# ArcCurvature

ArcCurvature[{x1,,xn},t]

gives the curvature of the parametrized curve whose Cartesian coordinates xi are functions of t.

ArcCurvature[{x1,,xn},t,chart]

interprets the xi as coordinates in the specified coordinate chart.

# Details • The arc curvature is sometimes referred to as the unsigned or Frenet curvature.
• The arc curvature of the curve in three-dimensional Euclidean space is given by .
• In a general space, the arc curvature of the curve is given by .
• In ArcCurvature[x,t], if x is a scalar expression, ArcCurvature gives the curvature of the parametric curve {t,x}.
• Coordinate charts in the third argument of ArcCurvature 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.

# Examples

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## Basic Examples(2)

A circle has constant curvature:

The curvature of Fermat's spiral expressed in polar coordinates:

Visualize both branches of the curve:

## Scope(5)

Curvature of loxodromes on a sphere:

A parabola has maximal curvature at its vertex, which decreases to 0 at infinity:

Curvature specifying metric, coordinate system, and parameters:

Parallels and meridians can be considered either curves in flat space or on the two-dimensional sphere:

As curves in three-space, they have the expected curvature inverse to their radius:

On the sphere, meridians, being geodesics, have zero curvature, but non-equatorial parallels do not:

Curvature in higher-dimensional Euclidean space:

## Applications(2)

Compute the radius of curvature of Bernoulli's lemniscate:

The FaryMilnor theorem states that a closed curve with total curvature less than cannot be knotted. Thus, a circle, which has total curvature , cannot be a knot:

The trefoil knot must have total curvature at least , and it does:

## Properties & Relations(2)

ArcCurvature returns only a single curvature:

FrenetSerretSystem returns all curvatures in dimension :

ArcCurvature is unsigned:

Extract the two-dimensional signed curvature from FrenetSerretSystem: