geodesic surface of revolution
/Length 10 CWk��H���R�(�^M��g��yX/��I`����b���R�1< >> Geodesics on surfaces of revolution 6 References 8 6. endobj endstream 1442 0 obj endstream endstream endstream stream endobj 8 0 obj 147 0 obj 1472 0 obj Examples, cont. <>1371 0 R]/P 1609 0 R/Pg 1606 0 R/S/InternalLink>> << 24 0 obj <>234 0 R]/P 1554 0 R/Pg 1553 0 R/S/InternalLink>> 6.10 Geodesics and Plate Development. endobj << /F1 2 0 R /Length 49 endobj 1487 0 obj endobj endstream /Filter /FlateDecode <> <>239 0 R]/P 1564 0 R/Pg 1553 0 R/S/InternalLink>> << endstream endobj <>219 0 R]/P 1530 0 R/Pg 1491 0 R/S/InternalLink>> endobj /Filter /FlateDecode of its geodesic lines. 8����f"� << 1464 0 obj <>885 0 R]/P 1597 0 R/Pg 1588 0 R/S/InternalLink>> <>210 0 R]/P 1502 0 R/Pg 1491 0 R/S/InternalLink>> endobj 9 0 obj The Direct and Inverse problems of the geodesic on an ellipsoidIn geodesy, the geodesic is a unique curve on the surface of an ellipsoid defining the shortest distance between two points. /Filter /FlateDecode <>200 0 R]/P 1526 0 R/Pg 1491 0 R/S/InternalLink>> /Font /Filter /FlateDecode 1458 0 obj V>1. endobj 1480 0 obj B���?G������~�Â�]9���K�X�`�pKe����,Ⲱ����;����vN��Fwǒ�sJ@ ��L��ӊ:��i��1&�|���yV2�H�51��J��b��Y`s����k�p�O�u�� endobj Like ellipses these … /Length 10 Like the sphere, a toroidal surface can have closed geodesics, but they are special cases. endstream <>240 0 R]/P 1570 0 R/Pg 1553 0 R/S/InternalLink>> /Length 10 endobj endobj stream >> endobj R(I
�7$� >> << >> <>226 0 R]/P 1551 0 R/Pg 1542 0 R/S/InternalLink>> <>236 0 R]/P 1566 0 R/Pg 1553 0 R/S/InternalLink>> /Length 10 endobj <>238 0 R]/P 1568 0 R/Pg 1553 0 R/S/InternalLink>> endobj 1435 0 obj For these pictures, 10'000 geodesics have been started from one point and integrated until time 10. 2020-06-03T12:29:44-07:00 >> <>1104 0 R]/P 1604 0 R/Pg 1599 0 R/S/InternalLink>> <> <>881 0 R]/P 1591 0 R/Pg 1588 0 R/S/InternalLink>> 1456 0 obj /Encoding /WinAnsiEncoding <>201 0 R]/P 1520 0 R/Pg 1491 0 R/S/InternalLink>> 1449 0 obj stream endobj >> <>1101 0 R]/P 1600 0 R/Pg 1599 0 R/S/InternalLink>> endstream <>211 0 R]/P 1506 0 R/Pg 1491 0 R/S/InternalLink>> 1474 0 obj <>218 0 R]/P 1532 0 R/Pg 1491 0 R/S/InternalLink>> We explore the n-body problem, n ≥ 3, on a surface of revolution with a general interaction depending on the pairwise geodesic distance.Using the geometric methods of classical mechanics we determine a large set of properties. Geodesics We will give de nitions of geodesics in terms of length minimising curves, in terms of the geodesic curvature vanishing, in terms of the covariant derivative of vector elds, and in terms of a set of equations. >> /Filter /FlateDecode endstream PLANE MODEL. �hQ�9���� <> >> A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. 25 0 obj 1455 0 obj �f�����Ԓ�p�ܠ�I�m�,M�I�:��. endobj 3 0 obj endobj Proposition /Filter /FlateDecode Since it is a complete negatively curved surface, there is exactly one geodesic connecting any two points. endobj <>882 0 R]/P 1593 0 R/Pg 1588 0 R/S/InternalLink>> 1450 0 obj -P˃��H'��d�/���lP8}o,U+륚N�iGx��:�\euR|Bv� endobj << Denition 1.1 (Surface of Revolution). ˑ 1 0 obj 21 0 obj <>880 0 R]/P 1589 0 R/Pg 1588 0 R/S/InternalLink>> ���Vx�jW��L��-n�� Geodesics on such a surface of rotation have a simple general structure. <>stream
