geodesic surface of revolution
14 0 obj 2020-06-03T12:29:44-07:00 endobj <> /Filter /FlateDecode << /BaseFont /Helvetica �������; �6��s�ѐ��$ 148 0 obj The geodesic curvature of a plane curve on the xy-plane is the signed curvature of the curve. It comes from the fact that by using a rectangle and flatten at most both long edges, you induce a Killing field. A surface similar to an ellipsoid can be generated by revolution of the ovals of Cassini. stream �^�>�#��� 19 0 obj <>228 0 R]/P 1547 0 R/Pg 1542 0 R/S/InternalLink>> 15 0 obj 146 0 obj 2020-06-03T12:29:44-07:00 /Length 48 <>369 0 R]/P 1579 0 R/Pg 1572 0 R/S/InternalLink>> << 1464 0 obj �
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�A���a�^:b�N The geodesic equations 3 6.6. The meridians of a surface of revolution are geodesics. /Filter /FlateDecode <> 11 0 obj <>434 0 R]/P 1582 0 R/Pg 1581 0 R/S/InternalLink>> endobj endobj (e) The pseudosphere is the surface of revolution parametrized by x(u, v) = 111 - cos u, -sinu, 11- - coshul, UER. 1486 0 obj 1446 0 obj <>211 0 R]/P 1506 0 R/Pg 1491 0 R/S/InternalLink>> /Length 10 �y�[:
�: 1448 0 obj <>216 0 R]/P 1538 0 R/Pg 1491 0 R/S/InternalLink>> 10 0 obj endobj endobj /BBox [0 0 504 720] 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. << 7 0 obj <>239 0 R]/P 1564 0 R/Pg 1553 0 R/S/InternalLink>> /Length 10 ˑ /Length 10 Z�8�*�2:L endobj endobj << <>217 0 R]/P 1534 0 R/Pg 1491 0 R/S/InternalLink>> endobj endobj stream For further reading we send the reader to the wide literature on Riemannian and Finsler geometry and topology, in particular the geodesic research. 1436 0 obj <> stream <>237 0 R]/P 1560 0 R/Pg 1553 0 R/S/InternalLink>> endobj >> endobj /Type /Font 1480 0 obj For these pictures, 10'000 geodesics have been started from one point and integrated until time 10. endobj >> <>220 0 R]/P 1524 0 R/Pg 1491 0 R/S/InternalLink>> 1481 0 obj The primary caustic can already be complicated for a rotationally symmetric torus of revolution. V>1. <>201 0 R]/P 1520 0 R/Pg 1491 0 R/S/InternalLink>> 1473 0 obj >> Denition 1.1 (Surface of Revolution). application/pdf -P˃��H'��d�/���lP8}o,U+륚N�iGx��:�\euR|Bv� - a geodesic of a surface is planar if and only if it is a curvature line. 1443 0 obj endstream Like the sphere, a toroidal surface can have closed geodesics, but they are special cases. stream endobj R(I
�7$� <>221 0 R]/P 1522 0 R/Pg 1491 0 R/S/InternalLink>> Examples of how to use “surface of revolution” in a sentence from the Cambridge Dictionary Labs An admissible surface 5 is formed by revolving about Oy a curve which rises monotonically from the origin to infinity as x increases, and which possesses a continuously turning tangent (save possibly at certain exceptional points). The relation remains valid for a geodesic on an arbitrary surface of revolution. /Filter /FlateDecode /Filter /FlateDecode endobj <>206 0 R]/P 1496 0 R/Pg 1491 0 R/S/InternalLink>> 1483 0 obj 8����f"� The Clairaut parameterization of a torus treats it as a surface of revolution. endobj << |ˉ��I�$��*�}d�V�[wˍn(�;�#N�ћi��Ě�6�8'�B�r endobj 1451 0 obj <>203 0 R]/P 1516 0 R/Pg 1491 0 R/S/InternalLink>> Geodesics on such a surface of rotation have a simple general structure. 20 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. <>364 0 R]/P 1573 0 R/Pg 1572 0 R/S/InternalLink>> ���Vx�jW��L��-n�� << stream For example, the geodesics of a sphere are its great circles. /Length 10 21 0 obj 147 0 obj << >> 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]. The Clairot integral rsin(φ) is the analogue of Snells integral g(x)sin(α) we have seen before. >> endobj <>233 0 R]/P 1556 0 R/Pg 1553 0 R/S/InternalLink>> 1467 0 obj 1478 0 obj 1. /Length 48 A parallel is a geodesic if and only if its tangent vector is parallel to the z-axis. 1452 0 obj /Filter /FlateDecode <> The geodesic is drawn by the line in the middle of the rectangle when you can flat at most the rectangle on the surface. endobj A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. 