223:
239:
395:
859:
199:
31:
978:. In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation
672:
It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to be
844:. However, one of the coordinate curves is reduced to a single point, the origin, which is often viewed as a circle of radius zero. Similarly, spherical and cylindrical coordinate systems have coordinate curves that are lines, circles or circles of radius zero.
774:, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to â3, so that the coordinate of each point becomes 3 more.
929:
is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map is a
677:. Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as the
504:
are the
Cartesian coordinates of the point. This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the
763:
Such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
766:
Such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)
878:
are the spheres with center at the origin. In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the
Cartesian coordinate system we may speak of
663:
are used to determine the position of a line in space. When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term
1004:
The Earth as a whole is one of the most common geometric spaces requiring the precise measurement of location, and thus coordinate systems. Starting with the Greeks of the
651:
Coordinates systems are often used to specify the position of a point, but they may also be used to specify the position of more complex figures such as lines, planes,
942:. It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an
708:, which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (
798:
Given a coordinate system, if one of the coordinates of a point varies while the other coordinates are held constant, then the resulting curve is called a
704:
There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems is described by
557:
represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin.
833:, all coordinates curves are lines, and, therefore, there are as many coordinate axes as coordinates. Moreover, the coordinate axes are pairwise
866:
In three-dimensional space, if one coordinate is held constant and the other two are allowed to vary, then the resulting surface is called a
320:(measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is
513:. In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values.
267:
planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create
950:
and additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example, a
263:
lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three mutually
699:
2085:
2234:
1159:
1562:
1503:
1396:
1331:
1306:
1275:
1149:
986:
of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three
2269:
1948:
954:
is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.
1728:
1684:
1619:
1586:
1537:
1444:
1360:
693:
2150:
1169:
110:. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered
1701:
191:, where the signed distance is the distance taken as positive or negative depending on which side of the line
134:. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and
2001:
1933:
1651:
1259:
812:
590:
385:
2026:
1043:
1012:
2372:
2264:
1646:
1089:
1047:
1033:
1016:
875:
830:
528:
are a generalization of coordinate systems generally; the system is based on the intersection of curves.
389:
252:
229:
212:
35:
24:
195:
lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.
2377:
2075:
1895:
1495:
1154:
1747:
1174:
1073:
1055:
430:). Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (
42:. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance
2229:
336:) there is a single point, but any point is represented by many pairs of coordinates. For example, (
2331:
2249:
2203:
1910:
999:
402:
There are two common methods for extending the polar coordinate system to three dimensions. In the
20:
2301:
1988:
1905:
1875:
1641:
1109:
951:
570:
564:
525:
467:
295:
1388:
1324:
Field Theory
Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions
2259:
2115:
2070:
1099:
971:
818:
531:
123:
1292:
660:
560:
2341:
2296:
1776:
1721:
1674:
1607:
1164:
1029:
983:
848:
680:
604:
584:
580:
554:
1659:
1326:(corrected 2nd, 3rd print ed.). New York: Springer-Verlag. pp. 9â11 (Table 1.01).
2316:
2244:
2130:
1996:
1958:
1890:
1298:
1059:
1008:, a variety of coordinate systems have been developed based on the types above, including:
979:
574:
1050:
that models the earth as an object, and are most commonly used for modeling the orbits of
8:
2193:
2016:
2006:
1855:
1840:
1796:
1135:
728:
axis, then the coordinate transformation from polar to
Cartesian coordinates is given by
95:
2326:
2183:
2036:
1850:
1786:
1381:
1104:
1005:
840:
A polar coordinate system is a curvilinear system where coordinate curves are lines or
612:
545:
535:
1213:
639:
563:
are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as
198:
2321:
2090:
2065:
1880:
1791:
1680:
1615:
1582:
1558:
1533:
1499:
1450:
1440:
1410:
1402:
1392:
1356:
1349:
1327:
1302:
1271:
1235:
1210:
1078:
792:
139:
2336:
2011:
1978:
1963:
1845:
1714:
1599:
1130:
1125:
1083:
943:
785:
665:
632:
628:
541:
506:
256:
131:
99:
970:, coordinate systems are used to describe the (linear) position of points and the
847:
Many curves can occur as coordinate curves. For example, the coordinate curves of
2306:
2254:
2198:
2178:
2080:
1968:
1835:
1806:
1432:
1428:
1238:
1094:
810:. A coordinate system for which some coordinate curves are not lines is called a
283:
162:
161:
The simplest example of a coordinate system is the identification of points on a
107:
825:
A coordinate line with all other constant coordinates equal to zero is called a
2346:
2311:
2208:
2041:
2031:
2021:
1943:
1915:
1900:
1885:
1801:
1670:
1483:
1376:
1114:
1037:
910:
368:) are all polar coordinates for the same point. The pole is represented by (0,
127:
2291:
410:-coordinate with the same meaning as in Cartesian coordinates is added to the
222:
2366:
2283:
2188:
2100:
1973:
931:
803:
316:, there is a single line through the pole whose angle with the polar axis is
264:
260:
1346:
2351:
2155:
2140:
2105:
1953:
1938:
1668:
1603:
1414:
858:
759:
from the space to itself two coordinate transformations can be associated:
594:
394:
238:
2239:
2213:
2135:
1824:
1763:
1267:
987:
669:
is used for any coordinate system that specifies the position of a line.