1467 0 obj <>214 0 R]/P 1536 0 R/Pg 1491 0 R/S/InternalLink>> integral. endstream >> 16 0 obj In the case of a Riemannian surface of revolution, one can study the behaviour of geodesic by using Clairaut relation, we can see that if the geodesic is neither a profile curve nor s parallel then it will be tangent to the some parallel. /Filter /FlateDecode The Clairaut parameterization of a torus treats it as a surface of revolution. << <>208 0 R]/P 1504 0 R/Pg 1491 0 R/S/InternalLink>> endstream << 1479 0 obj endobj << <>366 0 R]/P 1575 0 R/Pg 1572 0 R/S/InternalLink>> /Filter /FlateDecode <>213 0 R]/P 1494 0 R/Pg 1491 0 R/S/InternalLink>> 1484 0 obj 1454 0 obj ���l���"q endobj <>241 0 R]/P 1558 0 R/Pg 1553 0 R/S/InternalLink>> <>227 0 R]/P 1549 0 R/Pg 1542 0 R/S/InternalLink>> |ˉ��I�$��*�}d�V�[wˍn(�;�#N�ћi��Ě�6�8'�B�r stream endobj As Luther Eisenhart remarks, 2 Òthe geo desics up on a surface of rev olution referred to its meridians and parallels can b e found b y quadrature.Ó 3 There is, ho w ever, no guaran tee that the integral (6) is tractable = describable in terms of named functions, and in the case of the hexenh ut w e will Þnd that it is not. endstream 1433 0 obj W rite endobj stream /Filter /FlateDecode endobj For further reading we send the reader to the wide literature on Riemannian and Finsler geometry and topology, in particular the geodesic research. /BBox [0 0 504 720] Geodesics are curves on the surface which satisfy a certain second-order ordinary differential equation which is specified by the first fundamental form. endobj endobj %PDF-1.7
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One is visible with the default settings: experiment a bit to find others. 1438 0 obj The geodesic is drawn by the line in the middle of the rectangle when you can flat at most the rectangle on the surface. >> The curve (circle) generated by rotating the point given by g(u)=0, i.e., z =0, is a geodesic, which we call the equator.Ameridian isacurveu1 =constant. <>233 0 R]/P 1556 0 R/Pg 1553 0 R/S/InternalLink>> endobj <>1368 0 R]/P 1607 0 R/Pg 1606 0 R/S/InternalLink>> x��. 1611 0 obj 1445 0 obj endobj <>/Metadata 2 0 R/Outlines 5 0 R/Pages 3 0 R/StructTreeRoot 6 0 R/Type/Catalog/ViewerPreferences<>>> endobj 1477 0 obj endstream /Matrix [1 0 0 1 0 0] <>202 0 R]/P 1510 0 R/Pg 1491 0 R/S/InternalLink>> endobj /Length 10 << /Length 48 /Length 48 <> 15 0 obj >> endobj 1614 0 obj (But I could easily have made a mistake in the calculation anyway.) stream endobj /Resources 1462 0 obj 1478 0 obj - a geodesic of a surface is planar if and only if it is a curvature line. 1470 0 obj /Length 10 1483 0 obj >> endobj 1448 0 obj 1452 0 obj Any surface of revolution in $3$-space with poles will have this property. /Filter /FlateDecode The relation remains valid for a geodesic on an arbitrary surface of revolution. ) (d) Conversely, show that if Clairaut's relation is satisfied along a curve a : 1 + S on a surface of revolution, and there is no non-empty open interval J CI such that a(J) is contained in a parallel, then a is a geodesic. %PDF-1.4 1440 0 obj endobj Wenli Chang stream - the meridians of a surface of revolution are geodesics (but not the parallels, except those with extreme radius). endobj <>203 0 R]/P 1516 0 R/Pg 1491 0 R/S/InternalLink>> 1468 0 obj endstream 1437 0 obj endobj 1460 0 obj 12 0 obj 1446 0 obj endobj <>223 0 R]/P 1512 0 R/Pg 1491 0 R/S/InternalLink>>
Ʀ�=�w����WRt��ST�&�m��D����e���oQ%Q�E /Filter /FlateDecode /Name /F1 1441 0 obj /Filter /FlateDecode "surface of revolution" 어떻게 사용되는 지 Cambridge Dictionary Labs에 예문이 있습니다 endobj /Filter /FlateDecode /Length 48 /Filter /FlateDecode uuid:6197c564-ae8a-11b2-0a00-f0cf7d020000 1451 0 obj 1469 0 obj 1482 0 obj << stream endobj >> <>205 0 R]/P 1500 0 R/Pg 1491 0 R/S/InternalLink>> Always the first point was marked, where the Jacobi field is zero. stream The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks.The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere.A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. endobj <> /Filter /FlateDecode <>431 0 R]/P 1584 0 R/Pg 1581 0 R/S/InternalLink>> << /Filter /FlateDecode /Type /Font endobj endstream /Subtype /Type1 5 0 obj spherical 2-orbifold of revolution is a closed tw o-dimensional surface of revolution homeomorphic to S 2 that satisfies a certain special orbifold condition at its north and south poles. >> >> The codimension 1 coincides with the fact that the geodesic is of dimension 1. 