1254:
600:
2851:
1894:
400:
in brackets are usually omitted), abbreviated "S30°E", which is the bearing 30 degrees in the eastward direction from south, i.e. the bearing 150 degrees clockwise from north. The reference direction, stated first, is always north or south, and the turning direction, stated last, is east or west. The directions are chosen so that the angle, stated between them, is positive, between zero and 90 degrees. If the bearing happens to be exactly in the direction of one of the
40:
843:
348:
2911:
2152:
2887:
1249:{\displaystyle {\begin{aligned}e^{2}&=f(2-f)\\1-e^{2}&=(1-f)^{2}\\\Lambda &=\left(1-e^{2}\right){\frac {\tan \varphi _{2}}{\tan \varphi _{1}}}+e^{2}{\sqrt {\frac {1+\left(1-e^{2}\right)\left(\tan \varphi _{2}\right)^{2}}{1+\left(1-e^{2}\right)\left(\tan \varphi _{1}\right)^{2}}}}\\\tan \alpha &={\frac {\sin L}{(\Lambda -\cos L)\sin \varphi _{1}}}\end{aligned}}}
2863:
2899:
2875:
1414:
2839:
399:
Quite commonly, azimuths or compass bearings are stated in a system in which either north or south can be the zero, and the angle may be measured clockwise or anticlockwise from the zero. For example, a bearing might be described as "(from) south, (turn) thirty degrees (toward the) east" (the words
1883:. They all have the property that directions (the azimuths) from a central point are preserved. Some navigation systems use south as the reference plane. However, any direction can serve as the plane of reference, as long as it is clearly defined for everyone using that system.
1893:
396:) can be used. Moving clockwise on a 360 degree circle, east has azimuth 90°, south 180°, and west 270°. There are exceptions: some navigation systems use south as the reference vector. Any direction can be the reference vector, as long as it is clearly defined.
1858:
796:
1532:
1385:
1633:
94:
1755:
848:
1541:
and that the azimuth is clockwise relative to the north. This is the reason why the X and Y axis in the above formula are swapped. If the azimuth becomes negative, one can always add 360°.
834:(the shortest path on the surface of the spheroid from our viewpoint to Point 2). The difference is usually negligible: less than 0.03 arc second for distances less than 100 km.
1750:
695:
679:
652:
1700:
1668:
1443:
2803:
1302:
2020:
refers to the angle the sound source makes compared to the imaginary straight line that is drawn from within the head through the area between the eyes.
381:
has also been more generally defined as a horizontal angle measured clockwise from any fixed reference plane or easily established base direction line.
1900:
279:. The first known record in any Western language is in Spanish in the 1270s in an astronomy book that was largely derived from Arabic sources, the
2547:
265:= "the direction"). The Arabic word entered late medieval Latin in an astronomy context and in particular in the use of the Arabic version of the
616:
2946:
2651:
1554:
1405:
with declination and longitude difference with hour angle, and change the sign (since the hour angle is positive westward instead of east).
2319:
2388:
1903:
Comparison of some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii.
2783:
2688:
281:
2060:
153:
2209:
368:
198:. The star is the point of interest, the reference plane is the local area (e.g. a circular area with a 5 km radius at
2642:
1946:
620:
2829:
304:
298:
179:
2540:
2473:
1950:
143:
685:(positive eastward). We can get a fair approximation by assuming the Earth is a sphere, in which case the azimuth
47:
from the observer to a point of interest projected on the same plane as the reference direction orthogonal to the
2819:
275:
2468:
2383:
2055:
1434:
580:
175:); the angle between the projected vector and a reference vector on the reference plane is called the azimuth.
