27:
1475:
1856:
2635:
2752:
1164:
2152:
1049:
1565:
130:
2284:
1470:{\displaystyle {\begin{aligned}\Delta \sigma _{\text{c}}&=2{\sqrt {\sin ^{2}\left({\frac {\Delta \phi }{2}}\right)+\cos {\phi _{1}}\cdot \cos {\phi _{2}}\cdot \sin ^{2}\left({\frac {\Delta \lambda }{2}}\right)}}\ ,\\&=2{\sqrt {\left(\sin {\frac {\Delta \lambda }{2}}\cos \phi _{\textrm {m}}\right)^{2}+\left(\cos {\frac {\Delta \lambda }{2}}\sin {\frac {\Delta \phi }{2}}\right)^{2}}}\ ,\end{aligned}}}
1937:
1851:{\displaystyle {\begin{aligned}\Delta \sigma ={\operatorname {atan2} }{\Bigl (}&{\sqrt {\left(\cos \phi _{2}\sin \Delta \lambda \right)^{2}+\left(\cos \phi _{1}\sin \phi _{2}-\sin \phi _{1}\cos \phi _{2}\cos \Delta \lambda \right)^{2}}},\\&\quad {\sin \phi _{1}\sin \phi _{2}+\cos \phi _{1}\cos \phi _{2}\cos \Delta \lambda }{\Bigr )},\end{aligned}}}
878:
2630:{\displaystyle {\begin{aligned}\Delta {X}&=\cos \phi _{2}\cos \lambda _{2}-\cos \phi _{1}\cos \lambda _{1};\\\Delta {Y}&=\cos \phi _{2}\sin \lambda _{2}-\cos \phi _{1}\sin \lambda _{1};\\\Delta {Z}&=\sin \phi _{2}-\sin \phi _{1};\\\Delta \sigma _{\text{c}}&={\sqrt {(\Delta {X})^{2}+(\Delta {Y})^{2}+(\Delta {Z})^{2}}}.\end{aligned}}}
686:
2147:{\displaystyle {\begin{aligned}\Delta \sigma &=\arccos \left(\mathbf {n} _{1}\cdot \mathbf {n} _{2}\right)\\&=\arcsin \left|\mathbf {n} _{1}\times \mathbf {n} _{2}\right|\\&=\arctan {\frac {\left|\mathbf {n} _{1}\times \mathbf {n} _{2}\right|}{\mathbf {n} _{1}\cdot \mathbf {n} _{2}}}\\\end{aligned}}}
1044:{\displaystyle {\begin{aligned}\Delta \sigma &=\operatorname {archav} \left(\operatorname {hav} \left(\Delta \phi \right)+\left(1-\operatorname {hav} (\Delta \phi )-\operatorname {hav} (\phi _{1}+\phi _{2})\right)\operatorname {hav} \left(\Delta \lambda \right)\right).\end{aligned}}}
842:
417:
2902:, or 6399.594 km, a 1% difference. So long as a spherical Earth is assumed, any single formula for distance on the Earth is only guaranteed correct within 0.5% (though better accuracy is possible if the formula is only intended to apply to a limited area). Using the
568:
1554:
Although this formula is accurate for most distances on a sphere, it too suffers from rounding errors for the special (and somewhat unusual) case of antipodal points. A formula that is accurate for all distances is the following special case of the
92:(diametrically opposite) both lie on a unique great circle, which the points separate into two arcs; the length of the shorter arc is the great-circle distance between the points. This arc length is proportional to the
1108:
1549:
2973:
743:
294:
1153:
2689:
2829:
from the center of the spheroid to each pole is 6356.7523142 km. When calculating the length of a short north-south line at the equator, the circle that best approximates that line has a radius of
2737:
2289:
1942:
1570:
1169:
883:
573:
735:
2272:
560:
255:
1899:
2215:
are the normals to the sphere at the two positions 1 and 2. Similarly to the equations above based on latitude and longitude, the expression based on arctan is the only one that is
681:{\displaystyle {\begin{aligned}\Delta \sigma &=2\arcsin {\frac {\Delta \sigma _{\text{c}}}{2}},\\\Delta \sigma _{\text{c}}&=2\sin {\frac {\Delta \sigma }{2}}.\end{aligned}}}
215:
175:
66:
arc between them. This arc is the shortest path between the two points on the surface of the sphere. (By comparison, the shortest path passing through the sphere's interior is the
493:
2213:
2184:
2900:
2862:
524:
443:
278:
85:, curves which are locally straight with respect to the surface. Geodesics on the sphere are great circles, circles whose center coincides with the center of the sphere.
864:, the spherical law of cosines formula, given above, does not have serious rounding errors for distances larger than a few meters on the surface of the Earth. The
2827:
2807:
860:
if the distance is small (if the two points are a kilometer apart on the surface of the Earth, the cosine of the central angle is near 0.99999999). For modern
3108:
The errors introduced by assuming a spherical Earth based on the international nautical mile are not more than 0.5% for latitude, 0.2% for longitude.
