An intrinsic equation of a surface

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Summary:

Does such an equation exist?
Is there an equation that a surface dweller could developedto globally describe his or her surface?

Let us say a sphere?

If we determine that curvature is everywhere constant what would be an equation that would describe that surface ? (simply that curvature is everywhere constant?)

And if curvature is not constant (say an ellipse) would there be any way of expressing this in a global mathematical form and not just as an agglomeration of local measurements of local curvature?

Are global equations of surfaces almost by definition extrinsic?
 

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  • #2
lavinia
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Summary:: Does such an equation exist?

Is there an equation that a surface dweller could developedto globally describe his or her surface?

Let us say a sphere?

If we determine that curvature is everywhere constant what would be an equation that would describe that surface ? (simply that curvature is everywhere constant?)

And if curvature is not constant (say an ellipse) would there be any way of expressing this in a global mathematical form and not just as an agglomeration of local measurements of local curvature?

Are global equations of surfaces almost by definition extrinsic?
Every orientable two dimensional closed surface has a metric of constant Gauss curvature. The sphere has a metric of constant positive curvature, the torus constant zero curvature, all other orientable closed surfaces constant negative curvature. Constant positive curvature determines the sphere and constant zero curvature the torus. But other orientable closed surfaces are not determined by constant negative curvature. I think the only non-orientable closed surface of constant positive curvature is the projective plane and the only flat no-orientable closed surface is the Klein bottle.

The Gauss-Bonnet Theorem says that for a closed orientable surface, the integral of the Gauss curvature over the entire surface is 2π times its Euler characteristic. The sphere is the only one that has positive Euler characteristic and the torus the only one with Euler characteristic zero.
 
  • #3
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Thanks for that information


Actually the surfaces I had in mind were irregular (or just more complex) ones.I only mentioned the sphere and an ellipse because they seemed simplest, but I was thinking of surfaces that ,although continuous and differentiable could be quite complex.

I was also looking for a way to model the entire surface and not just its overall curvature(in case you gathered that)

I also have to say that my understanding of this whole area is extremely elemental (which I am sure you already know or suspect) and so I apologise in advance if what I am asking about comes across as pretentious or plain ignorant.....
 
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Orodruin
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Summary:: Does such an equation exist?

an ellipse)
Being one-dimensional, an ellipse has zero intrinsic curvature.

What you are looking for is the entire framework of differential geometry, not a single equation.
 
  • #5
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Being one-dimensional, an ellipse has zero intrinsic curvature.

What you are looking for is the entire framework of differential geometry, not a single equation.
To your second point,that is what I half expected.(although I wondered if it could be done by parametrization,which I am fairly new to)

To your first point,I should have said "ellipsoid".... sorry
(if that is the name for an oblong sphere)
 
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Orodruin
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Parametrizing the surface is just a part of it. It will only give you a local coordinate system on the surface, but it a priori tells you nothing about its curvature.
 

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