Function Repository Resource:

# BrioschiCurvature

Compute the Gaussian curvature for a metric

Contributed by: Wolfram Staff (original content by Alfred Gray)
 ResourceFunction["BrioschiCurvature"][m,{u,v}] computes the Gaussian curvature for a metric m with respect to variables u and v from the Brioschi formula. ResourceFunction["BrioschiCurvature"][{fx,fy,fz},{u,v}] computes the Gaussian curvature for a parametrization with respect to variables u and v. ResourceFunction["BrioschiCurvature"][e,f,g,{u,v}] computes the Gaussian curvature from the first fundamental form coefficients e,f and g.

## Details

The metric m is limited to a 2×2 matrix, being square and symmetric.
The Brioschi formula gives Gaussian curvature solely in terms of the first fundamental form.
The explicit form of the formula was worked out by Francesco Brioschi (1852).

## Examples

### Basic Examples (2)

Compute the Gaussian curvature of a sphere using the Brioschi formula:

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Define a funnel:

The covariant basis:

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The metric tensor:

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Compute the Brioschi curvature:

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### Properties and Relations (3)

The Brioschi formula may be slower than computing the Gaussian curvature directly:

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Apply the Brioschi formula for a parametrized surface:

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Apply using the first fundamental form coefficients:

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The covariant basis of a surface:

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The metric tensor:

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Compute the scalar curvature:

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This coincides with twice the Gaussian curvature:

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Enrique Zeleny

## Version History

• 1.0.0 – 16 March 2021