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# Curvature of a parametric surface

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How do I find the direction of maximum and minimum curvature, primary directions, of a surface? Then how do I find the curvature of the surface in that direction? I know how to find the curvature of a space curve. I also know how to find a gradiant, but not for a parametric surface.

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now we see where calculus comes in.

just dx/dz or dy/dz at that point and you''ll have the x-curvture and the y-curvture

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Erm... you may want the radius of curvature, if you want to know how curvy a curve is at a point. ( Or exactly, the radius of a circle that would coincide with the curve at that point, if you get what I mean... )

Anyway, the forumulas:

Intrinsic form:

Parametric form:

Cartesian form:

Death of one is a tragedy, death of a million is just a statistic.

[edited by - python_regious on June 10, 2002 1:56:30 PM]

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quote:
Original post by danz
just dx/dz or dy/dz at that point and you''ll have the x-curvture and the y-curvture

No, you''ll have the gradient in those directions.

Death of one is a tragedy, death of a million is just a statistic.

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i''m not good at english math terms, i''ve learned it all in hebrew

what would be a gradient ? and what is the curvture?

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Gradient would be something like the slope of the tangent to a point on a curve. The curvature is how bendy something is... sorta... The radius of curvature is the radius of the circle that would co-incide with the curve at that point ( centered on the normal to the curve at that point ).

Death of one is a tragedy, death of a million is just a statistic.

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ok

so i have to find a circle which is tangent at that point and has the same normal as the surface at that point, right ?

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By definition the primary directions of a parametric surface are the (normalized) eigenvectors of the two quadratic form II,I(First and Second fundamental form) i.e the eigenvectors of II/I,
while the Gaussian curvature is det II/det I.
Hope this helps.

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Ok.. I''ve done a little picture of what I mean...

There we go, rho is the radius of curvature...

Death of one is a tragedy, death of a million is just a statistic.

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This sounds like a homework question to me! If it is, please read the forum FAQ for the policy on asking such questions.

Thanks,

Timkin

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Python,

Those formulas are for a curve in two dimensions, not a surface in three. As I said I can find the curvature for a space curve and that would include a curve confined to a plane. There can be more than one curvature for a point on a surface in three dimensions. Basically picture your circle rotating around the normal.

Anon,

What is the first and second fundamental form?

Timkin,

Nope, not homework. If it was I would have some referance material to draw off of and thus not need to ask the question. It is for an adapative subdivision routine, i.e. where the curvature is high generate more polygons. If you could tell me what class or better a book that would cover this I would be more than happy to get such a book.

As it is the only thing I can think to do is to rotate a vector positioned in the tangent plane at the point around the normal. The point, that vector and the normal gives me a plane which provides me a relationship between the three equations in the parametric equation. That lets me find a relation between the two parameters for the trace of surface in that plane which in turn gives me a parametric equation of one variable for a space curve. I can then find the curvature for that space curve at that point. That should give me an equation for the curvature as a function of the rotation of that vector around the normal. I can then find the max and min of that equation which should give me the primary directions. That seems like that is going to be one extremely nasty equation. I suspect though that there is a far simplier way. Thus the reason for the question.

Oh yes, one more thing. A substantial short cut seems like it would be to find the curvatures when holding the two parameters in turn constant. I could then use the mean or product of those as the curvature of the point. It just isn't clear to me if that is a good measure of curvature. It seems like it should do well enough, but I don't have any definite basis for saying that.

[edited by - LilBudyWizer on June 10, 2002 11:57:35 PM]

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There is a good section in "3D Game Engine Design" about subdividing according to variation.

Death of one is a tragedy, death of a million is just a statistic.

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Ah, thank you, yes, there is a whole section on curvature of surfaces in fact. I almost forgot I had that book. It was one of the main reasons I decided I needed to go back to school. Bought it, started reading it and couldn''t even read the notation

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