Function Repository Resource:

# ShapeOperator

Compute the shape operator on a surface

Contributed by: Wolfram Staff (original content by Alfred Gray)
 ResourceFunction["ShapeOperator"][s,{u,v}] computes 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)

Shape operator on the sphere:

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The monkey saddle:

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The shape operator:

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### Properties and Relations (2)

A paraboloid:

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The shape operator:

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Alternatively, the resource function UnitNormal can be used to compute the shape operator:

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The Weingarten matrix can be computed using the shape operator with the metric:

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Check the calculation:

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A Monge patch:

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Gaussian and mean curvatures:

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The shape operator:

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Enrique Zeleny

## Version History

• 1.0.0 – 04 September 2020