Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 OsculatingPlane

Source Notebook

Compute the osculating plane of a space curve

Contributed by: Alfred Gray

ResourceFunction["OsculatingPlane"][c,t]

computes the osculating plane of space curve c with parameter t.

Details

The osculating plane is spanned by the tangent vector and normal vector.
The intersection of the osculating plane with the normal plane is the line containing the normal vector.
ResourceFunction["OsculatingPlane"] gives an InfinitePlane object.

Examples

Basic Examples (3) 

Define a helix curve:

In[1]:=
helix = Entity["SpaceCurve", "Helix"]["ParametricEquations"][1, 1][t]
Out[1]=

Compute the normal plane:

In[2]:=
ResourceFunction["OsculatingPlane"][helix, t]
Out[2]=

Plot the different planes while traversing the helix:

In[3]:=
Manipulate[ With[{helix = Entity["SpaceCurve", "Helix"]["ParametricEquations"][1, 1][t]}, Show[ParametricPlot3D[helix, {t, 0, 3}], Graphics3D[{Opacity[.5], ResourceFunction["OsculatingPlane"][helix, t], ResourceFunction["RectifyingPlane"][helix, t], ResourceFunction["NormalPlane"][helix, t]} /. t -> tf], PlotRange -> {{-1, 1}, {0, 1}, {0, 3}}]], {{tf, 1}, 0, 3}]
Out[3]=

Publisher

Enrique Zeleny

Version History

  • 1.0.1 – 08 January 2021
  • 1.0.0 – 31 March 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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