147 0 obj Examples of how to use “surface of revolution” in a sentence from the Cambridge Dictionary Labs uuid:6197c565-ae8a-11b2-0a00-00b5668fff7f 14 0 obj Like the sphere, a toroidal surface can have closed geodesics, but they are special cases. <>205 0 R]/P 1500 0 R/Pg 1491 0 R/S/InternalLink>> On every geodesic of 5 … endobj /Filter /FlateDecode <>224 0 R]/P 1514 0 R/Pg 1491 0 R/S/InternalLink>> >> >> integral. Any surface of revolution in $3$-space with poles will have this property. "surface of revolution" 어떻게 사용되는 지 Cambridge Dictionary Labs에 예문이 있습니다 15 0 obj The primary caustic can already be complicated for a rotationally symmetric torus of revolution. endstream Wenli Chang endobj << "E�$,[2 ���v�p 1443 0 obj << /Length 49 stream 1452 0 obj In the case of a Riemannian surface of revolution, one can study the behaviour of geodesic by using Clairaut relation, we can see that if the geodesic is neither a profile curve nor s parallel then it will be tangent to the some parallel. A formal mathematical statement of Clairaut's relation is: Let γ be a geodesic on a surface of revolution S, let ρ be the distance of a point of S from the axis of rotation, and let ψ be the angle between γ and the meridians of S. A surface of revolution is a surface created by rotating a plane curve in a circle. 20 0 obj <>236 0 R]/P 1566 0 R/Pg 1553 0 R/S/InternalLink>> endobj stream 6.5. ���l���"q The geodesic curve connecting two points on a surface of revolution as a boundary value problem (BVP) can be solved through the Euler–Lagrange (EL) equations [1]. << /Filter /FlateDecode /Name /F1 <> << 1470 0 obj endobj <>211 0 R]/P 1506 0 R/Pg 1491 0 R/S/InternalLink>> endobj stream <>/Metadata 2 0 R/Outlines 5 0 R/Pages 3 0 R/StructTreeRoot 6 0 R/Type/Catalog/ViewerPreferences<>>> stream << 1614 0 obj 1442 0 obj endobj >> endstream |ˉ��I�$��*�}d�V�[wˍn(�;�#N�ћi��Ě�6�8'�B�r R(I �7$� /Type /Font 17 0 obj 2020-06-03T12:29:44-07:00 endobj <>226 0 R]/P 1551 0 R/Pg 1542 0 R/S/InternalLink>> N7�|4���s� /Filter /FlateDecode endobj 25 0 obj 1 0 obj Proposition endobj << >> endobj V>1. >> A parallel is a geodesic if and only if its tangent vector is parallel to the z-axis. the Randers metric as an examples for the Finsler case. ��()�휧�.>,�]���Df�KצԄ /Filter /FlateDecode %PDF-1.7 %���� endobj It comes from the fact that by using a rectangle and flatten at most both long edges, you induce a Killing field. /FormType 1 1445 0 obj Mathematical formulation A general surface of revolution in a polar coordinate system with parameters ( , ) … 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 Examples, cont. <>235 0 R]/P 1562 0 R/Pg 1553 0 R/S/InternalLink>> /Length 48 �y�[: �: To send this article to your Kindle, first ensure no-reply@cambridge. Geodesics of surface of revolution << /Length 10 endobj stream One is visible with the default settings: experiment a bit to find others. 1484 0 obj <>222 0 R]/P 1528 0 R/Pg 1491 0 R/S/InternalLink>> /Length 48 ]�. <>240 0 R]/P 1570 0 R/Pg 1553 0 R/S/InternalLink>> <> 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. 11 0 obj 19 0 obj /Type /XObject 10 0 obj ���g7�n9c 9 0 obj endstream endstream 146 0 obj /Filter /FlateDecode <> endobj << 1480 0 obj 1453 0 obj /Length 48 Geodesics on a torus of revolution. 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. Prince 12.5 (www.princexml.com) 1462 0 obj Ʀ�=�w����WRt��ST�&�m��D����e���oQ%Q�E endobj /BBox [0 0 504 720] Adrian Biran, in Geometry for Naval Architects, 2019. >> <>216 0 R]/P 1538 0 R/Pg 1491 0 R/S/InternalLink>> For any geodesic ζ and a point p <>1101 0 R]/P 1600 0 R/Pg 1599 0 R/S/InternalLink>> 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. <>212 0 R]/P 1492 0 R/Pg 1491 0 R/S/InternalLink>> >> >> <>202 0 R]/P 1510 0 R/Pg 1491 0 R/S/InternalLink>> 2020-06-03T12:29:44-07:00 endobj /Filter /FlateDecode Geodesics on surfaces of revolution 6 References 8 6. 