31:
2339:
1904:
1853:{\displaystyle {\begin{aligned}X_{2}&=X_{1}+D\sin \alpha \\Y_{2}&=Y_{1}+D\cos \alpha \end{aligned}}}
1538:
599:
2583:
2533:
2256:
590:
110:
791:{\displaystyle \tan \alpha ={\frac {\sin L}{\cos \varphi _{1}\tan \varphi _{2}-\sin \varphi _{1}\cos L}}}
2441:
2276:
2184:
1286:
594:
2850:
2956:
2667:
2314:
2271:
2261:
2189:
2100:
2070:
2050:
2045:
831:
813:
373:
342:
317:
308:
253:
The word azimuth is used in all
European languages today. It originates from medieval Arabic السموت (
161:
157:
221:(°), in the positive range 0° to 360° or in the signed range -180° to +180°. The concept is used in
2936:
2595:
2356:
2194:
2120:
2080:
1970:
1433:(in decimal degrees) can be calculated when the coordinates of 2 points are known in a flat plane (
586:
191:
122:
2648:
657:
630:
2941:
2903:
2891:
2855:
2429:
2304:
2125:
2013:
1934:
135:
1925:
If, instead of measuring from and along the horizon, the angles are measured from and along the
2779:
2684:
2344:
2281:
2160:
2065:
1974:
1962:
1926:
312:
203:
150:
2951:
2867:
2843:
2514:
2463:
2228:
2172:
2110:
2105:
2024:
270:
211:
172:
62:
43:
The azimuth is the angle formed between a reference direction (in this example north) and a
39:
2717:, FM 21-26, Headquarters, Dept. of the Army, Washington, D.C. (28 March 1956), ch. 3, p. 63
1673:
1641:
1537:
Remark that the reference axes are swapped relative to the (counterclockwise) mathematical
347:
2655:
2574:
2519:
2424:
2334:
2309:
1930:
1527:{\displaystyle \alpha ={\frac {180}{\pi }}\operatorname {atan2} (X_{2}-X_{1},Y_{2}-Y_{1})}
803:
168:
101:
2813:
2630:
654:, longitude zero; we want to find the azimuth from our viewpoint to Point 2 at latitude
2931:
2915:
2797:
2704:, FM 21-26, Headquarters, Dept. of the Army, Washington, D.C. (7 May 1993), ch. 6, p. 2
2615:
2491:
2407:
2401:
2324:
2249:
2243:
2238:
2075:
1880:
401:
352:
322:
286:
218:
2925:
2496:
2329:
2286:
1864:
164:
2151:
1380:{\displaystyle \tan \alpha ={\frac {\sin L}{\left(1-e^{2}\right)\tan \varphi _{2}}}}
2910:
2879:
2214:
2204:
2199:
2028:
389:
210:. The azimuth is the angle between the north vector and the star's vector on the
2378:
2085:
1391:
393:
230:
2739:, Headquarters, War Department, Washington, D.C. (17 September 1941), pp. 24–25
17:
2458:
2414:
2373:
2002:
1395:
819:
385:
326:
242:
226:
222:
207:
128:
2610:
2090:
2032:
608:
604:
330:
266:
199:
44:
801:
A better approximation assumes the Earth is a slightly-squashed sphere (an
269:
astronomy instrument. Its first recorded use in
English is in the 1390s in
2752:, Headquarters, War Department, Washington, D.C. (23 December 1944), p. 15
1398:
at a specific location, modify the formula for a spherical Earth. Replace
2095:
612:
2636:
1933:
if referenced to the Vernal
Equinox, or hour angle if referenced to the
2812:
2115:
1628:{\displaystyle \alpha =\operatorname {atan2} (X_{2}-X_{1},Y_{2}-Y_{1})}
1418:
183:
2130:
1982:
1545:
321:, sometimes called elevation above the horizon. It is also used for
238:
48:
2874:
1413:
2179:
1954:
1868:
1702:
1412:
598:
360:
346:
139:
131:
38:
27:
Horizontal angle from north or other reference cardinal direction
187:
2796:
1421:, commonly used by geologists and surveyors to measure azimuth
234:
195:
2761:
Torge & Müller (2012) Geodesy, De
Gruyter, eq.6.70, p.248
822:
whose axis is perpendicular to the surface of the spheroid;
83:
80:
74:
404:, a different notation, e.g. "due east", is used instead.
595:
Geographical distance § Ellipsoidal-surface formulae
68:
1390:
To calculate the azimuth of the Sun or a star given its
811:
then has at least two very slightly different meanings.
388:, measured as a 0° azimuth, though other angular units (
2009:
refers to the angle between the tape head(s) and tape.
384:
Today, the reference plane for an azimuth is typically
2827:
2638:
Centre
National de Ressources Textuelles et Lexicales
1973:. A special case of an azimuth angle is the angle in
1753:
1676:
1644:
1557:
1446:
1305:
846:
837:
Normal-section azimuth can be calculated as follows:
698:
660:
633:
71:
2588:
1981:-plane, although this angle is normally measured in
261:), meaning "the directions" (plural of Arabic السمت
115:
77:
65:
1852:
1708:The opposite problem occurs when the coordinates (
1694:
1662:
1627:
1526:
1379:
1248:
790:
673:
646:
333:azimuth is nearly always measured from the north.
1867:and azimuth identification (AzID), especially in
1945:In mathematics, the azimuth angle of a point in
1744:) are known, one can calculate its coordinates:
1267:the eccentricity for the chosen spheroid (e.g.,
2632:New English Dictionary on Historical Principles
615:(the shortest route) changes from 141° to 42°.