1463 0 obj stream application/pdf << 20 0 obj /Filter /FlateDecode /Length 48 endobj The surface of revolution as the Earth’s model – sphere S2 or the spheroid is locally approximated by the Euclidean plane tangent in … The geodesic equations 3 6.6. <>222 0 R]/P 1528 0 R/Pg 1491 0 R/S/InternalLink>> To send this article to your Kindle, first ensure no-reply@cambridge. <>217 0 R]/P 1534 0 R/Pg 1491 0 R/S/InternalLink>> 1489 0 obj <>369 0 R]/P 1579 0 R/Pg 1572 0 R/S/InternalLink>> endobj /BaseFont /Helvetica the Randers metric as an examples for the Finsler case. � /Length 48 >> /Length 49 1465 0 obj For example, the geodesics of a sphere are its great circles. this project, I focus on the study of geodesics on a surface of revolution. endobj Then every u-parameter curve is a geodesic and a v-parameter curve with u = u 0 is a geodesic precisely when G u(u 0) = 0. Note in the figure above the difference in slant of the geodesic … ��()�휧�.>,�]���Df�KצԄ 6 0 obj endobj stream <>1375 0 R]/P 1611 0 R/Pg 1606 0 R/S/InternalLink>> endobj 6 0 obj <>1102 0 R]/P 1602 0 R/Pg 1599 0 R/S/InternalLink>> trajectories including geodesic, non-geodesic, constant winding angle and a combination of these trajectories have been generated for a conical shape. A theorem on geodesics of a surface of revolution is proved in chapter 8. 2020-06-03T12:29:44-07:00 The Geodesic Equation. <>229 0 R]/P 1545 0 R/Pg 1542 0 R/S/InternalLink>> endobj endobj 1606 0 obj A similar result holds for three dimensional Minkowski space for time-like geodesics on surfaces of revolution about the time axis. uuid:6197c565-ae8a-11b2-0a00-00b5668fff7f 6.5. endobj /Length 48 endobj <>224 0 R]/P 1514 0 R/Pg 1491 0 R/S/InternalLink>> endobj /Length 126 endobj endobj <>212 0 R]/P 1492 0 R/Pg 1491 0 R/S/InternalLink>> endobj Given a surface S and two points on it, the shortest path on S that connects them is along a geodesic of S.However, the definition of a geodesic as the line of shortest distance on a surface causes some difficulties. << 1434 0 obj It comes from the fact that by using a rectangle and flatten at most both long edges, you induce a Killing field. ���g7�n9c The lower bound on the arc length of the geodesic connecting S(pi) and S(pi+2) where S is a surface is the Euclidean distance kS(pi) − S(pi+2)k. Assuming that this path must also contain pi+1, the lower bound becomes LB(pi+1) where LB(x) = kS(pi)−S(x)k+kS(x)−S(pi+2)k. If the surface S is locally planar, and the points in the sequence are << endobj <> << 1485 0 obj 1453 0 obj 10 0 obj endstream <>204 0 R]/P 1518 0 R/Pg 1491 0 R/S/InternalLink>> stream >> ClairautÕ s Theorem . <>206 0 R]/P 1496 0 R/Pg 1491 0 R/S/InternalLink>> 1476 0 obj endobj 1443 0 obj In attempting some work on geodesics on a spheroid, I was led to work out the geodesic on a sphere, and it may be interesting to see how the usual Spherical Trigonometry results arise from the general equation of a geodesic on a surface of revolution. << endobj 1471 0 obj 1486 0 obj 1. 