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Topic: How to bend an ellipsoid

There is an ellipsoid, want to take it for two semiaxes and to start them to bend (concerning center). What should be a surface? On the one hand it is possible to intersect two ellipsoids at an angle, but here the turned out figure and will be as a result intersection of two ellipsoids. And how to construct a figure so that it remained an ellipsoid (let and deformed)? Well that is the mathematical description of such figure is necessary even more likely. Here there is a canonical equation of an ellipsoid x 2/a 2+y 2/b 2+z 2/c 2=1 Here there is a formula for the surface description in a case if to start semiaxes to shift at an angle? What to esteem?

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Re: How to bend an ellipsoid

Hello, zubactik, you wrote: Z> There is an ellipsoid, want to take it for two semiaxes and to start them to bend (concerning center). What means to "bend"? Ellipsoid semiaxes are imaginary lines. They always straight lines. How you represent "bend"? What should be a surface? On the one hand it is possible to intersect two ellipsoids at an angle, but here the turned out figure and will be as a result intersection of two ellipsoids. And how to construct a figure so that it remained an ellipsoid (let and deformed)? Z> Here there is a formula for the surface description in a case if to start semiaxes to shift at an angle? What to esteem? What means to "shift"? It is Meanwhile clear that you want to deform an ellipsoid somehow. For this purpose it is necessary to understand that remains invariable, and that - changes. Here, for example, if we take square-topped  1*1*L, where L>> 1 it is possible to present it as a figure which describes a plane square with the individual side at driving along a direct segment of length L. Then we can tell "and now let's take instead of a straight line segment - a fragment of a curved line of some form. Also we will move along it a square so that the normal to it coincided with a tangent to this curve." Then we receive "bent  square section".

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Re: How to bend an ellipsoid

Excellent  a question! Thanks! Here an example of a picture for an ellipse. On the lower part dark blue lines I tried to finish those parts which I want to approximate "correctly" There is a sensation that any properties (the topological should be saved!?) but as I am badly familiar with geometry (that that teaches Fomenko) and is not familiar with topology I can not formulate the equation which should describe such formed surface. If such generally is. About that that, probably, it there should be a curve on which the ellipse with  semiaxes I should move thought but how as much as possible "competently" to describe such curve and dependence between ellipse position on "" and the size of its semiaxes.

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Re: How to bend an ellipsoid

Hello, zubactik, you wrote: Z> About that that, probably, it there should be a curve on which the ellipse with  semiaxes I should move thought but how as much as possible "competently" to describe such curve and dependence between ellipse position on "" and the size of its semiaxes. I suspect that you smooth conjugation of surfaces interests. It is very deep subject; there is a chasm of candidate solutions even for more or less specific target. Esteem here: http://www.tflexcad.ru/help/cad/15/3de.htm

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Re: How to bend an ellipsoid

Yes! Cool! Thanks! Yet did not read, but here from the point of view of geometry there are no invariants which characterize an ellipse and which should be saved at its modification that the figure continued to remain an ellipse?

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Re: How to bend an ellipsoid

Hello, zubactik, you wrote: Z> Yet did not read, but here from the point of view of geometry there are no invariants which characterize an ellipse and which should be saved at its modification that the figure continued to remain an ellipse? Yes. x^2/a^2 + y^2/b^2 = 1. The figure "boomerang" drawn in your example, an ellipse is not.

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Re: How to bend an ellipsoid

Hello, zubactik, you wrote: Z> Here there is a formula for the surface description in a case if to start semiaxes to shift at an angle? What to esteem? It seems to me that the task more physical, than mathematical. It is not assured, but it seems that it is necessary to take the environment of physical modeling, to set materials (elasticity etc.) and to apply force. The result will be quite described to itself by the physical equations and, most likely, arranges.