Function Repository Resource:

# MeanCurvature

Compute the mean curvature of a surface

Contributed by: Wolfram Staff (original content by Alfred Gray)
 ResourceFunction["MeanCurvature"][s,{u,v}] computes the mean curvature of parametrized surface s with respect to parametrizing variables u and v. ResourceFunction["MeanCurvature"][eq,{x,y,z}] computes the mean curvature of the surface given by the implicit equation eq in variables x,y and z.

## Details

Mean curvature can be defined as the mean of the principal curvatures, while Gaussian curvature is the product of principal curvatures.
Mean curvature is an extrinsic measure of a surface and locally describes the curvature of an embedded surface.
Mean curvature can be derived as well from the first and second fundamental forms.

## Examples

### Basic Examples (1)

Mean curvature of a sphere:

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### Scope (2)

Plot the Kuen surface:

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Compute its mean curvature:

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Plot the mean curvature:

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Plot the surface with a color function in accordance with the mean curvature:

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The Gaussian curvature is constant:

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Define the implicit equation for the sine surface:

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The mean curvature:

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### Properties and Relations (3)

The mean curvature of a minimal surface is zero:

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The mean curvature for an implicit surface:

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The curvature can be obtained for named surfaces using entities:

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The expressions seem to be different, but they are equal:

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Enrique Zeleny

## Version History

• 1.1.0 – 19 July 2021
• 1.0.0 – 11 March 2020