359:In land navigation, azimuth is usually denoted
2541:
367:, and defined as a horizontal angle measured
8:
818:is the angle measured at our viewpoint by a
2750:Advanced Map and Aerial Photograph Reading
2737:Advanced Map and Aerial Photograph Reading
2649:EnglishWordsOfArabicAncestry.wordpress.com
2548:
2534:
2136:
591:Vincenty's formulae § Inverse problem
587:Earth section paths § Inverse problem
1825:
1808:
1779:
1762:
1754:
1752:
1675:
1643:
1616:
1603:
1590:
1577:
1556:
1515:
1502:
1489:
1476:
1453:
1445:
1368:
1347:
1318:
1304:
1233:
1188:
1158:
1147:
1120:
1091:
1080:
1053:
1028:
1022:
1006:
988:
975:
964:
929:
900:
855:
847:
845:
770:
751:
735:
711:
697:
665:
659:
638:
632:
2778:, University of Minnesota Press (2000),
2683:, University of Minnesota Press (2000),
493:
412:
2834:
2566:
2439:
2354:
2158:
2144:
1977:of the component of the vector in the
830:) is the angle between north and the
206:, and the reference vector points to
7:
2578:
1985:rather than degrees and denoted by
105:
2035:that can be rotated horizontally.
1205:
939:
25:
2573:The singular form of the noun is
2320:Kepler's laws of planetary motion
617:Azimuthal orthographic projection
142:, in a local or observer-centric
127:'the directions') is the
2909:
2897:
2885:
2873:
2861:
2849:
2837:
2668:pronunciation of "al-" in Arabic
2150:
2061:Azimuthal equidistant projection
2016:experiments and literature, the
1961:-axis and the projection of the
1892:
61:
2715:Map Reading and Land Navigation
2702:Map Reading and Land Navigation
217:Azimuth is usually measured in
1689:
1677:
1657:
1645:
1622:
1570:
1521:
1469:
1220:
1202:
926:
913:
883:
871:
282:Libros del saber de astronomía
1:
2644:Brill's Encyclopedia of Islam
1885:
1722:) of one point, the distance
621:Miller cylindrical projection
351:Azimuth marker, Mount Allen (
2947:Horizontal coordinate system
1905:(click for detail)
674:{\displaystyle \varphi _{2}}
647:{\displaystyle \varphi _{1}}
627:We are standing at latitude
311:, azimuth is one of the two
305:horizontal coordinate system
299:Horizontal coordinate system
160:) to a point of interest is
2776:The Wilderness Route Finder
2681:The Wilderness Route Finder
2589:
2474:Tsiolkovsky rocket equation
1879:There is a wide variety of
144:spherical coordinate system
116:
2973:
2820:Collier's New Encyclopedia
2443:Engineering and efficiency
2262:Bi-elliptic transfer orbit
1863:This is typically used in
1670:in contrast to the normal
1548:would be slightly easier:
1435:cartographical coordinates
584:
578:
371:from a north base line or
355:), southern California, US
340:
296:
29:
2616:Dictionary.com Unabridged
1881:azimuthal map projections
563:
555:
547:
539:
531:
523:
515:
507:
502:
499:
495:From north, western side
482:
474:
466:
458:
450:
442:
434:
426:
421:
418:
414:From north, eastern side
408:True north-based azimuths
276:Treatise on the Astrolabe
2658:. In Arabic the written
2654:January 2, 2014, at the
2600:'the direction'.
2469:Propellant mass fraction
2368:Gravitational influences
2056:Azimuthal quantum number
1929:, the angles are called
581:Inverse geodetic problem
325:installation (see also:
202:) around an observer on
32:Azimuth (disambiguation)
2804:Encyclopædia Britannica
2340:Specific orbital energy
1947:cylindrical coordinates
1539:polar coordinate system
1417:A standard Brunton Geo
2807:(11th ed.). 1911.
2257:Hohmann transfer orbit
1854:
1696:
1664:
1629:
1528:
1427:cartographical azimuth
1422:
1381:
1263:is the flattening and
1250:
792:
675:
648:
624:
356:
52:
2726:U.S. Army, ch. 6 p. 2
2662:is always pronounced
2453:Preflight engineering
2185:Argument of periapsis
1957:between the positive
1953:is the anticlockwise
1951:spherical coordinates
1855:
1697:
1695:{\displaystyle (y,x)}
1665:
1663:{\displaystyle (x,y)}
1630:
1529:
1416:
1382:
1251:
793:
676:
649:
602:
350:
182:, the azimuth is the
42:
2509:Propulsive maneuvers
2101:Magnetic declination
2071:Bearing (navigation)
2051:Angular displacement
2046:Altitude (astronomy)
2003:magnetic tape drives
1751:
1674:
1642:
1555:
1444:
1303:
844:
832:ellipsoidal geodesic
696:
658:
631:
603:The azimuth between
343:Azimuth (navigation)
309:celestial navigation
180:celestial coordinate
149:Mathematically, the
30:For other uses, see
2486:Efficiency measures
2389:Sphere of influence
2358:Celestial mechanics
2140:Part of a series on
2121:Solar azimuth angle
2081:Course (navigation)
1916:Related coordinates
496:
415:
192:astronomical object
2305:Dynamical friction
2126:Sound Localization
2014:sound localization
1935:celestial meridian
1850:
1848:
1730:to another point (
1726:, and the azimuth
1692:
1660:
1625:
1524:
1423:
1377:
1246:
1244:
788:
671:
644:
625:
494:
413:
357:
156:from an observer (
136:cardinal direction
53:
2599:
2587:
2558:
2557:
2408:Lagrangian points
2345:Vis-viva equation
2315:Kepler's equation
2162:Orbital mechanics
2066:Azimuth recording
1975:polar coordinates
1927:celestial equator
1913:
1912:
1638:Note the swapped
1461:
1375:
1240:
1166:
1165:
1013:
786:
571:
570:
490:
489:
151:relative position
126:
114:
16:(Redirected from
2964:
2914:
2913:
2902:
2901:
2900:
2890:
2889:
2888:
2878:
2877:
2866:
2865:
2864:
2854:
2853:
2842:
2841:
2840:
2833:
2824:
2816:
2808:
2800:
2774:Rutstrum, Carl,
2762:
2759:
2753:
2746:
2740:
2733:
2727:
2724:
2718:
2711:
2705:
2698:
2692:
2679:Rutstrum, Carl,
2677:
2671:
2627:
2621:
2620:
2607:
2601:
2594:
2592:
2582:
2580:
2571:
2550:
2543:
2536:
2515:Orbital maneuver
2464:Payload fraction
2444:
2425:Lissajous orbits
2359:
2330:Orbital velocity
2277:Hyperbolic orbit
2173:Orbital elements
2163:
2154:
2137:
2111:Relative bearing
2106:Panning (camera)
2025:azimuth thruster
1941:Polar coordinate
1896:
1886:
1859:
1857:
1856:
1851:
1849:
1830:
1829:
1813:
1812:
1784:
1783:
1767:
1766:
1701:
1699:
1698:
1693:
1669:
1667:
1666:
1661:
1634:
1632:
1631:
1626:
1621:
1620:
1608:
1607:
1595:
1594:
1582:
1581:
1533:
1531:
1530:
1525:
1520:
1519:
1507:
1506:
1494:
1493:
1481:
1480:
1462:
1454:
1386:
1384:
1383:
1378:
1376:
1374:
1373:
1372:
1357:
1353:
1352:
1351:
1330:
1319:
1284:
1283:
1282:
1281:
1278:
1271:
1255:
1253:
1252:
1247:
1245:
1241:
1239:
1238:
1237:
1200:
1189:
1167:
1164:
1163:
1162:
1157:
1153:
1152:
1151:
1130:
1126:
1125:
1124:
1097:
1096:
1095:
1090:
1086:
1085:
1084:
1063:
1059:
1058:
1057:
1030:
1029:
1027:
1026:
1014:
1012:
1011:
1010:
994:
993:
992:
976:
974:
970:
969:
968:
934:
933:
905:
904:
860:
859:
828:geodesic azimuth
824:geodetic azimuth
797:
795:
794:
789:
787:
785:
775:
774:
756:
755:
740:
739:
723:
712:
680:
678:
677:
672:
670:
669:
653:
651:
650:
645:
643:
642:
564:North-northwest
516:South-southwest
497:
483:South-southeast
435:North-northeast
416:
315:. The other is
285:commissioned by
271:Geoffrey Chaucer
212:horizontal plane
173:horizontal plane
