Function Repository Resource:

# NormalCurvature

Compute the normal curvature of a curve on a surface

Contributed by: Wolfram Staff (original content by Alfred Gray)
 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.

## Details and Options

The normal curvature κn(vp) measures how a regular surface M bends in the direction vp. This quantity is the curvature of the resulting plane curve from the intersection of M with a plane passing through a point p, with tangent v and normal , being perpendicular to M.
The normal curvature can be defined as , where is the unit normal.
The maximum and minimum values of the normal curvature are the principal curvatures.
The normal curvature κn in direction φ is given by the Euler curvature formula κn=κ1cos2φ+κ2sin2φ, where κ1 and κ2 are the principal curvatures.

## Examples

### Basic Examples (2)

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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### Properties and Relations (6)

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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Enrique Zeleny

## Version History

• 1.0.0 – 22 July 2020