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.
now we see where calculus comes in.
calculus it the answer
just dx/dz or dy/dz at that point and you''ll have the x-curvture and the y-curvture
calculus it the answer
just dx/dz or dy/dz at that point and you''ll have the x-curvture and the y-curvture
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]
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]
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.
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?
what would be a gradient ? and what is the curvture?
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.
Death of one is a tragedy, death of a million is just a statistic.
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 ?
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 ?
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.
while the Gaussian curvature is det II/det I.
Hope this helps.
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.
There we go, rho is the radius of curvature...
Death of one is a tragedy, death of a million is just a statistic.
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
Thanks,
Timkin
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