/Subtype /Type1 1439 0 obj endobj << endstream /Length 49 1451 0 obj <>233 0 R]/P 1556 0 R/Pg 1553 0 R/S/InternalLink>> 1467 0 obj A surface similar to an ellipsoid can be generated by revolution of the ovals of Cassini. endobj <>217 0 R]/P 1534 0 R/Pg 1491 0 R/S/InternalLink>> <> << stream �f�����Ԓ�p�ܠ�I�m�,M�I�:��. 1606 0 obj 1450 0 obj >> endobj 24 0 obj endobj <>/Metadata 2 0 R/Outlines 5 0 R/Pages 3 0 R/StructTreeRoot 6 0 R/Type/Catalog/ViewerPreferences<>>> 1475 0 obj >> /Filter /FlateDecode /Length 10 endobj endobj 1480 0 obj <> >> /Length 10 Examples of surfaces of revolution are the cylinder, the cone or the torus. 1452 0 obj ��Y�շ�H7#�f�-�z�2�s� >> 1456 0 obj 1458 0 obj <>238 0 R]/P 1568 0 R/Pg 1553 0 R/S/InternalLink>> /F1 2 0 R << endstream Send article to Kindle. <>371 0 R]/P 1577 0 R/Pg 1572 0 R/S/InternalLink>> Like the sphere, a toroidal surface can have closed geodesics, but they are special cases. <> endobj /Filter /FlateDecode endobj 2020-06-03T12:29:44-07:00 1482 0 obj endstream /Filter /FlateDecode 2020-06-03T12:29:44-07:00 20 0 obj endobj 1448 0 obj /BaseFont /Helvetica R(I �7$� 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). <>202 0 R]/P 1510 0 R/Pg 1491 0 R/S/InternalLink>> 1446 0 obj The geodesic curve connecting two points on a surface of revolution as a boundary value problem (BVP) can be solved through the Euler–Lagrange (EL) equations [1]. stream 6 0 obj <>228 0 R]/P 1547 0 R/Pg 1542 0 R/S/InternalLink>> 17 0 obj >> /Length 48 >> >> /Name /F1 PLANE MODEL. >> 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. 1473 0 obj 1478 0 obj 1 0 obj 1445 0 obj CWk��H���R�(�^M��g��yX/��I`����b���R�1< endobj 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. 3 0 obj Appligent AppendPDF Pro 6.3 A parallel is a geodesic if and only if its tangent vector is parallel to the z-axis. 21 0 obj trajectories including geodesic, non-geodesic, constant winding angle and a combination of these trajectories have been generated for a conical shape. <>366 0 R]/P 1575 0 R/Pg 1572 0 R/S/InternalLink>> endstream <>885 0 R]/P 1597 0 R/Pg 1588 0 R/S/InternalLink>> <>435 0 R]/P 1586 0 R/Pg 1581 0 R/S/InternalLink>> 9 0 obj endobj 148 0 obj endobj endobj endstream endobj Examples, cont. (e) The pseudosphere is the surface of revolution parametrized by x(u, v) = 111 - cos u, -sinu, 11- - coshul, UER. ��()�휧�.>,�]���Df�KצԄ endobj endobj <>208 0 R]/P 1504 0 R/Pg 1491 0 R/S/InternalLink>> << <>219 0 R]/P 1530 0 R/Pg 1491 0 R/S/InternalLink>> << V>1. >> endobj 1447 0 obj >> stream 1488 0 obj Primary caustic computation on a surface of revolution r = exp(-z^2). 1614 0 obj /Length 10 >> endstream <>205 0 R]/P 1500 0 R/Pg 1491 0 R/S/InternalLink>> 1471 0 obj endobj The stream <> 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. << A geodesic will cut meridians of an ellipsoid at angles α , known as azimuths and measured clockwise from north 0º to . A geodesic starting in a certain direction from a given point on the surface is an initial value problem (IVP) and can be solved through the canonical geodesic (CG) equations [2]. 16 0 obj <>1368 0 R]/P 1607 0 R/Pg 1606 0 R/S/InternalLink>> The relation remains valid for a geodesic on an arbitrary surface of revolution. << << endobj 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. stream 18 0 obj � qrH�G�v��V���PE�*�4|����cF �A���a�^:b�N /Filter /FlateDecode 2 0 obj uuid:6197c565-ae8a-11b2-0a00-00b5668fff7f /Filter /FlateDecode << <>431 0 R]/P 1584 0 R/Pg 1581 0 R/S/InternalLink>> /Filter /FlateDecode endobj %���� <>223 0 R]/P 1512 0 R/Pg 1491 0 R/S/InternalLink>> <> uuid:6197c564-ae8a-11b2-0a00-f0cf7d020000 /Length 49 of its geodesic lines. >> 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. <> )�v���I��c The geodesic is drawn by the line in the middle of the rectangle when you can flat at most the rectangle on the surface. There are directions, in which the