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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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