Function Repository Resource:

# NaturalEquations

Compute the natural equations of a curve

Contributed by: Wolfram Staff (original content by Eric W. Weisstein)
 ResourceFunction["NaturalEquations"][c,t] computes the natural equations of a curve c with parameter t.

## Details and Options

A natural equation is an equation that specifies a curve independent of any choice of coordinates or parametrization.
Natural equations are computed using the tangential angle , where κ is the curvature, s is the arc length, and and are the coordinates.
ResourceFunction["NaturalEquations"] accepts the same options as Integrate.

## Examples

### Basic Examples (3)

Natural equations for a circle:

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Natural equations for several curves:

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Natural equations for some curves can be solved in terms of elementary functions. Get the equation for a logarithmic spiral:

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Get the natural equations:

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### Options (1)

#### Assumptions (1)

Use the option Assumptions to specify assumptions:

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### Properties and Relations (2)

The Cornu spiral:

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NaturalEquations gives the same result:

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Alfred Gray’s generalization of the Cornu spiral:

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In this case, the resulting natural equations contain special functions:

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Using assumptions can simplify the resulting expressions:

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Get equations for the clothoid prime curve (the velocity of the nth clothoid):

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Simplifying:

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### Possible Issues (2)

Some cases can give large results:

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But they can often be simplified:

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Cassini curve:

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Some cases need some time to evaluate:

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Enrique Zeleny

## Version History

• 1.0.0 – 12 June 2020