geodesic winds around the torus several times before the Jacobi field reaches a … endobj - a geodesic of a surface is planar if and only if it is a curvature line. In chapter 7, I derive the differential equations for a curve being a geodesic. /Filter /FlateDecode The Geodesic Equation. endstream endobj <> Geodesics of surface of revolution 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 Since it is a complete negatively curved surface, there is exactly one geodesic connecting any two points. Mathematical formulation A general surface of revolution in a polar coordinate system with parameters ( , ) … endobj <>881 0 R]/P 1591 0 R/Pg 1588 0 R/S/InternalLink>> endobj <>206 0 R]/P 1496 0 R/Pg 1491 0 R/S/InternalLink>> stream 1485 0 obj 1481 0 obj Examples of how to use “surface of revolution” in a sentence from the Cambridge Dictionary Labs 1487 0 obj ���Vx�jW��L��-n�� <>209 0 R]/P 1498 0 R/Pg 1491 0 R/S/InternalLink>> endstream endstream /Font endobj endobj The meridians of a surface of revolution are geodesics. /Length 10 - the straight lines of a surface are geodesics (and they are the only one to be geodesics and asymptotic lines). <>stream 1469 0 obj >> /Length 126 << 10 0 obj endobj It is standard differential geometry to find the differential equation for the geodesics on this surface. /Filter /FlateDecode stream <>369 0 R]/P 1579 0 R/Pg 1572 0 R/S/InternalLink>> Ʀ�=�w����WRt��ST�&�m��D����e���oQ%Q�E /ProcSet [/PDF /Text] 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. <>229 0 R]/P 1545 0 R/Pg 1542 0 R/S/InternalLink>> 8 0 obj /Filter /FlateDecode >> stream Any surface of revolution in $3$-space with poles will have this property. /Filter /FlateDecode 1435 0 obj endobj 1474 0 obj <>221 0 R]/P 1522 0 R/Pg 1491 0 R/S/InternalLink>> <> 1 0 obj /Length 10 4 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�� 6 0 obj << /Filter /FlateDecode ) (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. << endobj <>1101 0 R]/P 1600 0 R/Pg 1599 0 R/S/InternalLink>> <>1104 0 R]/P 1604 0 R/Pg 1599 0 R/S/InternalLink>> 1476 0 obj << AppendPDF Pro 6.3 Linux 64 bit Aug 30 2019 Library 15.0.4 On every geodesic of 5 … endobj endobj %PDF-1.4 endobj <> 1464 0 obj << endstream endobj Geodesics on such a surface of rotation have a simple general structure. <>241 0 R]/P 1558 0 R/Pg 1553 0 R/S/InternalLink>> 1477 0 obj endobj 1479 0 obj endobj The Clairaut parameterization of a torus treats it as a surface of revolution. endobj The geodesic curvature of a plane curve on the xy-plane is the signed curvature of the curve. <> W rite /Resources 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. Length minimising curves 4 6.7. ClairautÕ s Theorem . endobj <>234 0 R]/P 1554 0 R/Pg 1553 0 R/S/InternalLink>> endstream endobj 12 0 obj 2 0 obj 22 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. N7�|4���s� /Length 10 <>204 0 R]/P 1518 0 R/Pg 1491 0 R/S/InternalLink>> << "E�$,[2 ���v�p endobj 6.10 Geodesics and Plate Development. << Theorem 5.2 Let Mbe a surface with a u-Clairaut patch x(u,v). endobj endobj endobj 1434 0 obj Since a geodesic can pass through any point on the surface, we call these unbounded geodesics. 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. >> 1461 0 obj endobj endobj Geodesics on a torus of revolution. << 1449 0 obj >> 147 0 obj -P˃��H'��d�/���lP8}o,U+륚N�iGx��:�\euR|Bv� /Length 48 1470 0 obj ���g7�n9c Z�8�*�2:L endobj I first introduce some of the key concepts in differential geometry in the first 6 chapters. endobj << Proposition 1442 0 obj Wenli Chang stream endstream /Filter /FlateDecode stream <>stream 3 0 obj 1468 0 obj It comes from the fact that by using a rectangle and flatten at most both long edges, you induce a Killing field. /Filter /FlateDecode /Length 48 /Filter /FlateDecode 7 0 obj <>200 0 R]/P 1526 0 R/Pg 1491 0 R/S/InternalLink>>
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