# 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 Automatic 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 :

The length of a circle with radius :

Circumference of a parameterized unit circle:

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

## Scope(16)

### Special Regions(3)

Line:

Lines can be used in any number of dimensions:

Only a 1D Simplex has meaningful arc length:

It can be embedded in any dimension:

### Formula Regions(2)

The arc length of a circle represented as an ImplicitRegion:

An ellipse:

The arc length of a circle represented as a ParametricRegion:

Using a rational parameterization of the circle:

### Mesh Regions(2)

The arc length of a MeshRegion:

In 3D:

The arc length of a BoundaryMeshRegion in 1D:

### Derived Regions(4)

The portion of a circle intersecting a disk:

The arc length of a Circle intersected with a Triangle:

The arc length of a TransformedRegion:

The measure of a RegionBoundary:

### Parametric Formulas(5)

An infinite curve in polar coordinates with finite length:

The length of the parabola between and :

Arc length specifying metric, coordinate system, and parameters:

Arc length of a curve in higher-dimensional Euclidean space:

The length of a meridian on the two-sphere expressed in stereographic coordinates:

## Options(3)

### Assumptions(1)

The length of a cardioid with arbitrary parameter a:

### WorkingPrecision(2)

Compute the ArcLength using machine arithmetic:

In some cases, the exact answer cannot be computed:

Find the ArcLength using 30 digits of precision:

## Applications(8)

The length of a function curve :

Equivalently:

Compute the length of a knot:

Compute the length of Jupiter's orbit in meters:

The length can be computed using the polar representation of an ellipse:

Alternatively, use elliptic coordinates with half focal distance and constant :

Extract lines from a graphic and compute their coordinate length:

Color a Lissajous curve by distance traversed:

Color Viviani's curve on the sphere by the fraction of distance traversed:

Find mean linear charge density along a circular wire:

Compute the perimeter length of a Polygon:

## Properties & Relations(6)

ArcLength is a non-negative quantity:

ArcLength[r] is the same as for any one-dimensional region:

ArcLength for a parametric form is defined as an integral:

ArcLength[x,t,c] is equivalent to RegionMeasure[x,{t},c]:

For a 1D region, ArcLength is defined as the integral of 1 over that region:

The circumference of a 2D region is the ArcLength of its RegionBoundary:

## Possible Issues(2)

The parametric form or ArcLength computes the length of possibly multiple coverings:

The region version computes the length of the image:

The length of a region of dimension other than one is Undefined: