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Mathematics Forum
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| Author | Message / Information |
| davidlaz Quote | Reply | | Intrinsic (Natural) Geometry posted on: 11/1/2002 12:29:53 AM Do any members use Intrinsic Coordinates and Intrinsic Geometry to study Plane and Space Curves? This was a special development of Ernesto Cesaro, although many mathematicians of 19th Century made some use of the discipline. |
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herbert
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Intrinsic (Natural) Geometry
replied on: 12/15/2003 12:21:59 PM 1) go to the GOOGLE and feed him with "intrisic geometry" or "differential geometry" That´s no joke. It is up to you to get a first impression of the subject. Any good teacher can give you hints to proceed afterwards if there are some specific questions to be answered. 2) It is a custom in mathematics to talk about "noneuclidean geometry (NG)"; well, somehow we forgot that centuries before Lobacevsky the navigators used to navigate around the earth and for this purpose they had to use SPHERICAL TRIGONOMETRY - nothing else but the simplest form of a NG. Every geometrical manifold that cannot use the fact of euclidean geometry namely a*a + b*b = c*c in the NONDIFFERENTIAL CASE must be handled by some noneuclidean geometry. After the sphere there comes the geometry of an ellipsoid. There are also theories concerning OPEN surfaces - there is no end at all.. So for every surface you are after you need a special geometry for macroscopic distances(and areas) With the help of differential geometry you will get the intrinsic geometry of your special surface and there are of lot of good textbooks explaining the details. The basics of any NG are not so difficult if the assumption that in the differential case a^2 + b^2 = c^2 is assumed to be true. In connection with the GAUSSIAN definition of coordinates it gives the usually well known analytical form of surface areas ( again in the differential case ) for example df = sqrt (EG - F^2 )dudv df ... differential element of area E,G and F are the intrinsic parameters of the surface considered at a certain point and (u,v) constitute a set of Gaussian coordinates on the surface. If I were you I would start to acquire some basics of spherical trigonometry and then proceed to the general theory. Herbert |
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Davrin
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Intrinsic (Natural) Geometry
replied on: 1/8/2006 6:11:17 AM Thanks for your reply. Sorry that I took so long to see it (Jan 2006) Actually, I have translated all the plane intrinsic geometry, that Cesaro published in 1906, and it has some extremely powerful tools, not found in Cartesian and other extrinsic frameworks. I was hoping to find like-minded mathematicians, to share the knowledge. This year on 12 September marks the centenary of Cesaro's drowning. Regards Davrin |
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