1061:
133:
An illustration of the central angle, ฮฯ, between two points, P and Q. ฮป and ฯ are the longitudinal and latitudinal angles of P respectively
104:
to obtain the arc length. Two antipodal points both lie on infinitely many great circles, each of which they divide into two arcs of length
1483:
837:{\displaystyle \Delta \sigma =\Delta \sigma _{\text{c}}\left(1+{\frac {1}{24}}\left(\Delta \sigma _{\text{c}}\right)^{2}+\cdots \right).}
2909:
412:{\displaystyle \Delta \sigma =\arccos {\bigl (}\sin \phi _{1}\sin \phi _{2}+\cos \phi _{1}\cos \phi _{2}\cos \Delta \lambda {\bigr )}.}
2216:
1113:
2643:
121:, great-circle distance formulas applied to longitude and geodetic latitude of points on Earth are accurate to within about 0.5%.
2694:
2239:
between the two points can be determined from the chord length. The great circle distance is proportional to the central angle.
3237:
2986:
For distances smaller than 500 kilometers and outside of the poles, an
Euclidean approximation of an ellipsoidal Earth (
694:
3301:
3101:
2245:
533:
228:
3357:
3208:
1866:
2987:
180:
140:
462:
2275:
30:
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two
2189:
2160:
3367:
3025:
20:
2871:
2833:
506:
425:
260:
117:, which also computes the azimuths at the end points and intermediate way-points. Because the Earth
19:
This article is about shortest-distance on a sphere. For the shortest distance on an ellipsoid, see
3352:
2868:), or 6335.439 km, while the spheroid at the poles is best approximated by a sphere of radius
2219:. The expression based on arctan requires the magnitude of the cross product over the dot product.
1920:
285:
3127:
2903:
3194:"Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations"
114:
3065:
1556:
3091:
3045:
3035:
2764:
2980:
1054:
Historically, the use of this formula was simplified by the availability of tables for the
8:
3121:
26:
3060:
2812:
2792:
2865:
113:
The determination of the great-circle distance is part of the more general problem of
3362:
3189:
3149:
3097:
1919:
instead of latitude and longitude to describe the positions, is found by means of 3D
865:
3279:
3252:
3212:
3010:
3005:
2232:
1158:
The following shows the equivalent formula expressing the chord length explicitly:
869:
527:
67:
55:
31:
3030:
3015:
2228:
89:
3333:
3216:
3193:
3000:
857:
853:
3256:
2274:, may be calculated as follows for the corresponding unit sphere, by means of
3346:
3283:
2236:
1928:
281:
93:
78:
3050:
3040:
2776:
118:
63:
2751:
2739:
this formula can be algebraically manipulated to the form shown above in
1924:
1103:{\displaystyle \operatorname {hav} \theta =\sin ^{2}{\frac {\theta }{2}}}
3313:
3150:"Calculate distance, bearing and more between Latitude/Longitude points"
2979:
ellipsoid) means that in the limit of small flattening, the mean square
422:
The problem is normally expressed in terms of finding the central angle
3055:
2227:
A line through three-dimensional space between points of interest on a
446:
288:
if one of the poles is used as an auxiliary third point on the sphere:
3337:
1544:{\displaystyle \phi _{\text{m}}={\tfrac {1}{2}}(\phi _{1}+\phi _{2})}
1055:
218:
2968:{\textstyle R_{1}={\frac {1}{3}}(2a+b)\approx 6371.009{\text{ km}}}
2786:
1916:
861:
222:
82:
74:
51:
498:
3020:
129:
3070:
856:
precision, the spherical law of cosines formula can have large
101:
97:
59:
16:
Shortest distance between two points on the surface of a sphere
3170:
Sinnott, Roger W. (August 1984). "Virtues of the
Haversine".
2976:
2782:
1904:
1148:{\displaystyle \operatorname {archav} x=2\arcsin {\sqrt {x}}}
2684:{\displaystyle \lambda _{1}=-{\tfrac {1}{2}}\Delta \lambda }
2732:{\displaystyle \lambda _{2}={\tfrac {1}{2}}\Delta \lambda }
1907:. Using atan2 ensures that the correct quadrant is chosen.
3120:
Kells, Lyman M.; Kern, Willis F.; Bland, James R. (1940).
872:
for small distances by using the chord-length relation:
105:
2912:
2874:
2836:
2712:
2664:
1915:
Another representation of similar formulas, but using
1501:
2815:
2795:
2697:
2646:
2287:
2248:
2192:
2163:
1940:
1869:
1568:
1486:
1167:
1116:
1064:
881:
746:
697:
571:
536:
509:
465:
428:
297:
263:
231:
183:
143:
3238:"A non-singular horizontal position representation"
2967:
2894:
2856:
2821:
2801:
2731:
2683:
2629:
2266:
2207:
2178:
2146:
1893:
1850:
1559:for an ellipsoid with equal major and minor axes:
1543:
1469:
1147:
1102:
1043:
836:
729:
680:
554:
518:
487:
437:
411:
272:
249:
209:
169:
2263:
1836:
1589:
551:
88:Any two distinct points on a sphere that are not
3344:
3270:McCaw, G. T. (1932). "Long lines on the Earth".
