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 MovingOrthonormalFrame

Source Notebook

Compute the moving orthonormal frame of a surface

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

ResourceFunction["MovingOrthonormalFrame"][x,{u,v}]

computes the moving orthonormal frame of a parametrized 3D surface x with respect to variables u and v.

Details and Options

Let be a principal patch with metric . Put and . The resulting triple (E1,E2,U), where U denotes the unit normal of x, is called the moving orthonormal frame of x.

Examples

Basic Examples (3) 

Define the Chebyshev principal patch on the pseudosphere:

In[1]:=
pseudospheretcheb[a_][u_, v_] := a {Cos[v]/Cosh[u], Sin[v]/Cosh[u], u - Tanh[u]}

The moving frame associated to a patch x:

In[2]:=
mof = ResourceFunction["MovingOrthonormalFrame"][ pseudospheretcheb[a][u, v], {u, v}]
Out[2]=
In[3]:=
mof = mof // PowerExpand // Simplify
Out[3]=

Plot the moving frame as u and v are varied:

In[4]:=
Manipulate[Show[ParametricPlot3D[ pseudospheretcheb[1][u, v], {u, -2.8, 2.8}, {v, 0, 3 (Pi/2)}, PlotStyle -> Opacity[0.5], PlotRange -> All], Graphics3D[ Arrow[{pseudospheretcheb[1][uf, vf], pseudospheretcheb[1][uf, vf] + #}] & /@ (mof /. {a -> 1, u -> uf, v -> vf})]], {{uf, 1.}, -2.8, 2.8}, {{vf, .35}, 0, 3 \[Pi]/2}]
Out[4]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 01 December 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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