138:, most commonly
121:
119:
109:
107:
99:
98:
97:
96:
89:
86:
85:
82:
79:
76:
73:
70:
67:
21:
2972:
2971:
2967:
2966:
2965:
2963:
2962:
2961:
2922:
2921:
2920:
2908:
2898:
2896:
2886:
2884:
2872:
2862:
2860:
2848:
2838:
2836:
2828:
2814:"Azimuth"
2811:
2798:"Azimuth"
2795:
2792:
2771:
2769:Further reading
2766:
2765:
2760:
2756:
2747:
2743:
2734:
2730:
2725:
2721:
2712:
2708:
2699:
2695:
2678:
2674:
2656:Wayback Machine
2647:; "azimuth" at
2641:; "al-Samt" at
2628:
2624:
2609:
2608:
2604:
2572:
2568:
2563:
2554:
2525:
2524:
2520:Orbit insertion
2510:
2502:
2501:
2487:
2479:
2478:
2454:
2446:
2442:
2435:
2434:
2430:Lyapunov orbits
2421:
2420:
2404:
2394:
2393:
2369:
2361:
2357:
2350:
2349:
2335:Surface gravity
2310:Escape velocity
2300:
2292:
2291:
2272:Parabolic orbit
2268:
2267:
2234:
2232:
2229:two-body orbits
2220:
2219:
2210:Semi-major axis
2175:
2165:
2161:
2135:
2041:
1999:
1943:
1931:right ascension
1923:
1921:Right ascension
1918:
1909:
1908:
1907:
1902:
1897:
1877:
1875:Map projections
1847:
1846:
1821:
1814:
1804:
1801:
1800:
1775:
1768:
1758:
1749:
1748:
1743:
1736:
1721:
1714:
1672:
1671:
1640:
1639:
1612:
1599:
1586:
1573:
1553:
1552:
1544:The formula in
1511:
1498:
1485:
1472:
1442:
1441:
1411:
1404:
1364:
1343:
1336:
1332:
1331:
1320:
1301:
1300:
1295:
1279:
1276:
1274:
1273:
1269:
1268:
1243:
1242:
1229:
1201:
1190:
1181:
1169:
1168:
1143:
1136:
1132:
1131:
1116:
1109:
1105:
1098:
1076:
1069:
1065:
1064:
1049:
1042:
1038:
1031:
1018:
1002:
995:
984:
977:
960:
953:
949:
942:
936:
935:
925:
906:
896:
887:
886:
861:
851:
842:
841:
804:oblate spheroid
766:
747:
731:
724:
713:
694:
693:
661:
656:
655:
634:
629:
628:
597:
583:
577:
572:
548:West-northwest
532:West-southwest
491:
467:East-southeast
451:East-northeast
410:
402:cardinal points
345:
339:
301:
295:
251:
204:Earth's surface
186:direction of a
178:When used as a
169:reference plane
165:perpendicularly
93:
92:
91:
64:
60:
35:
28:
23:
22:
18:Azimuthal angle
15:
12:
11:
5:
2970:
2968:
2960:
2959:
2954:
2949:
2944:
2939:
2934:
2924:
2923:
2919:
2918:
2906:
2894:
2882:
2870:
2858:
2846:
2826:
2825:
2809:
2791:
2790:External links
2788:
2787:
2786:
2770:
2767:
2764:
2763:
2754:
2741:
2728:
2719:
2706:
2693:
2672:
2635:; "azimut" at
2622:
2619:(Online). n.d.
2602:
2565:
2564:
2562:
2559:
2556:
2555:
2553:
2552:
2545:
2538:
2530:
2527:
2526:
2523:
2522:
2517:
2511:
2508:
2507:
2504:
2503:
2500:
2499:
2494:
2492:Gravity assist
2488:
2485:
2484:
2481:
2480:
2477:
2476:
2471:
2466:
2461:
2455:
2452:
2451:
2448:
2447:
2440:
2437:
2436:
2433:
2432:
2427:
2419:
2418:
2410:
2406:
2405:
2400:
2399:
2396:
2395:
2392:
2391:
2386:
2381:
2376:
2370:
2367:
2366:
2363:
2362:
2355:
2352:
2351:
2348:
2347:
2342:
2337:
2332:
2327:
2325:Orbital period
2322:
2317:
2312:
2307:
2301:
2298:
2297:
2294:
2293:
2290:
2289:
2287:Decaying orbit
2284:
2279:
2274:
2266:
2265:
2259:
2252:
2250:Transfer orbit
2248:
2247:
2246:
2244:Elliptic orbit
2241:
2239:Circular orbit
2235:
2226:
2225:
2222:
2221:
2218:
2217:
2212:
2207:
2202:
2197:
2192:
2187:
2182:
2176:
2171:
2170:
2167:
2166:
2159:
2156:
2155:
2147:
2146:
2142:
2141:
2134:
2133:
2128:
2123:
2118:
2113:
2108:
2103:
2098:
2093:
2088:
2083:
2078:
2076:Clock position
2073:
2068:
2063:
2058:
2053:
2048:
2042:
2040:
2037:
1998:
1995:
1942:
1939:
1922:
1919:
1917:
1914:
1911:
1910:
1899:
1898:
1891:
1890:
1889:
1876:
1873:
1871:applications.