730:{\displaystyle |\Delta \sigma _{\text{c}}|\ll 1}
3302:"Fast geodesic approximations with Cheap Ruler"
2990:) is both simpler and more accurate (to 0.1%).
499:Relation between central angle and chord length
3119:
2763:) and mean Earth radii as defined in the 1984
2746:
2267:{\displaystyle \Delta \sigma _{\text{c}}\,\!}
555:{\displaystyle \Delta \sigma _{\text{c}}\,\!}
401:
315:
2983:in the estimates for distance is minimized.
2235:of the great circle between the points. The
250:{\displaystyle \Delta \lambda ,\Delta \phi }
3096:, The Stationery Office, 1987, p. 10,
2740:
1894:{\displaystyle \operatorname {atan2} (y,x)}
3251:(3). Cambridge University Press: 395โ417.
3126:. McGraw Hill Book Company, Inc. pp.
445:. Given this angle in radians, the actual
3299:
2262:
847:
550:
475:
96:between the points, which if measured in
81:is replaced by a more general concept of
3207:(176). Kingston Road, Tolworth, Surrey:
3188:
3093:Admiralty Manual of Navigation, Volume 1
2785:closely resembles a flattened sphere (a
2750:
128:
25:
3169:
3077:
3345:
3300:Agafonkin, Vladimir (30 August 2017).
3263:
210:{\displaystyle \lambda _{2},\phi _{2}}
170:{\displaystyle \lambda _{1},\phi _{1}}
3269:
3235:
2222:
488:{\displaystyle d=r\,\Delta \sigma .}
257:be their absolute differences; then
13:
2723:
2675:
2600:
2576:
2552:
2527:
2470:
2381:
2292:
2249:
1945:
1827:
1725:
1626:
1573:
1432:
1411:
1352:
1298:
1217:
1172:
1018:
955:
921:
886:
796:
756:
747:
703:
691:For short-distance approximation (
659:
627:
601:
576:
537:
510:
476:
429:
393:
298:
264:
241:
232:
14:
3379:
3327:
1910:
100:can be scaled up by the sphere's
3123:Plane And Spherical Trigonometry
2208:{\displaystyle \mathbf {n} _{2}}
2195:
2179:{\displaystyle \mathbf {n} _{1}}
2166:
2127:
2112:
2094:
2079:
2039:
2024:
1986:
1971:
3209:Directorate of Overseas Surveys
2895:{\textstyle {\frac {a^{2}}{b}}}
2857:{\textstyle {\frac {b^{2}}{a}}}
2242:The great circle chord length,
2217:well-conditioned for all angles
1752:
3290:
3229:
3182:
3163:
3142:
3113:
3084:
2951:
2936:
2809:of 6378.137 km; distance
2609:
2597:
2585:
2573:
2561:
2549:
1888:
1876:
1538:
1512:
999:
973:
961:
952:
870:numerically better-conditioned
717:
699:
519:{\displaystyle \Delta \sigma }
438:{\displaystyle \Delta \sigma }
284:between them, is given by the
273:{\displaystyle \Delta \sigma }
1:
2864:(which equals the meridian's
2741:ยง Computational formulae
862:64-bit floating-point numbers
852:On computer systems with low
456:can be trivially computed as
7:
2993:
225:of two points 1 and 2, and
124:
10:
3384:
3217:10.1179/sre.1975.23.176.88
2774:
2747:Radius for spherical Earth
18:
3257:10.1017/S0373463309990415
3245:The Journal of Navigation
3026:Geodesics on an ellipsoid
2789:) with equatorial radius
34:, u and v are also shown.
21:geodesics on an ellipsoid
3284:10.1179/sre.1932.1.6.259
286:spherical law of cosines
1905:two-argument arctangent
115:great-circle navigation
3358:Spherical trigonometry
3316:. Mapbox. 10 May 2024.
3236:Gade, Kenneth (2010).
3066:Spherical trigonometry
2969:
2896:
2858:
2823:
2803:
2772:
2733:
2685:
2631:
2268:
2209:
2180:
2148:
1895:
1852:
1545:
1471:
1149:
1104:
1045:
848:Computational formulae
838:
731:
682:
556:
530:length of unit sphere
520:
489:
452:on a sphere of radius
439:
413:
274:
251:
211:
171:
134:
35:
3046:Loxodromic navigation
3036:Geographical distance
2970:
2897:
2859:
2824:
2804:
2765:World Geodetic System
2754:
2734:
2686:
2632:
2276:Cartesian subtraction
2269:
2210:
2181:
2149:
1896:
1853:
1546:
1472:
1150:
1105:
1046:
839:
732:
683:
557:
521:
490:
440:
414:
275:
252:
212:
172:
132:
70:between the points.)
