Function Repository Resource:

# WeingartenMatrix

Compute the Weingarten matrix of a surface

Contributed by: Wolfram Staff (original content by Alfred Gray)
 ResourceFunction["WeingartenMatrix"][s,{u,v}] is the matrix of the shape operator of surface s with respect to variables u and v.

## Details and Options

The negative derivative of the unit normal of a surface is called the shape operator and measures how the surface bends in different directions (here, v is a tangent vector). Weingarten is the matrix of the shape operator of the local surface given in terms of the components of the fundamental forms with respect to variables u and v.

## Examples

### Basic Examples (2)

Define the monkey saddle:

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Compute the Weingarten matrix:

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### Properties and Relations (2)

A paraboloid:

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The Weingarten matrix can be computed using the shape operator:

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Compute the shape operator of the paraboloid:

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The product with the inverse metric:

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The Weingarten matrix:

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We will be comparing with the derivatives of the unit normals:

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Derivatives of the unit normal:

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The Gaussian and the mean curvature:

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The Gaussian and mean curvatures can be computed from the Weingarten matrix. The Gaussian curvature is equal to the determinant of the Weingarten matrix:

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The mean curvature is equal to half the trace of the Weingarten matrix:

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The principal curvatures are minus the eigenvalues of the Weingarten matrix:

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A Monge patch:

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The Weingarten matrix:

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Enrique Zeleny

## Version History

• 1.0.0 – 04 September 2020