1861:
1860:
1845:
1842:
1839:
1836:
1833:
1828:
1824:
1820:
1817:
1815:
1811:
1807:
1803:
1802:
1799:
1796:
1793:
1790:
1787:
1782:
1778:
1774:
1771:
1769:
1765:
1761:
1757:
1756:
1741:
1734:
1719:
1712:
1691:
1688:
1685:
1682:
1679:
1659:
1656:
1653:
1650:
1647:
1636:
1635:
1624:
1619:
1615:
1611:
1606:
1602:
1598:
1593:
1589:
1585:
1580:
1576:
1572:
1569:
1566:
1563:
1560:
1535:
1534:
1523:
1518:
1514:
1510:
1505:
1501:
1497:
1492:
1488:
1484:
1479:
1475:
1471:
1468:
1465:
1460:
1457:
1452:
1449:
1410:
1409:In cartography
1407:
1402:
1388:
1387:
1371:
1367:
1363:
1360:
1356:
1350:
1346:
1342:
1339:
1335:
1329:
1326:
1323:
1317:
1314:
1311:
1308:
1293:
1257:
1256:
1236:
1232:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1199:
1196:
1193:
1187:
1184:
1182:
1180:
1177:
1174:
1171:
1170:
1161:
1156:
1150:
1146:
1142:
1139:
1135:
1129:
1123:
1119:
1115:
1112:
1108:
1104:
1101:
1094:
1089:
1083:
1079:
1075:
1072:
1068:
1062:
1056:
1052:
1048:
1045:
1041:
1037:
1034:
1025:
1021:
1017:
1009:
1005:
1001:
998:
991:
987:
983:
980:
973:
967:
963:
959:
956:
952:
948:
945:
943:
941:
938:
937:
932:
928:
924:
921:
918:
915:
912:
909:
907:
903:
899:
895:
892:
889:
888:
885:
882:
879:
876:
873:
870:
867:
864:
862:
858:
854:
850:
849:
814:Normal-section
799:
798:
784:
781:
778:
773:
769:
765:
762:
759:
754:
750:
746:
743:
738:
734:
730:
727:
722:
719:
716:
710:
707:
704:
701:
668:
664:
641:
637:
579:Main article:
576:
573:
569:
568:
565:
561:
560:
557:
553:
552:
549:
545:
544:
541:
537:
536:
533:
529:
528:
525:
521:
520:
517:
513:
512:
509:
505:
504:
501:
492:
488:
487:
484:
480:
479:
476:
472:
471:
468:
464:
463:
460:
456:
455:
452:
448:
447:
444:
440:
439:
436:
432:
431:
428:
424:
423:
420:
411:
409:
406:
353:Sandstone Peak
341:Main article:
338:
335:
323:satellite dish
297:Main article:
294:
291:
287:King Alfonso X
250:
247:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2969:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2929:
2927:
2917:
2912:
2907:
2905:
2895:
2893:
2883:
2881:
2876:
2871:
2869:
2859:
2857:
2852:
2847:
2845:
2835:
2831:
2822:
2821:
2815:
2810:
2806:
2805:
2799:
2794:
2793:
2789:
2785:
2784:0-8166-3661-3
2781:
2777:
2773:
2772:
2768:
2758:
2755:
2751:
2745:
2742:
2738:
2732:
2729:
2723:
2720:
2716:
2710:
2707:
2703:
2697:
2694:
2690:
2689:0-8166-3661-3
2686:
2682:
2676:
2673:
2669:
2665:
2661:
2657:
2653:
2650:
2646:
2645:
2640:
2639:
2634:
2633:
2629:"Azimuth" at
2626:
2623:
2618:
2617:
2612:
2606:
2603:
2597:
2591:
2585:
2576:
2570:
2567:
2560:
2551:
2546:
2544:
2539:
2537:
2532:
2531:
2529:
2528:
2521:
2518:
2516:
2513:
2512:
2506:
2505:
2498:
2497:Oberth effect
2495:
2493:
2490:
2489:
2483:
2482:
2475:
2472:
2470:
2467:
2465:
2462:
2460:
2457:
2456:
2450:
2449:
2445:
2438:
2431:
2428:
2426:
2423:
2422:
2416:
2412:
2411:
2409:
2403:
2402:N-body orbits
2398:
2397:
2390:
2387:
2385:
2384:Perturbations
2382:
2380:
2377:
2375:
2372:
2371:
2365:
2364:
2360:
2353:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2306:
2303:
2302:
2296:
2295:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2269:
2263:
2260:
2258:
2254:
2253:
2251:
2245:
2242:
2240:
2237:
2236:
2230:
2224:
2223:
2216:
2213:
2211:
2208:
2206:
2205:Orbital nodes
2203:
2201:
2198:
2196:
2193:
2191:
2188:
2186:
2183:
2181:
2178:
2177:
2174:
2169:
2168:
2164:
2157:
2153:
2149:
2148:
2145:Astrodynamics
2143:
2139:
2138:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2067:
2064:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2043:
2038:
2036:
2034:
2030:
2026:
2021:
2019:
2015:
2010:
2008:
2004:
1996:
1994:
1992:
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1960:
1956:
1952:
1948:
1940:
1938:
1936:
1932:
1928:
1920:
1915:
1906:
1901:
1895:
1888:
1887:
1884:
1882:
1874:
1872:
1870:
1866:
1865:triangulation
1843:
1840:
1837:
1834:
1831:
1826:
1822:
1818:
1816:
1809:
1805:
1797:
1794:
1791:
1788:
1785:
1780:
1776:
1772:
1770:
1763:
1759:
1747:
1746:
1745:
1740:
1733:
1729:
1725:
1718:
1711:
1706:
1705:input order.