62:, measured along the
40:great-circle distance
29:
3314:"mapbox/cheap-ruler"
3272:Empire Survey Review
3078:References and notes
2910:
2872:
2834:
2813:
2793:
2695:
2644:
2285:
2246:
2190:
2161:
1938:
1931:, or a combination:
1867:
1566:
1484:
1165:
1114:
1062:
879:
744:
695:
569:
534:
526:is related with the
507:
463:
426:
295:
261:
229:
217:be the geographical
181:
141:
44:orthodromic distance
119:is nearly spherical
3190:Vincenty, Thaddeus
3061:Spherical geometry
2965:
2892:
2854:
2819:
2799:
2783:shape of the Earth
2773:
2729:
2721:
2681:
2673:
2627:
2625:
2264:
2205:
2176:
2144:
2142:
1891:
1848:
1846:
1541:
1510:
1467:
1465:
1145:
1100:
1041:
1039:
834:
727:
678:
676:
552:
516:
503:The central angle
485:
435:
409:
270:
247:
207:
167:
135:
110:times the radius.
48:spherical distance
36:
3172:Sky and Telescope
2963:
2934:
2904:mean Earth radius
2890:
2866:semi-latus rectum
2852:
2822:{\displaystyle b}
2802:{\displaystyle a}
2720:
2672:
2618:
2537:
2259:
2223:From chord length
2138:
1742:
1509:
1494:
1459:
1455:
1442:
1421:
1378:
1362:
1318:
1314:
1308:
1227:
1182:
1143:
1098:
866:haversine formula
806:
788:
766:
713:
669:
637:
618:
611:
547:
77:, the concept of
3375:
3368:Spherical curves
3321:
3317:
3309:
3294:
3288:
3287:
3267:
3261:
3260:
3242:
3233:
3227:
3226:
3224:
3223:
3198:
3186:
3180:
3179:
3167:
3161:
3160:
3158:
3156:
3146:
3140:
3139:
3137:
3135:
3117:
3111:
3110:
3088:
3011:Circumnavigation
3006:Angular distance
2974:
2972:
2971:
2966:
2964:
2961:
2935:
2927:
2922:
2921:
2901:
2899:
2898:
2893:
2891:
2886:
2885:
2876:
2863:
2861:
2860:
2855:
2853:
2848:
2847:
2838:
2828:
2826:
2825:
2820:
2808:
2806:
2805:
2800:
2738:
2736:
2735:
2730:
2722:
2713:
2707:
2706:
2690:
2688:
2687:
2682:
2674:
2665:
2656:
2655:
2636:
2634:
2633:
2628:
2626:
2619:
2617:
2616:
2607:
2593:
2592:
2583:
2569:
2568:
2559:
2548:
2539:
2538:
2535:
2519:
2518:
2500:
2499:
2477:
2462:
2461:
2446:
2445:
2427:
2426:
2411:
2410:
2388:
2373:
2372:
2357:
2356:
2338:
2337:
2322:
2321:
2299:
2273:
2271:
2270:
2265:
2261:
2260:
2257:
2214:
2212:
2211:
2206:
2204:
2203:
2198:
2185:
2183:
2182:
2177:
2175:
2174:
2169:
2153:
2151:
2150:
2145:
2143:
2139:
2137:
2136:
2135:
2130:
2121:
2120:
2115:
2108:
2104:
2103:
2102:
2097:
2088:
2087:
2082:
2071:
2057:
2053:
2049:
2048:
2047:
2042:
2033:
2032:
2027:
2004:
2000:
1996:
1995:
1994:
1989:
1980:
1979:
1974:
1902:
1900:
1898:
1897:
1892:
1857:
1855:
1854:
1849:
1847:
1840:
1839:
1833:
1820:
1819:
1804:
1803:
1785:
1784:
1769:
1768:
1750:
1743:
1741:
1740:
1735:
1731:
1718:
1717:
1702:
1701:
1683:
1682:
1667:
1666:
1642:
1641:
1636:
1632:
1619:
1618:
1597:
1593:
1592:
1586:
1557:Vincenty formula
1550:
1548:
1547:
1542:
1537:
1536:
1524:
1523:
1511:
1502:
1496:
1495:
1492:
1476:
1474:
1473:
1468:
1466:
1457:
1456:
1454:
1453:
1448:
1444:
1443:
1438:
1430:
1422:
1417:
1409:
1392:
1391:
1386:
1382:
1381:
1380:
1379:
1376:
1363:
1358:
1350:
1336:
1325:
1316:
1315:
1313:
1309:
1304:
1296:
1287:
1286:
1274:
1273:
1272:
1253:
1252:
1251:
1232:
1228:
1223:
1215:
1206:
1205:
1196:
1184:
1183:
1180:
1154:
1152:
1151:
1146:
1144:
1139:
1109:
1107:
1106:
1101:
1099:
1091:
1086:
1085:
1050:
1048:
1047:
1042:
1040:
1033:
1029:
1028:
1024:
1006:
1002:
998:
997:
985:
984:
931:
927:
843:
841:
840:
835:
830:
826:
819:
818:
813:
809:
808:
807:
804:
789:
781:
768:
