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Compute the normal curvature of a curve on a surface
ResourceFunction["NormalCurvature"][s,c,{u,v},t] computes the normal curvature of the plane curve c with respect to variable t over the surface s parameterized by u and v. |
A sphere:
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Normal curvature of the sphere:
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A hyperbolic paraboloid:
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A normal section of the hyperbolic paraboloid:
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Plot of the normal sections of the hyperbolic paraboloid:
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The normal curvature:
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The change in direction happens when the curvature crosses zero:
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Choose c1 to get zeros in the interval (0,π):
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The principal curvatures:
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The maximum and minimum values of the normal curvature are the principal curvatures:
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The normal curvature when t=0:
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The normal curvature varying φ (principal curvatures have opposite signs in a hyperbolic point):
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The asymptotic directions correspond to the angles for the zeros:
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In these directions, the normal curvature vanishes:
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Plot the principal directions and the principal curves:
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The normal sections and the normal curvature:
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A cylinder:
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The normal curvature:
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Computing the normal curvature manually, first find the derivatives:
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Derive again when t=0:
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The dot product with the unit normal gives:
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Another way to compute the normal curvature using the principal curvatures:
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