167:
156:
119:
2120:
975:
967:
834:
620:
1347:
Finney, Ross; George Thomas; Franklin Demana; Bert Waits (June 1994).
862:
Coordinate surfaces of the three-dimensional paraboloidal coordinates.
2095:
2046:
1243:
1218:
1051:
1024:
771:
756:
616:
946:
covering the space. A space equipped with such an atlas is called a
897:-dimensional spaces resulting from fixing a single coordinate of an
822:
are a special but extremely common case of curvilinear coordinates.
2125:
2110:
1487:
1119:
1020:
963:
916:
852:
598:
510:
103:
79:
1614:. American Institute of Aeronautics and Astronautics. p. 71.
1355:(Single Variable Version ed.). Addison-Wesley Publishing Co.
1819:
1781:
1555:
A Computational
Differential Geometry Approach to Grid Generation
1454:
1406:
1322:
Moon P, Spencer DE (1988). "Rectangular
Coordinates (x, y, z)".
2145:
1737:
1598:
841:
656:
652:
611:
There are ways of describing curves without coordinates, using
379:
87:
30:
19:"Coordinate" redirects here. For coordinates on the Earth, see
777:
111:
51:
1492:
Methods of
Algebraic Geometry, Volume I (Book II)
1122:, graphical representations of different coordinate systems
870:. For example, the coordinate surfaces obtained by holding
791:"Coordinate plane" redirects here. Not to be confused with
724:) have the same origin, and the polar axis is the positive
1706:
1208:
784:"Coordinate line" redirects here. Not to be confused with
244:
The
Cartesian coordinate system in three-dimensional space
67:
59:
521:
Some other common coordinate systems are the following:
251:
The prototypical example of a coordinate system is the
179:) is chosen on a given line. The coordinate of a point
1291:
Anton, Howard; Bivens, Irl C.; Davis, Stephen (2021).
300:
Another common coordinate system for the plane is the
1639:
1233:
1040:
to create a planar surface of the world or a region.
1258:
646:
1380:
1348:
1142:
1530:Mathematical Methods for Engineers and Scientists
2364:
1290:
957:
461:
130:or elements of a more abstract system such as a
1375:
516:
206:
118:-coordinate". The coordinates are taken to be
1722:
1482:
1427:
1640:Voitsekhovskii, M.I.; Ivanov, A.B. (2001) ,
982:, which includes, in its three columns, the
380:Cylindrical and spherical coordinate systems
278:Depending on the direction and order of the
145:
1321:
778:Coordinate lines/curves and planes/surfaces
472:A point in the plane may be represented in
1729:
1715:
1387:. New York City: D. van Nostrand. p.
308:and a ray from this point is taken as the
289:
1351:Calculus: Graphical, Numerical, Algebraic
1262:; Redlin, Lothar; Watson, Saleem (2008).
700:List of common coordinate transformations
2086:Covariance and contravariance of vectors
1552:
1383:The Mathematics of Physics and Chemistry
857:
393:
282:, the three-dimensional system may be a
29:
1470:An Introduction to Algebraical Geometry
183:is defined as the signed distance from
2365:
1439:. New York: McGraw-Hill. p. 658.
1437:Methods of Theoretical Physics, Part I
615:that use invariant quantities such as
597:and more generally in the analysis of
114:and sometimes by a letter, as in "the
1710:
1657:
1612:Analytical Mechanics of Space Systems
1532:. Vol. 2. Springer. p. 13.