767:
764:
736:
734:
733:
728:
720:
715:
714:
711:
702:
687:
685:
684:
679:
677:
670:
665:
657:
639:
638:
635:
619:
614:
613:
612:
609:
599:
561:
559:
558:
553:
549:
548:
545:
525:
523:
522:
517:
494:
492:
491:
486:
444:
442:
441:
436:
418:
416:
415:
410:
405:
404:
386:
385:
370:
369:
351:
350:
335:
334:
319:
318:
279:
277:
276:
271:
256:
254:
253:
248:
216:
214:
213:
208:
206:
205:
193:
192:
176:
174:
173:
168:
166:
165:
153:
152:
108:
32:antipodal points
3383:
3382:
3378:
3377:
3376:
3374:
3373:
3372:
3353:Metric geometry
3343:
3342:
3330:
3325:
3324:
3320:
3312:
3295:
3291:
3268:
3264:
3240:
3234:
3230:
3221:
3219:
3196:
3187:
3183:
3168:
3164:
3154:
3152:
3148:
3147:
3143:
3133:
3131:
3118:
3114:
3104:
3090:
3089:
3085:
3080:
3075:
3031:Geodetic system
3016:Flight planning
2996:
2960:
2926:
2917:
2913:
2911:
2908:
2907:
2881:
2877:
2875:
2873:
2870:
2869:
2843:
2839:
2837:
2835:
2832:
2831:
2814:
2811:
2810:
2794:
2791:
2790:
2779:
2749:
2711:
2702:
2698:
2696:
2693:
2692:
2663:
2651:
2647:
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2642:
2641:
2624:
2623:
2612:
2608:
2603:
2588:
2584:
2579:
2564:
2560:
2555:
2547:
2540:
2534:
2530:
2524:
2523:
2514:
2510:
2495:
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2478:
2473:
2467:
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2457:
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2441:
2437:
2422:
2418:
2406:
2402:
2389:
2384:
2378:
2377:
2368:
2364:
2352:
2348:
2333:
2329:
2317:
2313:
2300:
2295:
2288:
2286:
2283:
2282:
2256:
2252:
2247:
2244:
2243:
2229:spherical Earth
2225:
2199:
2194:
2193:
2191:
2188:
2187:
2170:
2165:
2164:
2162:
2159:
2158:
2141:
2140:
2131:
2126:
2125:
2116:
2111:
2110:
2109:
2098:
2093:
2092:
2083:
2078:
2077:
2076:
2072:
2070:
2055:
2054:
2043:
2038:
2037:
2028:
2023:
2022:
2021:
2017:
2002:
2001:
1990:
1985:
1984:
1975:
1970:
1969:
1968:
1964:
1951:
1941:
1939:
1936:
1935:
1913:
1868:
1865:
1864:
1862:
1845:
1844:
1835:
1834:
1815:
1811:
1799:
1795:
1780:
1776:
1764:
1760:
1753:
1748:
1747:
1736:
1713:
1709:
1697:
1693:
1678:
1674:
1662:
1658:
1651:
1647:
1646:
1637:
1614:
1610:
1603:
1599:
1598:
1596:
1594:
1588:
1587:
1582:
1569:
1567:
1564:
1563:
1532:
1528:
1519:
1515:
1500:
1491:
1487:
1485:
1482:
1481:
1464:
1463:
1449:
1431:
1429:
1410:
1408:
1401:
1397:
1396:
1387:
1375:
1374:
1370:
1351:
1349:
1342:
1338:
1337:
1335:
1323:
1322:
1297:
1295:
1291:
1282:
1278:
1268:
1264:
1263:
1247:
1243:
1242:
1216:
1214:
1210:
1201:
1197:
1195:
1185:
1179:
1175:
1168:
1166:
1163:
1162:
1138:
1115:
1112:
1111:
1090:
1081:
1077:
1063:
1060:
1059:
1038:
1037:
1017:
1013:
993:
989:
980:
976:
939:
935:
920:
916:
909:
905:
892:
882:
880:
877:
876:
858:rounding errors
850:
814:
803:
799:
795:
791:
790:
780:
773:
769:
763:
759:
745:
742:
741:
716:
710:
706:
698:
696:
693:
692:
675:
674:
658:
656:
640:
634:
630:
624:
623:
608:
604:
600:
598:
582:
572:
570:
567:
566:
544:
540:
535:
532:
531:
508:
505:
504:
501:
464:
461:
460:
427:
424:
423:
400:
399:
381:
377:
365:
361:
346:
342:
330:
326:
314:
313:
296:
293:
292:
262:
259:
258:
230:
227:
226:
201:
197:
188:
184:
182:
179:
178:
161:
157:
148:
144:
142:
139:
138:
127:
106:
24:
17:
12:
11:
5:
3381:
3371:
3370:
3365:
3360:
3355:
3341:
3340:
3329:
3328:External links
3326:
3323:
3322:
3319:
3318:
3310:
3296:
3289:
3278:(6): 259โ263.
