ArcLength

ArcLength[reg]
gives the length of the one-dimensional region reg.

ArcLength[{x1,,xn},{t,tmin,tmax}]
gives the length of the parametrized curve whose Cartesian coordinates are functions of t.

ArcLength[{x1,,xn},{t,tmin,tmax},chart]
interprets the as coordinates in the specified coordinate chart.

Details and OptionsDetails and Options

  • ArcLength is also known as length or curve length.
  • A one-dimensional region can be embedded in any dimension greater than or equal to one.
  • The ArcLength of a curve in Cartesian coordinates is .
  • In a general coordinate chart, the ArcLength of a parametric curve is given by , where is the metric.
  • In ArcLength[x,t], if x is a scalar, ArcLength returns the length of the parametric curve .
  • Coordinate charts in the third argument of ArcLength can be specified as triples in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsAutomaticwhether to generate answers that involve conditions on parameters
    MethodAutomaticmethod to use
  • Specific methods include:
  • Automaticautomatic method selection
    "Integrate"exact symbolic integration
    "NIntegrate"numeric integration
  • Additional method suboptions can be given in the form Method->{,opts}.
  • Any option of NIntegrate can be passed as a method suboption to the method.
  • Symbolic limits of integration are assumed to be real and ordered. Symbolic coordinate chart parameters are assumed to be in range given by the property of CoordinateChartData.

ExamplesExamplesopen allclose all

Basic Examples  (4)Basic Examples  (4)

The length of the line connecting the points , , and :

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Circumference of a parameterized unit circle:

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Length of one revolution of the helix , , expressed in cylindrical coordinates:

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The length of a region of dimension zero is :

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The length of a region of dimension two or higher is :

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Introduced in 2014
(10.0)