1467:
1234:
1209:
993:
607:are used in the context of triangles.
171:. In this system, an arbitrary point
16:Method for specifying point positions
1527:
279:
418:polar coordinates giving a triple (
328:. For a given pair of coordinates (
13:
1949:Tensors in curvilinear coordinates
904:
687:
197:
86:is a system that uses one or more
14:
2389:
1695:
1669:Shigeyuki Morita; Teruko Nagase;
1150:EddingtonâFinkelstein coordinates
694:Active and passive transformation
642:relates arc length and curvature.
102:or other geometric elements on a
901:-dimensional coordinate system.
647:Coordinates of geometric objects
237:
221:
1592:
1571:
1546:
1521:
1512:
1476:
1461:
1160:GullstrandâPainlevĂŠ coordinates
1143:Relativistic coordinate systems
934:from an open subset of a space
1676:Geometry of Differential Forms
1553:Liseikin, Vladimir D. (2007).
1468:Jones, Alfred Clement (1912).
1421:
1369:
1340:
1315:
1284:
1252:
1227:
1202:
1193:
275:-dimensional Euclidean space.
150:
1:
2002:Exterior covariant derivative
1934:Tensor (intrinsic definition)
1679:. AMS Bookstore. p. 12.
1181:
958:Orientation-based coordinates
813:curvilinear coordinate system
802:. If a coordinate curve is a
591:Barycentric coordinate system
462:Homogeneous coordinate system
404:cylindrical coordinate system
398:Cylindrical coordinate system
386:Cylindrical coordinate system
271:coordinates for any point in
2027:Raising and lowering indices
1702:Hexagonal Coordinate Systems
1664:. Ginn and Co. pp. 1ff.
1658:Woods, Frederick S. (1922).
1379:; Murphy, George M. (1956).
1186:
1170:KruskalâSzekeres coordinates
1044:Geocentric coordinate system
1034:cartesian coordinate systems
1030:Projected coordinate systems
1013:Geographic coordinate system
165:with real numbers using the
94:, to uniquely determine the
7:
2265:Gluon field strength tensor
1736:
1647:Encyclopedia of Mathematics
1090:Celestial coordinate system
1066:
1048:cartesian coordinate system
876:spherical coordinate system
831:Cartesian coordinate system
631:relates arc length and the
555:log-polar coordinate system
517:Other commonly used systems
390:Spherical coordinate system
304:. A point is chosen as the
253:Cartesian coordinate system
230:Cartesian coordinate system
213:Cartesian coordinate system
207:Cartesian coordinate system
70:) is often used instead of
36:spherical coordinate system
25:Coordinate (disambiguation)
10:
2394:
2076:Cartan formalism (physics)
1896:Penrose graphical notation
1632:
1496:Cambridge University Press
1155:Gaussian polar coordinates
997:
914:
908:
790:
783:
706:coordinate transformations
697:
691:
465:
383:
293:
210:
154:
18:
2282:
2222:
2171:
2164:
2056:
1987:
1924:
1868:
1815:
1762:
1755:
1748:Glossary of tensor theory
1744:
1577:Munkres, James R. (2000)
1175:Schwarzschild coordinates
1074:Absolute angular momentum
1056:Global Positioning System
1032:, including thousands of
990:aligned with those axes.
716:) and polar coordinates (
286:or a left-handed system.
146:Common coordinate systems
2332:Gregorio Ricci-Curbastro
2204:Riemann curvature tensor
1911:Van der Waerden notation
1557:. Springer. p. 38.
1000:Spatial reference system
887:coordinate hypersurfaces
438:) to polar coordinates (
21:Spatial reference system
2302:Elwin Bruno Christoffel
2235:Angular momentum tensor
1906:Tetrad (index notation)
1876:Abstract index notation
1608:"Rigid body kinematics"
1294:Calculus: Multivariable
1110:Galilean transformation
952:differentiable manifold
587:treatment of mechanics.
577:treatment of mechanics.
