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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 DarbouxVector

Source Notebook

Compute the Darboux vector field of a curve

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

ResourceFunction["DarbouxVector"][α,t]

computes the Darboux vector field of a curve.

Details and Options

The Darboux vector of a space curve with unit speed is the angular velocity vector of the Frenet frame.

Examples

Basic Examples (3) 

Define and plot a helix:

In[1]:=
helix[a_, b_, c_][t_] := {a*Cos[t], b*Sin[t], c*t}
In[2]:=
ParametricPlot3D[Evaluate[helix[1, 2, 3][t]], {t, 0, 2 \[Pi]}, AspectRatio -> 2, ImageSize -> 100]
Out[2]=

Compute the Darboux vector of the helix and plot its components:

In[3]:=
dv = ResourceFunction["DarbouxVector"][helix[1, 2, 3], t] // PowerExpand // FullSimplify
Out[3]=
In[4]:=
Plot[dv, {t, 0, 2 Pi}]
Out[4]=

Plot the Darboux vector in space:

In[5]:=
ParametricPlot3D[ Evaluate[ResourceFunction["DarbouxVector"][helix[1, 2, 3], t]], {t, 0, 2 \[Pi]}]
Out[5]=

Properties and Relations (1) 

The Darboux vector can be expressed in terms of the Frenet-Serret system of the curve:

In[6]:=
helix[a_, b_, c_][t_] := {a*Cos[t], b*Sin[t], c*t}
In[7]:=
fss = FrenetSerretSystem[helix[a, b, c][t], t] // Simplify
Out[7]=
In[8]:=
(Cross[ResourceFunction["DarbouxVector"][helix[a, b, c], t], #] == D[#, t] // PowerExpand // Simplify) & /@ fss[[2]]
Out[8]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 20 February 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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