# 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 xi are functions of t.

ArcLength[{x1,,xn},{t,tmin,tmax},chart]

interprets the xi as coordinates in the specified coordinate chart.

# Details 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,tmin,tmax}], if x is a scalar, ArcLength returns the length of the parametric curve {t,x}.
• Coordinate charts in the third argument of ArcLength can be specified as triples {coordsys,metric,dim} 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:
•  AccuracyGoal Infinity digits of absolute accurary sought Assumptions \$Assumptions assumptions to make about parameters GenerateConditions Automatic whether to generate conditions on parameters PerformanceGoal \$PerformanceGoal aspects of performance to try to optimize PrecisionGoal Automatic digits of precision sought WorkingPrecision Infinity the precision used in internal computations
• Symbolic limits of integration are assumed to be real and ordered. Symbolic coordinate chart parameters are assumed to be in range given by the "ParameterRangeAssumptions" property of CoordinateChartData.
• ArcLength can be used with symbolic regions in GeometricScene.

# Examples

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## Basic Examples(3)

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

 In:= Out= The length of a circle with radius :

 In:= Out= Circumference of a parameterized unit circle:

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

 In:= Out= ## Possible Issues(2)

Introduced in 2014
(10.0)
|
Updated in 2019
(12.0)