571:Generalized coordinates
565:homogeneous coordinates
526:Curvilinear coordinates
474:homogeneous coordinates
468:Homogeneous coordinates
302:polar coordinate system
296:Polar coordinate system
290:Polar coordinate system
138:; this is the basis of
54:), and azimuthal angle
2116:Levi-Civita connection
1100:Fractional coordinates
1046:, a three-dimensional
863:
819:Orthogonal coordinates
532:Orthogonal coordinates
399:
203:
124:elementary mathematics
75:
23:. For other uses, see
2342:Jan Arnoldus Schouten
2297:Augustin-Louis Cauchy
1777:Differential geometry
1299:John Wiley & Sons
1165:Isotropic coordinates
1017:spherical coordinates
984:Cartesian coordinates
974:of axes, planes, and
938:to an open subset of
915:Further information:
861:
849:parabolic coordinates
605:Trilinear coordinates
581:Canonical coordinates
397:
201:
33:
2317:Carl Friedrich Gauss
2250:stressâenergy tensor
2245:Cauchy stress tensor
1997:Covariant derivative
1959:Antisymmetric tensor
1891:Multi-index notation
1528:Tang, K. T. (2006).
1060:satellite navigation
538:meet at right angles
312:. For a given angle
38:is commonly used in
2194:Nonmetricity tensor
2049:(2nd-order tensors)
2017:Hodge star operator
2007:Exterior derivative
1856:Transport phenomena
1841:Continuum mechanics
1797:Multilinear algebra
1214:"Coordinate System"
1136:Translation of axes
661:PlĂźcker coordinates
613:intrinsic equations
561:PlĂźcker coordinates
546:coordinate surfaces
536:coordinate surfaces
509:without the use of
446:) giving a triple (
372:) for any value of
2373:Coordinate systems
2327:Tullio Levi-Civita
2270:Metric tensor (GR)
2184:Levi-Civita symbol
2037:Tensor contraction
1851:General relativity
1787:Euclidean geometry
1270:. pp. 13â19.
1236:Weisstein, Eric W.
1211:Weisstein, Eric W.
1105:Frame of reference
1036:, each based on a
1006:Hellenistic period
994:Geographic systems
868:coordinate surface
864:
548:are not orthogonal
400:
204:
76:
2378:Analytic geometry
2360:
2359:
2322:Hermann Grassmann
2278:
2277:
2230:Moment of inertia
2091:Differential form
2066:Affine connection
1881:Einstein notation
1864:
1863:
1792:Exterior calculus
1772:Coordinate system
1581:. Prentice Hall.
1564:978-3-540-34235-9
1505:978-0-521-46900-5
1398:978-0-88275-423-9
1333:978-0-387-18430-2
1308:978-1-119-77798-4
1277:978-0-495-56521-5
1260:Stewart, James B.
1079:Alphanumeric grid
921:The concept of a
880:coordinate planes
806:, it is called a
793:Plane coordinates
623:. These include:
324:for given number
140:analytic geometry
84:coordinate system
2385:
2337:Bernhard Riemann
2169:
2168:
2012:Exterior product
1979:Two-point tensor
1964:Symmetric tensor
1846:Electromagnetism
1760:
1759:
1731:
1724:
1717:
1708:
1707:
1690:
1665:
1654:
1626:
1625:
1600:Hanspeter Schaub
1596:
1590:
1575:
1569:
1568:
1550:
1544:
1543:
1525:
1519:
1516:
1510:
1509:
1480:
1474:
1473:
1465:
1459:
1458:
1425:
1419:
1418:
1386:
1373:
1367:
1366:
1354:
1344:
1338:
1337:
1319:
1313:
1312:
1288:
1282:
1281:
1266:(5th ed.).
1256:
1250:
1249:
1248:
1231:
1225:
1224:
1223:
1206:
1200:
1197:
1131:Rotation of axes
1126:Reference system
1084:Axes conventions
1054:, including the
972:angular position
927:coordinate chart
896:
874:constant in the
800:coordinate curve
786:Line coordinates
770:For example, in
666:line coordinates
633:tangential angle
629:Whewell equation
583:are used in the
573:are used in the
542:Skew coordinates
507:projective plane
241:
225:
132:commutative ring
2393:
2392:
2388:
2387:
2386:
2384:
2383:
2382:
2363:
2362:
2361:
2356:
2307:Albert Einstein
2274:
2255:Einstein tensor
2218:
2199:Ricci curvature
2179:Kronecker delta
2165:Notable tensors
2160:
2081:Connection form
2058:
2052:
1983:
1969:Tensor operator
1926:
1920:
1860:
1836:Computer vision
1829:
1811:
1807:Tensor calculus
1751:
1740:
1735:
1698:
1693:
1687:
1661:Higher Geometry
1635:
1630:
1629:
1622:
1604:John L. Junkins
1597:
1593:
1576:
1572:
1565:
1551:
1547:
1540:
1526:
1522:
1517:
1513:
1506:
1481:
1477:
1466:
1462:
1447:
1426:
1422:
1399:
1377:Margenau, Henry
1374:
1370:
1363:
1345:
1341:
1334:
1320:
1316:
1309:
1301:. p. 657.