1704:
1686:
1683:
1680:
1654:
1651:
1648:
1617:
1613:
1609:
1604:
1600:
1596:
1591:
1587:
1583:
1578:
1574:
1567:
1564:
1561:
1558:
1551:
1550:
1549:
1547:
1542:
1540:
1516:
1512:
1508:
1503:
1499:
1495:
1490:
1486:
1482:
1477:
1473:
1466:
1463:
1458:
1455:
1450:
1447:
1440:
1439:
1438:
1436:
1432:
1428:
1420:
1415:
1408:
1406:
1401:
1397:
1393:
1369:
1365:
1361:
1358:
1354:
1348:
1344:
1340:
1337:
1333:
1327:
1324:
1321:
1315:
1312:
1309:
1306:
1299:
1298:
1297:
1292:
1288:
1266:
1262:
1234:
1230:
1226:
1223:
1217:
1214:
1211:
1208:
1197:
1194:
1191:
1185:
1183:
1178:
1175:
1172:
1159:
1154:
1148:
1144:
1140:
1137:
1133:
1127:
1121:
1117:
1113:
1110:
1106:
1102:
1099:
1092:
1087:
1081:
1077:
1073:
1070:
1066:
1060:
1054:
1050:
1046:
1043:
1039:
1035:
1032:
1023:
1019:
1015:
1007:
1003:
999:
996:
989:
985:
981:
978:
971:
965:
961:
957:
954:
950:
946:
944:
930:
922:
919:
916:
910:
908:
901:
897:
893:
890:
880:
877:
874:
868:
865:
863:
856:
852:
840:
839:
838:
835:
833:
829:
825:
821:
817:
815:
810:
806:
805:
782:
779:
776:
771:
767:
763:
760:
757:
752:
748:
744:
741:
736:
732:
728:
725:
720:
717:
714:
708:
705:
702:
699:
692:
691:
690:
688:
684:
666:
662:
639:
635:
622:
618:
614:
610:
606:
601:
596:
592:
588:
582:
574:
566:
562:
558:
554:
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
498:
485:
481:
477:
473:
469:
465:
461:
457:
453:
449:
445:
441:
437:
433:
429:
425:
417:
407:
405:
403:
397:
395:
391:
387:
382:
380:
376:
375:
370:
366:
362:
354:
349:
344:
337:In navigation
336:
334:
332:
329:). In modern
328:
324:
320:
319:
314:
310:
306:
300:
292:
290:
288:
284:
283:
278:
277:
272:
268:
264:
260:
257:, pronounced
256:
248:
246:
244:
240:
236:
232:
228:
224:
220:
215:
213:
209:
205:
201:
197:
193:
189:
185:
181:
176:
174:
170:
166:
163:
159:
155:
152:
147:
145:
141:
137:
133:
130:
124:
118:
112:
103:
95:
88:
58:
50:
46:
41:
37:
33:
19:
2818:
2802:
2775:
2757:
2749:
2744:
2736:
2731:
2722:
2714:
2709:
2701:
2696:
2680:
2675:
2663:
2659:
2643:
2637:
2631:
2625:
2614:
2605:
2569:
2282:Radial orbit
2233:eccentricity
2215:True anomaly
2200:Mean anomaly
2190:Eccentricity
2029:shipbuilding
2022:
2017:
2011:
2006:
2000:
1990:
1989:rather than
1986:
1978:
1966:
1958:
1944:
1924:
1878:
1862:
1738:
1731:
1727:
1723:
1716:
1709:
1707:
1637:
1543:
1536:
1431:grid azimuth
1430:
1426:
1424:
1399:
1389:
1290:
1264:
1260:
1258:
836:
827:
823:
812:
808:
802:
800:
689:is given by
686:
682:
681:, longitude
626:
398:
383:
378:
372:
364:
358:
316:
302:
293:In astronomy
289:of Castile.
280:
274:
262:
258:
254:
252:
216:
177:
148:
56:
54:
36:
2957:Cartography
2904:Outer space
2892:Spaceflight
2856:Mathematics
2748:U.S. Army,
2735:U.S. Army,
2713:U.S. Army,
2700:U.S. Army,
2415:Halo orbits
2379:Hill sphere
2195:Inclination
2086:Inclination
1392:declination
313:coordinates
231:engineering
2937:Navigation
2926:Categories
2561:References
2459:Mass ratio
2374:Barycenter
1997:Other uses
1396:hour angle
820:theodolite
611:along the
585:See also:
575:In geodesy
556:Northwest
524:Southwest
500:Direction
475:Southeast
443:Northeast
419:Direction
386:true north
327:sat finder
307:, used in
243:ballistics
223:navigation
208:true north
184:horizontal
129:horizontal
106:اَلسُّمُوت
2942:Surveying
2868:Astronomy
2844:Geography
2611:"azimuth"
2584:romanized
2299:Equations
2227:Types of
2091:Longitude
2033:propeller
1965:onto the
1844:α
1841:
1798:α
1795:
1610:−
1584:−
1568:
1559:α
1509:−
1483:−
1467:
1459:π
1448:α
1366:φ
1362:
1341:−
1325:
1313:α
1310:
1296:= 0 then
1231:φ
1227:
1215:
1209:−
1206:Λ
1195:
1179:α
1176:
1145:φ
1141:
1114:−
1078:φ
1074:
1047:−
1004:φ
1000:
986:φ
982:
958:−
940:Λ
920:−
894:−
878:−
780:
768:φ
764:
758:−
749:φ
745:
733:φ
729:
718:
706:α
703:
663:φ
636:φ
609:Melbourne
605:Cape Town
369:clockwise
331:astronomy
267:astrolabe
249:Etymology
227:astronomy
200:sea level
190:or other
162:projected
111:romanized
2691:, p. 194
2664:as-sumūt
2660:al-sumūt
2652:Archived
2579:السَّمْت
2096:Latitude
2039:See also
613:geodesic
503:Azimuth
422:Azimuth
374:meridian
318:altitude
259:as-sumūt
255:al-sumūt
117:as-sumūt
2952:Geodesy
2916:Science
2830:Portals
2823:. 1921.