3262:
3228:
3192:(1975-04-01).
3181:
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3079:
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3033:
3028:
3023:
3018:
3013:
3008:
3003:
3001:Air navigation
2997:
2995:
2992:
2981:relative error
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2933:
2930:
2925:
2920:
2916:
2889:
2884:
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2818:
2798:
2775:Main article:
2748:
2745:
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2716:
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2680:
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2671:
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2291:
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2255:
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2224:
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2197:
2173:
2168:
2155:
2154:
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2096:
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2075:
2069:
2066:
2063:
2060:
2058:
2056:
2052:
2046:
2041:
2036:
2031:
2026:
2020:
2016:
2013:
2010:
2007:
2005:
2003:
1999:
1993:
1988:
1983:
1978:
1973:
1967:
1963:
1960:
1957:
1954:
1952:
1950:
1947:
1944:
1943:
1921:vector algebra
1917:normal vectors
1912:
1911:Vector version
1909:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1859:
1858:
1843:
1838:
1832:
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1826:
1823:
1818:
1814:
1810:
1807:
1802:
1798:
1794:
1791:
1788:
1783:
1779:
1775:
1772:
1767:
1763:
1759:
1756:
1751:
1749:
1746:
1739:
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1727:
1724:
1721:
1716:
1712:
1708:
1705:
1700:
1696:
1692:
1689:
1686:
1681:
1677:
1673:
1670:
1665:
1661:
1657:
1654:
1650:
1645:
1640:
1635:
1631:
1628:
1625:
1622:
1617:
1613:
1609:
1606:
1602:
1595:
1591:
1585:
1581:
1578:
1575:
1572:
1571:
1540:
1535:
1531:
1527:
1522:
1518:
1514:
1508:
1505:
1499:
1490:
1478:
1477:
1462:
1452:
1447:
1441:
1437:
1434:
1428:
1425:
1420:
1416:
1413:
1407:
1404:
1400:
1395:
1390:
1385:
1373:
1369:
1366:
1361:
1357:
1354:
1348:
1345:
1341:
1334:
1331:
1328:
1326:
1324:
1321:
1312:
1307:
1303:
1300:
1294:
1290:
1285:
1281:
1277:
1271:
1267:
1262:
1259:
1256:
1250:
1246:
1241:
1238:
1235:
1231:
1226:
1222:
1219:
1213:
1209:
1204:
1200:
1194:
1191:
1188:
1186:
1178:
1174:
1171:
1170:
1142:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1097:
1094:
1089:
1084:
1080:
1076:
1073:
1070:
1067:
1052:
1051:
1036:
1032:
1027:
1023:
1020:
1016:
1012:
1009:
1005:
1001:
996:
992:
988:
983:
979:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
938:
934:
930:
926:
923:
919:
915:
912:
908:
904:
901:
898:
895:
893:
891:
888:
885:
884:
854:floating point
849:
846:
845:
844:
833:
829:
825:
822:
817:
812:
802:
798:
794:
787:
784:
779:
776:
772:
762:
758:
755:
752:
749:
726:
723:
719:
709:
705:
701:
689:
688:
673:
668:
664:
661:
655:
652:
649:
646:
643:
641:
633:
629:
626:
625:
622:
617:
607:
603:
597:
594:
591:
588:
585:
583:
581:
578:
575:
574:
543:
539:
515:
512:
500:
497:
496:
495:
484:
481:
478:
474:
471:
468:
434:
431:
420:
419:
408:
403:
398:
395:
392:
389:
384:
380:
376:
373:
368:
364:
360:
357:
354:
349:
345:
341:
338:
333:
329:
325:
322:
317:
312:
309:
306:
303:
300:
269:
266:
246:
243:
240:
237:
234:
204:
200:
196:
191:
187:
164:
160:
156:
151:
147:
126:
123:
79:straight lines
75:curved surface
15:
9:
6:
4:
3:
2:
3380:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3350:
3348:
3339:
3335:
3332:
3331:
3315:
3311:
3307:
3303:
3298:
3297:
3293:
3285:
3281:
3277:
3273:
3266:
3258:
3254:
3250:
3246:
3239:
3232:
3218:
3214:
3210:
3206:
3202:
3201:Survey Review
3195:
3191:
3185:
3177:
3173:
3166:
3151:
3145:
3129:
3125:
3124:
3116:
3109:
3105:
3103:9780117728806
3099:
3095:
3094:
3087:
3083:
3072:
3069:
3067:
3064:
3062:
3059:
3057:
3054:
3052:
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2998:
2991:
2989:
2988:FCC's formula
2984:
2982:
2978:
2957:
2954:
2948:
2945:
2942:
2939:
2931:
2928:
2923:
2918:
2914:
2905:
2887:
2882:
2878:
2867:
2849:
2844:
2840:
2816:
2796:
2788:
2784:
2778:
2770:
2766:
2762:
2758:
2753:
2744:
2742:
2726:
2717:
2714:
2708:
2703:
2699:
2678:
2669:
2666:
2660:
2657:
2652:
2648:
2640:Substituting
2620:
2613:
2604:
2594:
2589:
2580:
2570:
2565:
2556:
2544:
2542:
2531:
2520:
2515:
2511:
2507:
2504:
2501:
2496:
2492:
2488:
2485:
2482:
2480:
2474:
2463:
2458:
2454:
2450:
2447:
2442:
2438:
2434:
2431:
2428:
2423:
2419:
2415:
2412:
2407:
2403:
2399:
2396:
2393:
2391:
2385:
2374:
2369:
2365:
2361:
2358:
2353:
2349:
2345:
2342:
2339:
2334:
2330:
2326:
2323:
2318:
2314:
2310:
2307:
2304:
2302:
2296:
2281:
2280:
2279:
2277:
2253:
2240:
2238:
2237:central angle
2234:
2230:
2220:
2218:
2200:
2171:
2132:
2122:
2117:
2105:
2099:
2089:
2084:
2073:
2067:
2064:
2061:
2059:
2050:
2044:
2034:
2029:
2018:
2014:
2011:
2008:
2006:
1997:
1991:
1981:
1976:
1965:
1961:
1958:
1955:
1953:
1948:
1934:
1933:
1932:
1930:
1929:cross product
1926:
1922:
1918:
1908:
1906:
1885:
1882:
1879:
1873:
1870:
1841:
1830:
1824:
1821:
1816:
1812:
1808:
1805:
1800:
1796:
1792:
1789:
1786:
1781:
1777:
1773:
1770:
1765:
1761:
1757:
1754:
1744:
1737:
1732:
1728:
1722:
1719:
1714:
1710:
1706:
1703:
1698:
1694:
1690:
1687:
1684:
1679:
1675:
1671:
1668:
1663:
1659:
1655:
1652:
1648:
1643:
1638:
1633:
1629:
1623:
1620:
1615:
1611:
1607:
1604:
1600:
1583:
1579:
1576:
1562:
1561:
1560:
1558:
1552:
1533:
1529:
1525:
1520:
1516:
1506:
1503:
1497:
1488:
1460:
1450:
1445:
1439:
1435:
1426:
1423:
1418:
1414:
1405:
1402:
1398:
1393:
1388:
1383:
1371:
1367:
1364:
1359:
1355:
1346:
1343:
1339:
1332:
1329:
1327:
1319:
1310:
1305:
1301:
1292:
1288:
1283:
1279:
1275:
1269:
1265:
1260:
1257:
1254:
1248:
1244:
1239:
1236:
1233:
1229:
1224:
1220:
1211:
1207:
1202:
1198:
1192:
1189:
1187:
1176:
1161:
1160:
1159:
1156:
1140:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1095:
1092:
1087:
1082:
1078:
1074:
1071:
1068:
1065:
1057:
1034:
1030:
1025:
1021:
1014:
1010:
1007:
1003:
994:
990:
986:
981:
977:
970:
967:
964:
958:
949:
946:
943:
940:
936:
932:
928:
924:
917:
913:
910:
906:
902:
899:
896:
894:
889:
875:
874:
873:
871:
867:
863:
859:
855:
831:
827:
823:
820:
815:
810:
800:
792:
785:
782:
777:
774:
770:
760:
753:
750:
740:
739:
738:
724:
721:
707:
671:
666:
662:
653:
650:
647:
644:
642:
631:
620:
615:
605:
595:
592:
589:
586:
584:
579:
565:
564:
563:
541:
529:
513:
482:
479:
472:
469:
466:
459:
458:
457:
455:
451:
448:
432:
406:
396:
390:
387:
382:
378:
374:
371:
366:
362:
358:
355:
352:
347:
343:
339:
336:
331:
327:
323:
320:
310:
307:
304:
301:
291:
290:
289:
287:
283:
282:central angle
267:
244:
238:
235:
224:
220:
202:
198:
194:
189:
185:
162:
158:
154:
149:
145:
131:
122:
120:
116:
111:
109:
103:
99:
95:
94:central angle
91:
86:
84:
80:
76:
71:
69:
65:
61:
57:
53:
49:
45:
41:
33:
28:
22:
3305:
3292:
3275:
3271:
3265:
3248:
3244:
3231:
3220:. Retrieved
3204:
3200:
3184:
3175:
3171:
3165:
3153:. Retrieved
3144:
3132:. Retrieved
3122:
3115:
3107:
3092:
3086:
3051:Meridian arc
3041:Isoazimuthal
2985:
2780:
2777:Earth radius
2769:Not to scale
2768:
2760:
2756:
2755:Equatorial (
2639:
2241:
2226:
2156:
1923:, using the
1914:
1860:
1553:
1479:
1157:
1053:
851:
690:
502:
453:
449:
421:
136:
112:
87:
72:
64:great-circle
54:between two
47:
43:
39:
37:
3334:GreatCircle
2767:revision. (
1925:dot product
3347:Categories
3222:2008-07-21
3056:Rhumb line
2759:), polar (
1058:function:
447:arc length
3338:MathWorld
3211:: 88โ93.