1289:
1285:
1278:
1264:College Algebra
1257:
1253:
1232:
1228:
1207:
1203:
1198:
1194:
1189:
1184:
1179:
1145:
1140:
1095:Coordinate-free
1069:
1002:
996:
960:
919:
913:
907:
905:Coordinate maps
890:
827:coordinate axis
808:coordinate line
796:
789:
780:
702:
696:
690:
688:Transformations
659:. For example,
649:
640:CesĂ ro equation
519:
470:
464:
392:
384:Main articles:
382:
298:
292:
280:coordinate axes
249:
248:
247:
246:
245:
242:
234:
233:
226:
215:
209:
202:The number line
159:
153:
148:
128:complex numbers
108:Euclidean space
28:
17:
12:
11:
5:
2391:
2381:
2380:
2375:
2358:
2357:
2355:
2354:
2349:
2347:Woldemar Voigt
2344:
2339:
2334:
2329:
2324:
2319:
2314:
2312:Leonhard Euler
2309:
2304:
2299:
2294:
2288:
2286:
2284:Mathematicians
2280:
2279:
2276:
2275:
2273:
2272:
2267:
2262:
2257:
2252:
2247:
2242:
2237:
2232:
2226:
2224:
2220:
2219:
2217:
2216:
2211:
2209:Torsion tensor
2206:
2201:
2196:
2191:
2186:
2181:
2175:
2173:
2166:
2162:
2161:
2159:
2158:
2153:
2148:
2143:
2138:
2133:
2128:
2123:
2118:
2113:
2108:
2103:
2098:
2093:
2088:
2083:
2078:
2073:
2068:
2062:
2060:
2054:
2053:
2051:
2050:
2044:
2042:Tensor product
2039:
2034:
2032:Symmetrization
2029:
2024:
2022:Lie derivative
2019:
2014:
2009:
2004:
1999:
1993:
1991:
1985:
1984:
1982:
1981:
1976:
1971:
1966:
1961:
1956:
1951:
1946:
1944:Tensor density
1941:
1936:
1930:
1928:
1922:
1921:
1919:
1918:
1916:Voigt notation
1913:
1908:
1903:
1901:Ricci calculus
1898:
1893:
1888:
1886:Index notation
1883:
1878:
1872:
1870:
1866:
1865:
1862:
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1802:Tensor algebra
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1782:Dyadic algebra
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1697:
1696:External links
1694:
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1671:Katsumi Nomizu
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1115:Grid reference
1112:
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1097:
1092:
1087:
1086:in engineering
1081:
1076:
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1041:
1038:map projection
1027:
998:Main article:
995:
992:
959:
956:
923:coordinate map
911:Coordinate map
909:Main article:
906:
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779:
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698:Main article:
689:
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466:Main article:
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294:Main article:
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211:Main article:
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62:). The symbol
46:, polar angle
15:
9:
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4:
3:
2:
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2197:
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2189:Metric tensor
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2101:Exterior form
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1974:Tensor bundle
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1642:"Coordinates"
1638:
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1623:
1621:1-56347-563-4
1617:
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1595:
1588:
1587:0-13-181629-2
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1535:
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1485:
1484:Hodge, W.V.D.
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804:straight line
801:
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732: =
731:
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711:
707:
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695:
685:
683:
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679:principle of
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41:
37:
32:
26:
22:
2352:Hermann Weyl
2156:Vector space
2141:Pseudotensor
2106:Fiber bundle
2059:abstractions
1954:Mixed tensor
1939:Tensor field
1771:
1746:
1675:
1660:
1645:
1611:
1594:
1578:
1573:
1554:
1548:
1529:
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1491:
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1472:. Clarendon.