2598:
2590:as-samt
2586::
2116:Sextant
2018:azimuth
2007:azimuth
1983:radians
1546:radians
1419:compass
1275:298.257
1272:⁄
816:azimuth
809:azimuth
567:337.5°
551:292.5°
535:247.5°
519:202.5°
486:157.5°
470:112.5°
379:Azimuth
303:In the
263:al-samt
235:mapping
219:degrees
194:in the
167:onto a
134:from a
125:
113::
100:; from
57:azimuth
2782:
2687:
2575:Arabic
2131:Zenith
1963:vector
1289:). If
1259:where
593:, and
508:South
454:67.5°
438:22.5°
427:North
241:, and
239:mining
158:origin
154:vector
102:Arabic
49:zenith
2932:Angle
2880:Stars
2666:(see
2180:Apsis
2031:is a
1971:plane
1955:angle
1869:radar
1703:atan2
1565:atan2
1464:atan2
1287:WGS84
559:315°
543:270°
540:West
527:225°
511:180°
478:135°
459:East
361:alpha
171:(the
140:north
132:angle
90:
2780:ISBN
2685:ISBN
2596:lit.
2001:For
1425:The
1394:and
1285:for
826:(or
619:and
607:and
462:90°
446:45°
390:grad
188:star
123:lit.
45:line
2231:by
2027:in
2023:An
2012:In
1949:or
1838:cos
1792:sin
1456:180
1437:):
1429:or
1359:tan
1322:sin
1307:tan
1280:563
1277:223
1224:sin
1212:cos
1192:sin
1173:tan
1138:tan
1071:tan
997:tan
979:tan
807:);
777:cos
761:sin
742:tan
726:cos
715:sin
700:tan
430:0°
394:mil
273:'s
196:sky
146:.
55:An
2928::
2817:.
2801:.
2670:).
2613:.
2593:,
2581:,
2577::
2005:,
1993:.
1979:xy
1967:xy
1937:.
1737:,
1715:,
589:,
392:,
377:.
363:,
245:.
237:,
233:,
229:,
225:,
214:.
120:,
108:,
104::
2832::
2549:e
2542:t
2535:v
2417:)
2413:(
2264:)
2255:(
1991:φ
1987:θ
1969:-
1959:x
1835:D
1832:+
1827:1
1823:Y
1819:=
1810:2
1806:Y
1789:D
1786:+
1781:1
1777:X
1773:=
1764:2
1760:X
1742:2
1739:Y
1735:2
1732:X
1728:α
1724:D
1720:1
1717:Y
1713:1
1710:X
1690:)
1687:x
1684:,
1681:y
1678:(
1658:)
1655:y
1652:,
1649:x
1646:(
1623:)
1618:1
1614:Y
1605:2
1601:Y
1597:,
1592:1
1588:X
1579:2
1575:X
1571:(
1562:=
1522:)
1517:1
1513:Y
1504:2
1500:Y
1496:,
1491:1
1487:X
1478:2
1474:X
1470:(
1451:=
1403:2
1400:φ
1370:2
1355:)
1349:2
1345:e
1338:1
1334:(
1328:L
1316:=
1294:1
1291:φ
1270:1
1265:e
1261:f
1235:1
1221:)
1218:L
1203:(
1198:L
1186:=
1160:2
1155:)
1149:1
1134:(
1128:)
1122:2
1118:e
1111:1
1107:(
1103:+
1100:1
1093:2
1088:)
1082:2
1067:(
1061:)
1055:2
1051:e
1044:1
1040:(
1036:+
1033:1
1024:2
1020:e
1016:+
1008:1
990:2
972:)
966:2
962:e
955:1
951:(
947:=
931:2
927:)
923:f
917:1
914:(
911:=
902:2
898:e
891:1
884:)
881:f
875:2
872:(
869:f
866:=
857:2
853:e
783:L
772:1
753:2
737:1
721:L
709:=
687:α
683:L
667:2
640:1
623:.
365:α
87:/
84:θ
81:ə
78:m
75:ə
72:z
69:æ
66:ˈ
63:/
59:(
51:.
34:.
20:)
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