Function Repository Resource:

Curvature

Source Notebook

Compute the curvature of a curve

Contributed by: Wolfram Staff (original content by Alfred Gray)

ResourceFunction["Curvature"][c,t]

computes the curvature of curve c parametrized by t.

Details and Options

The curvature κ is also called the "first curvature".

The curvature of a curve is defined as κ=(x'y''-y'x'')/(x'²+y'²)^3/2.

Examples

Basic Examples (4)

Plot the twisted cubic curve:

In[1]:=

$ParametricPlot3D[{t, t^2, t^3}, {t, -5, 5}]$

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Compute the curvature of the twisted cubic curve:

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$ResourceFunction["Curvature"][{t, t^2, t^3}, t]$

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Compute the torsion with the resource function CurveTorsion:

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$ResourceFunction["CurveTorsion"][{t, t^2, t^3}, t]$

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Plot them:

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Scope (4)

For a plane curve, the curvature and torsion are the same:

In[5]:=

$ResourceFunction["Curvature"][{3 t - t^3, 3 t^2, 3 t + t^3}, t] // Simplify$

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$Simplify[ ResourceFunction["CurveTorsion"][{3 t - t^3, 3 t^2, 3 t + t^3}, t], t > 0]$

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Make a plot:

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A curve that is qualitatively similar to a torus knot:

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$torusknot[a_, b_, c_][p_, q_][ t_] := {(a + b Cos[q t]) Cos[p t], (a + b Cos[q t]) Sin[p t], c Sin[q t]}$

Plot the knot:

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$ParametricPlot3D[torusknot[8, 3, 5][2, 5][t], {t, 0, 2 \[Pi]}]$

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Find the curvature:

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Plot it:

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$Plot[%, {t, 0, 2 \[Pi]}]$

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Find the torsion with the resource function CurveTorsion:

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Plot the torsion:

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$Plot[%, {t, 0, 2 \[Pi]}]$

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Define a loxodrome:

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$loxodromes[a_, b_] := {2 a E^(b u) Cos[u], 2 a E^(b u) Sin[u], a^2 E^(2 b u) - 1}/(1 + a^2 E^(2 b u))$

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Compute its curvature:

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Plot the curvature:

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$gr = Plot[cur, {u, -4 \[Pi], 4 \[Pi]}, PlotRange -> All]$

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A curve colored according to its curvature value:

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range = Last[PlotRange /. AbsoluteOptions[gr, PlotRange]];

Show[ParametricPlot3D[loxodromes[1, 0.1], {u, -4 Pi, 4 Pi}, ColorFunction -> Function[{x, y, z, u, v}, ColorData["Rainbow"][Rescale[cur, range]]], ColorFunctionScaling -> False, PlotStyle -> Thickness[0.01]], Graphics3D[{Opacity[0.2], Sphere[]}]]

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A plane curve in polar coordinates:

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Plot it:

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$PolarPlot[archimedesspiralpolar[2, 1][t], {t, 0, 5 \[Pi]}]$

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The curvature:

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Properties and Relations (4)

The curvature of a circle:

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The curvature of the Cornu spiral:

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Define a conical spiral:

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Compute the curvature:

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$\[Kappa] = ResourceFunction["Curvature"][cs, t] // FullSimplify$

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Definition of a unit speed helix:

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$uhelix = {a Cos[t/Sqrt[a^2 + b^2]], a Sin[t/Sqrt[a^2 + b^2]], (b t)/ Sqrt[a^2 + b^2]};$

The curvature:

In[28]:=

$\[Kappa] = Simplify[ResourceFunction["Curvature"][uhelix, t], a > 0 && b > 0 && t > 0]$

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The tangent vector, via the resource function TangentVector:

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Derivative of the tangent vector:

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The normal vector, via the resource function NormalVector:

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The curvature times the normal vector is equal to the derivative of the tangent vector:

In[32]:=

$\[Kappa] n$

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The torsion, via the resource function CurveTorsion:

In[33]:=

$\[Tau] = Simplify[ResourceFunction["CurveTorsion"][uhelix, t], a > 0 && b > 0 && t > 0]$

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In the Frenet–Serret system, the curvature and the torsion are the first two quantities:

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Publisher

Version History

2.0.0 – 30 April 2020
1.0.0 – 28 April 2020

Source Metadata

Citation:
- Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.

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License Information

This work is licensed under a Creative Commons Attribution 4.0 International License