1469:
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1382:
1371:
1350:
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1323:
1317:
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1204:
1195:
1003:
988:unit vectors
976:rigid bodies
961:
947:
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920:
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729:
725:
721:
717:
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709:
705:
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678:
674:
671:
664:
650:
610:
593:as used for
520:
501:
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489:
485:
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477:
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451:
447:
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373:
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349:
345:
341:
337:
333:
329:
325:
321:
317:
313:
309:
305:
301:
299:
284:right-handed
277:
272:
268:
250:
232:in the plane
192:
188:
184:
180:
176:
172:
166:
160:
135:
120:real numbers
115:
91:
83:
77:
71:
63:
55:
47:
43:
39:
2292:Ălie Cartan
2240:Spin tensor
2214:Weyl tensor
2172:Mathematics
2136:Multivector
1927:definitions
1825:Engineering
1764:Mathematics
1433:Feshbach, H
1268:Brooks Cole
885:Similarly,
755:With every
585:Hamiltonian
168:number line
157:Number line
151:Number line
92:coordinates
2367:Categories
2121:Linear map
1989:Operations
1518:Woods p. 2
1199:Woods p. 1
1182:References
1058:and other
1052:satellites
968:kinematics
835:orthogonal
692:See also:
621:arc length
575:Lagrangian
310:polar axis
265:orthogonal
136:vice versa
2260:EM tensor
2096:Dimension
2047:Transpose
1652:EMS Press
1490:(1994) .
1429:Morse, PM
1244:MathWorld
1219:MathWorld
1187:Citations
1025:longitude
853:parabolas
757:bijection
748: sin
736: cos
675:dualistic
617:curvature
599:triangles
255:. In the
2126:Manifold
2111:Geodesic
1869:Notation
1673:(2001).
1606:(2003).
1579:Topology
1488:D. Pedoe
1455:52011515
1435:(1953).
1407:55010911
1120:Nomogram
1067:See also
1062:systems.
1021:latitude
964:geometry
948:manifold
917:Manifold
889:are the
829:. In a
511:infinity
488:) where
356:) and (â
106:such as
104:manifold
96:position
80:geometry
2223:Physics
2057:Related
1820:Physics
1738:Tensors
1633:Sources
1415:3017486
842:circles
720:,
712:,
681:duality
657:spheres
653:circles
484:,
480:,
454:,
450:,
442:,
434:,
426:,
422:,
360:,
348:,
340:,
332:,
98:of the
88:numbers
40:physics
2151:Vector
2146:Spinor
2131:Matrix
1925:Tensor
1683:
1618:
1585:
1561:
1536:
1502:
1453:
1443:
1413:
1405:
1395:
1359:
1330:
1305:
1274:
1015:, the
980:matrix
259:, two
177:origin
100:points
2071:Basis
1756:Scope
944:atlas
925:, or
257:plane
175:(the
112:tuple
90:, or
52:theta
1681:ISBN
1616:ISBN
1583:ISBN
1559:ISBN
1534:ISBN
1500:ISBN
1451:LCCN
1441:ISBN
1411:OCLC
1403:LCCN
1393:ISBN
1357:ISBN
1328:ISBN
1303:ISBN
1272:ISBN
1023:and
966:and
895:â 1)
851:are
740:and
638:The
627:The
619:and
553:The
496:and
414:and
406:, a
388:and
344:), (
306:pole
228:The
163:line
82:, a
34:The
1389:178
1019:of
962:In
837:.
655:or
458:).
187:to
122:in
78:In
68:rho
60:phi
2369::
1650:,
1644:,
1610:.
1602:;
1498:.
1494:.
1486:;
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1431:;
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1401:.
1391:.
1297:.
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1216:.
882:.
855:.
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772:1D
752:.
684:.
544::
534::
376:.
352:+2
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1730:e
1723:t
1716:v
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1589:.
1567:.
1542:.
1508:.
1457:.
1417:.
1365:.
1336:.
1311:.
1280:.
1247:.
1222:.
940:R
936:X
899:n
893:n
891:(
872:Ď
795:.
788:.
750:θ
746:r
742:y
738:θ
734:r
730:x
726:x
722:θ
718:r
714:y
710:x
635:.
601:.
567:.
502:z
500:/
498:y
494:z
492:/
490:x
486:z
482:y
478:x
456:Ď
452:θ
448:Ď
444:Ď
440:Ď
436:z
432:r
428:z
424:θ
420:r
416:θ
412:r
408:z
374:θ
370:θ
366:Ď
364:+
362:θ
358:r
354:Ď
350:θ
346:r
342:θ
338:r
334:θ
330:r
326:r
322:r
318:θ
314:θ
273:n
269:n
193:P
189:P
185:O
181:P
173:O
116:x
74:.
72:r
66:(
64:Ď
58:(
56:Ď
50:(
48:θ
44:r
27:.
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