) 갑자기 보다[찾다] [본문으로]