1459 0 obj <>216 0 R]/P 1538 0 R/Pg 1491 0 R/S/InternalLink>> <> /Filter /FlateDecode <>435 0 R]/P 1586 0 R/Pg 1581 0 R/S/InternalLink>> Geodesics on a torus of revolution. Length minimising curves 4 6.7. /Subtype /Form endobj 146 0 obj (e) The pseudosphere is the surface of revolution parametrized by x(u, v) = 111 - cos u, -sinu, 11- - coshul, UER. stream /Type /XObject stream /Filter /FlateDecode The reason is that, in this case, any geodesic either goes through a pole (i.e., a point where the axis of revolution meets the surface) and is a profile curve that lies in a plane or else, because of the Clairaut integral, it avoids that pole by some positive distance. endobj << �������; �6��s�ѐ��$ /Length 49 endobj <>207 0 R]/P 1508 0 R/Pg 1491 0 R/S/InternalLink>> A surface of revolution is a surface created by rotating a plane curve in a circle. Since a geodesic can pass through any point on the surface, we call these unbounded geodesics. >> << 1473 0 obj �
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�A���a�^:b�N 1444 0 obj A formal mathematical statement of Clairaut's relation is: Let γ be a geodesic on a surface of revolution S, let ρ be the distance of a point of S from the axis of rotation, and let ψ be the angle between γ and the meridians of S. 5 0 obj In Euclidean space, the geodesics on a surface of revolution can be characterized by mean of Clairauts theorem, which essentially says that the geodesics are curves of fixed angular momentum. >> endobj <>230 0 R]/P 1543 0 R/Pg 1542 0 R/S/InternalLink>> endstream endobj << <>371 0 R]/P 1577 0 R/Pg 1572 0 R/S/InternalLink>> << endobj /Filter /FlateDecode The primary caustic can already be complicated for a rotationally symmetric torus of revolution. ]�. �^�>�#��� <> - the straight lines of a surface are geodesics (and they are the only one to be geodesics and asymptotic lines). stream <>434 0 R]/P 1582 0 R/Pg 1581 0 R/S/InternalLink>> 13 0 obj If we write the torus as part of the plane with a space dependent metric which depends only on one coordinate, we have a geodesic flow on a surface of revolution. 1.1 Surfaces of Revolution Since our goal is to create a tube and a tube is a surface of revolution, we start by dening and exploring surfaces of revolution. stream stream An admissible surface 5 is formed by revolving about Oy a curve which rises monotonically from the origin to infinity as x increases, and which possesses a continuously turning tangent (save possibly at certain exceptional points). <> ��T�����
_���[HJ�%��Ph-�+>$�H�hc� >> Nw|��� endstream endobj <> 1466 0 obj There are directions, in which the geodesic winds around the torus several times before the Jacobi field reaches a … "E�$,[2 ���v�p Z�8�*�2:L << << endobj endobj Examples of how to use “surface of revolution” in a sentence from the Cambridge Dictionary Labs 7 0 obj stream I first introduce some of the key concepts in differential geometry in the first 6 chapters. endstream endobj 1447 0 obj 2020-06-03T12:29:44-07:00 /Length 10 )�v���I��c endobj /Filter /FlateDecode Appligent AppendPDF Pro 6.3 several times before the Jacobi field reaches a zero. It is standard differential geometry to find the differential equation for the geodesics on this surface. 1457 0 obj endobj <>228 0 R]/P 1547 0 R/Pg 1542 0 R/S/InternalLink>> 14 0 obj Geodesics of surface of revolution 18 0 obj endobj Theorem 5.2 Let Mbe a surface with a u-Clairaut patch x(u,v). /Length 10 /FormType 1 A surface similar to an ellipsoid can be generated by revolution of the ovals of Cassini. %���� endobj 1439 0 obj N7�|4���s� The meridians of a surface of revolution are geodesics. A geodesic will cut meridians of an ellipsoid at angles α , known as azimuths and measured clockwise from north 0º to . /Length 10 1461 0 obj In chapter 7, I derive the differential equations for a curve being a geodesic. 2 0 obj stream endobj >> <>220 0 R]/P 1524 0 R/Pg 1491 0 R/S/InternalLink>>