Function Repository Resource:

# Curvature

Compute the curvature of a curve

Contributed by: Wolfram Staff (original content by Alfred Gray)
 ResourceFunction["Curvature"][c,t] computes the curvature of curve c parametrized by t.

## Details and Options

The curvature κ is also called the "first curvature".
The curvature of a curve is defined as κ=(x'y''-y'x'')/(x'2+y'2)3/2.

## Examples

### Basic Examples (4)

Plot the twisted cubic curve:

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Compute the curvature of the twisted cubic curve:

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Compute the torsion with the resource function CurveTorsion:

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Plot them:

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### Scope (4)

For a plane curve, the curvature and torsion are the same:

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Make a plot:

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A curve that is qualitatively similar to a torus knot:

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Plot the knot:

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Find the curvature:

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Plot it:

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Find the torsion with the resource function CurveTorsion:

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Plot the torsion:

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Define a loxodrome:

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Compute its curvature:

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Plot the curvature:

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A curve colored according to its curvature value:

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A plane curve in polar coordinates:

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Plot it:

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The curvature:

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### Properties and Relations (4)

The curvature of a circle:

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The curvature of the Cornu spiral:

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Define a conical spiral:

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Compute the curvature:

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Definition of a unit speed helix:

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The curvature:

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The tangent vector, via the resource function TangentVector:

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Derivative of the tangent vector:

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The normal vector, via the resource function NormalVector:

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The curvature times the normal vector is equal to the derivative of the tangent vector:

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The torsion, via the resource function CurveTorsion:

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In the Frenet–Serret system, the curvature and the torsion are the first two quantities:

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Enrique Zeleny

## Version History

• 2.0.0 – 30 April 2020
• 1.0.0 – 28 April 2020