6 0 obj 1476 0 obj endstream <>223 0 R]/P 1512 0 R/Pg 1491 0 R/S/InternalLink>> Since a geodesic can pass through any point on the surface, we call these unbounded geodesics. endobj endobj >> /Filter /FlateDecode <>224 0 R]/P 1514 0 R/Pg 1491 0 R/S/InternalLink>> endstream <>240 0 R]/P 1570 0 R/Pg 1553 0 R/S/InternalLink>> <>213 0 R]/P 1494 0 R/Pg 1491 0 R/S/InternalLink>> � stream ���g7�n9c >> <>882 0 R]/P 1593 0 R/Pg 1588 0 R/S/InternalLink>> 18 0 obj endstream endobj endobj the Randers metric as an examples for the Finsler case. endstream <>1101 0 R]/P 1600 0 R/Pg 1599 0 R/S/InternalLink>> On every geodesic of 5 … stream Since it is a complete negatively curved surface, there is exactly one geodesic connecting any two points. 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. << endobj endobj endobj /Matrix [1 0 0 1 0 0] <>238 0 R]/P 1568 0 R/Pg 1553 0 R/S/InternalLink>> endobj <>371 0 R]/P 1577 0 R/Pg 1572 0 R/S/InternalLink>> <>207 0 R]/P 1508 0 R/Pg 1491 0 R/S/InternalLink>> <>/Metadata 2 0 R/Outlines 5 0 R/Pages 3 0 R/StructTreeRoot 6 0 R/Type/Catalog/ViewerPreferences<>>> endobj endobj endobj 1606 0 obj endobj /Filter /FlateDecode <>229 0 R]/P 1545 0 R/Pg 1542 0 R/S/InternalLink>> /Filter /FlateDecode x��. >> endobj In chapter 7, I derive the differential equations for a curve being a geodesic. 1487 0 obj <> 1437 0 obj - the meridians of a surface of revolution are geodesics (but not the parallels, except those with extreme radius). <>227 0 R]/P 1549 0 R/Pg 1542 0 R/S/InternalLink>> ClairautÕ s Theorem . 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. Nw|��� endobj 1460 0 obj << 1459 0 obj Length minimising curves 4 6.7. %PDF-1.7
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endobj /Filter /FlateDecode 1457 0 obj /Filter /FlateDecode Send article to Kindle. A surface of revolution is a surface created by rotating a plane curve in a circle. 1614 0 obj A similar result holds for three dimensional Minkowski space for time-like geodesics on surfaces of revolution about the time axis. >> /Subtype /Form ]�. << ��T�����
_���[HJ�%��Ph-�+>$�H�hc� /Length 48 << 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. endobj of its geodesic lines. 3 0 obj <> integral. 1 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�� <>208 0 R]/P 1504 0 R/Pg 1491 0 R/S/InternalLink>> endobj stream 5 0 obj >> endobj << endobj 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. >> endstream 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. One is visible with the default settings: experiment a bit to find others. << endobj <>218 0 R]/P 1532 0 R/Pg 1491 0 R/S/InternalLink>> AppendPDF Pro 6.3 Linux 64 bit Aug 30 2019 Library 15.0.4 "surface of revolution" 어떻게 사용되는 지 Cambridge Dictionary Labs에 예문이 있습니다 <>200 0 R]/P 1526 0 R/Pg 1491 0 R/S/InternalLink>> <>236 0 R]/P 1566 0 R/Pg 1553 0 R/S/InternalLink>> >> Examples of surfaces of revolution are the cylinder, the cone or the torus. 1 endobj 8 0 obj The codimension 1 coincides with the fact that the geodesic is of dimension 1. 1463 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. endobj stream endobj A geodesic will cut meridians of an ellipsoid at angles α , known as azimuths and measured clockwise from north 0º to . <>883 0 R]/P 1595 0 R/Pg 1588 0 R/S/InternalLink>> << endobj 17 0 obj Theorem 5.2 Let Mbe a surface with a u-Clairaut patch x(u,v). <>1368 0 R]/P 1607 0 R/Pg 1606 0 R/S/InternalLink>> /Length 10 /Filter /FlateDecode /Length 10 /Resources The Geodesic Equation. 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. 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 <>215 0 R]/P 1540 0 R/Pg 1491 0 R/S/InternalLink>> <> 1475 0 obj <>885 0 R]/P 1597 0 R/Pg 1588 0 R/S/InternalLink>> /Length 48 endobj Geodesics of surface of revolution <>205 0 R]/P 1500 0 R/Pg 1491 0 R/S/InternalLink>> /Subtype /Type1 /Length 10 endobj <>241 0 R]/P 1558 0 R/Pg 1553 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. 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 1472 0 obj endobj endstream >> endobj endstream <>226 0 R]/P 1551 0 R/Pg 1542 0 R/S/InternalLink>> /Encoding /WinAnsiEncoding endobj <> 1439 0 obj Like ellipses these … endobj 6 0 obj 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. <>214 0 R]/P 1536 0 R/Pg 1491 0 R/S/InternalLink>> To send this article to your Kindle, first ensure no-reply@cambridge. 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]. 