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]. >> << endobj endobj endstream ˑ endstream << 1456 0 obj /ProcSet [/PDF /Text] /Length 10 endobj stream <>234 0 R]/P 1554 0 R/Pg 1553 0 R/S/InternalLink>> <>882 0 R]/P 1593 0 R/Pg 1588 0 R/S/InternalLink>> endstream 1463 0 obj endobj 1471 0 obj <> /Length 10 <>stream << 1482 0 obj <> << endobj endobj endobj I first introduce some of the key concepts in differential geometry in the first 6 chapters. stream endobj The relation remains valid for a geodesic on an arbitrary surface of revolution. <>435 0 R]/P 1586 0 R/Pg 1581 0 R/S/InternalLink>> endobj << /Filter /FlateDecode 1458 0 obj endstream endobj (e) The pseudosphere is the surface of revolution parametrized by x(u, v) = 111 - cos u, -sinu, 11- - coshul, UER. endobj <>1371 0 R]/P 1609 0 R/Pg 1606 0 R/S/InternalLink>> /Encoding /WinAnsiEncoding 1434 0 obj /Filter /FlateDecode endobj Length minimising curves 4 6.7. <>228 0 R]/P 1547 0 R/Pg 1542 0 R/S/InternalLink>> stream << <>1368 0 R]/P 1607 0 R/Pg 1606 0 R/S/InternalLink>> endobj 1 endobj endobj endstream 1606 0 obj ���Vx�jW��L��-n�� endobj 1486 0 obj 8����f"� >> endobj 22 0 obj 1485 0 obj <>241 0 R]/P 1558 0 R/Pg 1553 0 R/S/InternalLink>> << ClairautÕ s Theorem . /Filter /FlateDecode The codimension 1 coincides with the fact that the geodesic is of dimension 1. The Geodesic Equation. endobj endobj <>214 0 R]/P 1536 0 R/Pg 1491 0 R/S/InternalLink>> AppendPDF Pro 6.3 Linux 64 bit Aug 30 2019 Library 15.0.4 1488 0 obj /Filter /FlateDecode %���� stream <>1104 0 R]/P 1604 0 R/Pg 1599 0 R/S/InternalLink>> <>209 0 R]/P 1498 0 R/Pg 1491 0 R/S/InternalLink>> Appligent AppendPDF Pro 6.3 <>215 0 R]/P 1540 0 R/Pg 1491 0 R/S/InternalLink>> endobj <>218 0 R]/P 1532 0 R/Pg 1491 0 R/S/InternalLink>> >> 148 0 obj 6.10 Geodesics and Plate Development. 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. /Filter /FlateDecode endobj stream 8 0 obj 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. stream 10 0 obj stream /Length 10 For further reading we send the reader to the wide literature on Riemannian and Finsler geometry and topology, in particular the geodesic research. <>880 0 R]/P 1589 0 R/Pg 1588 0 R/S/InternalLink>> Like ellipses these … <>221 0 R]/P 1522 0 R/Pg 1491 0 R/S/InternalLink>> A surface similar to an ellipsoid can be generated by revolution of the ovals of Cassini. The surface of revolution as the Earth’s model – sphere S2 or the spheroid is locally approximated by the Euclidean plane tangent in … endobj endobj � /Length 10 << /Length 48 <> endobj 1611 0 obj Always the first point was marked, where the Jacobi field is zero. /Length 48 <>1375 0 R]/P 1611 0 R/Pg 1606 0 R/S/InternalLink>> <>431 0 R]/P 1584 0 R/Pg 1581 0 R/S/InternalLink>> endstream <>233 0 R]/P 1556 0 R/Pg 1553 0 R/S/InternalLink>> 6 0 obj Since a geodesic can pass through any point on the surface, we call these unbounded geodesics. endobj <>stream Since it is a complete negatively curved surface, there is exactly one geodesic connecting any two points. 18 0 obj CWk��H���R�(�^M��g��yX/��I`����b���R�1< endobj /Length 48 <>213 0 R]/P 1494 0 R/Pg 1491 0 R/S/InternalLink>> <>230 0 R]/P 1543 0 R/Pg 1542 0 R/S/InternalLink>> endobj >> >> 1 0 obj endobj endobj - a geodesic of a surface is planar if and only if it is a curvature line. - the meridians of a surface of revolution are geodesics (but not the parallels, except those with extreme radius). /F1 2 0 R endobj It is standard differential geometry to find the differential equation for the geodesics on this surface. endstream There are directions, in which the geodesic winds around the torus several times before the Jacobi field reaches a … >> /Length 10 Any meridian is perpendicular to the equator. /Filter /FlateDecode 1441 0 obj trajectories including geodesic, non-geodesic, constant winding angle and a combination of these trajectories have been generated for a conical shape. endobj /Font endobj <> 1466 0 obj �������; �6��s�ѐ��$ 1448 0 obj 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. <>208 0 R]/P 1504 0 R/Pg 1491 0 R/S/InternalLink>> <>201 0 R]/P 1520 0 R/Pg 1491 0 R/S/InternalLink>> /Filter /FlateDecode Geodesics on such a surface of rotation have a simple general structure. 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. << 1433 0 obj <>206 0 R]/P 1496 0 R/Pg 1491 0 R/S/InternalLink>> stream 1454 0 obj endstream <>238 0 R]/P 1568 0 R/Pg 1553 0 R/S/InternalLink>> stream <> 1479 0 obj 1474 0 obj endobj <>364 0 R]/P 1573 0 R/Pg 1572 0 R/S/InternalLink>> Send article to Kindle. <>219 0 R]/P 1530 0 R/Pg 1491 0 R/S/InternalLink>> 21 0 obj stream << <>200 0 R]/P 1526 0 R/Pg 1491 0 R/S/InternalLink>> For example, the geodesics of a sphere are its great circles. /Filter /FlateDecode 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. << The Clairot integral rsin(φ) is the analogue of Snells integral g(x)sin(α) we have seen before. endobj � qrH�G�v��V���PE�*�4|����cF �A���a�^:b�N /Length 10 <>229 0 R]/P 1545 0 R/Pg 1542 0 R/S/InternalLink>> 2 0 obj /Filter /FlateDecode endobj /Length 48 endobj application/pdf Examples of surfaces of revolution are the cylinder, the cone or the torus. 4 0 obj endobj 1436 0 obj endstream stream endobj - the straight lines of a surface are geodesics (and they are the only one to be geodesics and asymptotic lines). endstream Geodesics are curves on the surface which satisfy a certain second-order ordinary differential equation which is specified by the first fundamental form. endobj <>371 0 R]/P 1577 0 R/Pg 1572 0 R/S/InternalLink>> endobj endobj << 1478 0 obj A geodesic will cut meridians of an ellipsoid at angles α , known as azimuths and measured clockwise from north 0º to . stream >> 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 1460 0 obj x��. >> Theorem 5.2 Let Mbe a surface with a u-Clairaut patch x(u,v). 1435 0 obj The geodesic curvature of a plane curve on the xy-plane is the signed curvature of the curve. endobj /BaseFont /Helvetica 1489 0 obj <>883 0 R]/P 1595 0 R/Pg 1588 0 R/S/InternalLink>> 1444 0 obj 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. (But I could easily have made a mistake in the calculation anyway.) endobj A theorem on geodesics of a surface of revolution is proved in chapter 8. <> Denition 1.1 (Surface of Revolution). endobj >> Primary caustic computation on a surface of revolution r = exp(-z^2). <>1102 0 R]/P 1602 0 R/Pg 1599 0 R/S/InternalLink>> 1472 0 obj <>885 0 R]/P 1597 0 R/Pg 1588 0 R/S/InternalLink>> endobj /Filter /FlateDecode >> 1438 0 obj >> 13 0 obj 5 0 obj The 1469 0 obj 1455 0 obj <>210 0 R]/P 1502 0 R/Pg 1491 0 R/S/InternalLink>> endobj >> /Filter /FlateDecode <>239 0 R]/P 1564 0 R/Pg 1553 0 R/S/InternalLink>> endobj �f�����Ԓ�p�ܠ�I�m�,M�I�:��. 16 0 obj endobj 1475 0 obj endobj 1483 0 obj �hQ�9���� endobj <>207 0 R]/P 1508 0 R/Pg 1491 0 R/S/InternalLink>> << endobj 3 0 obj << endobj /Subtype /Form 1461 0 obj /Filter /FlateDecode uuid:6197c564-ae8a-11b2-0a00-f0cf7d020000 stream ��T����� _���[HJ�%��Ph-�+>$�H�hc� 1446 0 obj 6 0 obj �^�>�#��� /Length 126 endobj 24 0 obj -P˃��H'��d�/���lP8}o,U+륚N�iGx��:�\euR|Bv� /Resources << endstream 12 0 obj 1447 0 obj 3 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. 2 0 obj 7 0 obj << 1440 0 obj endobj <>237 0 R]/P 1560 0 R/Pg 1553 0 R/S/InternalLink>> <>217 0 R]/P 1534 0 R/Pg 1491 0 R/S/InternalLink>> /Length 49 endstream endobj /Length 10 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). 1468 0 obj 1451 0 obj
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