3178:(2): 159.
2975:(for the
2955:≈
2727:λ
2724:Δ
2700:λ
2679:λ
2676:Δ
2661:−
2649:λ
2601:Δ
2577:Δ
2553:Δ
2532:σ
2528:Δ
2512:ϕ
2508:
2502:−
2493:ϕ
2489:
2471:Δ
2455:λ
2451:
2439:ϕ
2435:
2429:−
2420:λ
2416:
2404:ϕ
2400:
2382:Δ
2366:λ
2362:
2350:ϕ
2346:
2340:−
2331:λ
2327:
2315:ϕ
2311:
2293:Δ
2254:σ
2250:Δ
2123:⋅
2090:×
2068:
2035:×
2015:
1982:⋅
1962:
1949:σ
1946:Δ
1874:
1831:λ
1828:Δ
1825:
1813:ϕ
1809:
1797:ϕ
1793:
1778:ϕ
1774:
1762:ϕ
1758:
1729:λ
1726:Δ
1723:
1711:ϕ
1707:
1695:ϕ
1691:
1685:−
1676:ϕ
1672:
1660:ϕ
1656:
1630:λ
1627:Δ
1624:
1612:ϕ
1608:
1577:σ
1574:Δ
1530:ϕ
1517:ϕ
1489:ϕ
1436:ϕ
1433:Δ
1427:
1415:λ
1412:Δ
1406:
1372:ϕ
1368:
1356:λ
1353:Δ
1347:
1302:λ
1299:Δ
1289:
1276:⋅
1266:ϕ
1261:
1255:⋅
1245:ϕ
1240:
1221:ϕ
1218:Δ
1208:
1177:σ
1173:Δ
1136:
1121:
1093:θ
1088:
1072:θ
1069:
1056:haversine
1022:λ
1019:Δ
1011:
991:ϕ
978:ϕ
971:
965:−
959:ϕ
956:Δ
950:
944:−
925:ϕ
922:Δ
914:
903:
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887:Δ
824:⋯
801:σ
797:Δ
761:σ
757:Δ
751:σ
748:Δ
722:≪
708:σ
704:Δ
663:σ
660:Δ
654:
632:σ
628:Δ
606:σ
602:Δ
596:
580:σ
577:Δ
542:σ
538:Δ
514:σ
511:Δ
480:σ
477:Δ
433:σ
430:Δ
397:λ
394:Δ
391:
379:ϕ
375:
363:ϕ
359:
344:ϕ
340:
328:ϕ
324:
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302:σ
299:Δ
268:σ
265:Δ
245:ϕ
242:Δ
236:λ
233:Δ
219:longitude
199:ϕ
186:λ
159:ϕ
146:λ
90:antipodal
83:geodesics
3363:Distance
3134:July 13,
2994:See also
2962: km
2958:6371.009
2787:spheroid
223:latitude
125:Formulae
52:distance
3021:Geodesy
2231:is the
1903:is the
1901:
1863:
98:radians
50:is the
3306:Mapbox
3155:10 Aug
3100:
3071:Versor
2157:where
2065:arctan
2012:arcsin
1959:arccos
1861:where
1480:where
1458:
1317:
1133:arcsin
1118:archav
900:archav
593:arcsin
308:arccos
280:, the
102:radius
60:sphere
56:points
3241:(PDF)
3197:(PDF)
2977:WGS84
2233:chord
1871:atan2
1584:atan2
528:chord
73:On a
68:chord
58:on a
46:, or
3157:2013
3136:2018
3130:-326
3098:ISBN
2781:The
2691:and
2186:and
1110:and
221:and
177:and
137:Let
38:The
3336:at
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3253:doi
3213:doi
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2486:sin
2448:sin
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2324:cos
2308:cos
1822:cos
1806:cos
1790:cos
1771:sin
1755:sin
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1704:cos
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1008:hav
968:hav
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