5 0 obj Geodesics are curves on the surface which satisfy a certain second-order ordinary differential equation which is specified by the first fundamental form. endobj << 2020-06-03T12:29:44-07:00 endobj endobj 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. 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. 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. The surface of revolution as the Earth’s model – sphere S2 or the spheroid is locally approximated by the Euclidean plane tangent in … endobj /Font <>230 0 R]/P 1543 0 R/Pg 1542 0 R/S/InternalLink>> endobj 1433 0 obj /Filter /FlateDecode endobj endobj /Type /XObject endobj endstream this project, I focus on the study of geodesics on a surface of revolution. /Filter /FlateDecode <>1104 0 R]/P 1604 0 R/Pg 1599 0 R/S/InternalLink>> << 1471 0 obj 25 0 obj ��()�휧�.>,�]���Df�KצԄ 1462 0 obj /Length 48 endobj ��Y�շ�H7#�f�-�z�2�s� /F1 2 0 R << 13 0 obj ���l���"q endobj endobj 3 0 obj 1455 0 obj 1466 0 obj /Length 10 endobj >> /Filter /FlateDecode Primary caustic computation on a surface of revolution r = exp(-z^2). It is standard differential geometry to find the differential equation for the geodesics on this surface. N7�|4���s� A theorem on geodesics of a surface of revolution is proved in chapter 8. 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. <>881 0 R]/P 1591 0 R/Pg 1588 0 R/S/InternalLink>> >> uuid:6197c564-ae8a-11b2-0a00-f0cf7d020000 �f�����Ԓ�p�ܠ�I�m�,M�I�:��. /Length 49 Wenli Chang <>235 0 R]/P 1562 0 R/Pg 1553 0 R/S/InternalLink>> << /Length 48 <>222 0 R]/P 1528 0 R/Pg 1491 0 R/S/InternalLink>> endobj stream endobj Examples, cont. /Filter /FlateDecode endobj 1456 0 obj endstream Note in the figure above the difference in slant of the geodesic … endstream 1441 0 obj endobj 6.5. ) (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. PLANE MODEL. Appligent AppendPDF Pro 6.3
Ʀ�=�w����WRt��ST�&�m��D����e���oQ%Q�E 1485 0 obj stream endobj endstream endstream /Length 49 stream 1449 0 obj 2 0 obj endobj Geodesics on a torus of revolution. endobj 1435 0 obj 1458 0 obj /Length 126 endobj /Filter /FlateDecode Any meridian is perpendicular to the equator. In particular, we show that Saari's conjecture fails on surfaces of revolution admitting a geodesic circle. endstream endobj <>431 0 R]/P 1584 0 R/Pg 1581 0 R/S/InternalLink>> Geodesics on surfaces of revolution 6 References 8 6. 1434 0 obj << <>366 0 R]/P 1575 0 R/Pg 1572 0 R/S/InternalLink>> endstream Prince 12.5 (www.princexml.com) /Filter /FlateDecode <>435 0 R]/P 1586 0 R/Pg 1581 0 R/S/InternalLink>> << 4 0 obj )�v���I��c stream 1477 0 obj 1469 0 obj endobj This dynamical system is integrable as in any surface of revolution. 1440 0 obj >> Always the first point was marked, where the Jacobi field is zero. 1438 0 obj several times before the Jacobi field reaches a zero. stream endobj endstream <>210 0 R]/P 1502 0 R/Pg 1491 0 R/S/InternalLink>> Adrian Biran, in Geometry for Naval Architects, 2019. 1468 0 obj "E�$,[2 ���v�p >> >> 2 0 obj W rite <>202 0 R]/P 1510 0 R/Pg 1491 0 R/S/InternalLink>> %���� 1484 0 obj >> 1442 0 obj endobj << 1611 0 obj <>880 0 R]/P 1589 0 R/Pg 1588 0 R/S/InternalLink>> 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. <>stream
1444 0 obj stream endobj endobj stream /Length 10 1489 0 obj I first introduce some of the key concepts in differential geometry in the first 6 chapters. 1447 0 obj 6.10 Geodesics and Plate Development. Proposition 1470 0 obj << endobj stream endobj �hQ�9���� <>212 0 R]/P 1492 0 R/Pg 1491 0 R/S/InternalLink>> 1453 0 obj 1461 0 obj Mathematical formulation A general surface of revolution in a polar coordinate system with parameters ( , ) … Any surface of revolution in $3$-space with poles will have this property. endobj endobj <>stream
1479 0 obj <>219 0 R]/P 1530 0 R/Pg 1491 0 R/S/InternalLink>> There are directions, in which the geodesic winds around the torus several times before the Jacobi field reaches a … stream endobj /Filter /FlateDecode 1454 0 obj /Length 10 9 0 obj /Filter /FlateDecode endobj <>204 0 R]/P 1518 0 R/Pg 1491 0 R/S/InternalLink>> uuid:6197c565-ae8a-11b2-0a00-00b5668fff7f <>1102 0 R]/P 1602 0 R/Pg 1599 0 R/S/InternalLink>> stream <> endobj 12 0 obj