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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 Bjorling

Source Notebook

Compute the Björling formula

Contributed by: Alfred Gray

ResourceFunction["Bjorling"][c,n,z]

computes the Björling minimal surface parametrization for the curve c and normal vector field n in the variable z.

Details and Options

The curve c has a normal vector field n if the tangent to the curve, ∂zc(z), is everywhere perpendicular to n(z).

Examples

Basic Examples (2) 

Get the Björling minimal surface parametrization of a helix:

In[1]:=
helix = ResourceFunction["Bjorling"][{0, 0, z}, {Cos[z], Sin[z], 0}, z]
Out[1]=

Plot it:

In[2]:=
ParametricPlot3D[ Evaluate[Re[helix /. z -> x + I y]], {x, 0, 2 Pi}, {y, -1, 1}]
Out[2]=

Get the Björling minimal surface parametrization of a Möbius band:

In[3]:=
mobius = ResourceFunction[ "Bjorling"][{Cos[z], Sin[z], 0}, {Cos[z/2] Cos[z], Cos[z/2] Sin[z], Sin[z/2]}, z]
Out[3]=

Plot it:

In[4]:=
ParametricPlot3D[ Evaluate[Re[mobius /. z -> x + I y]], {x, 0, 2 Pi}, {y, -1, 1}]
Out[4]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 26 February 2020

Source Metadata

  • Citation:
    • Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.
    • Peternell, M., Pottmann, H., "Computing Rational Parametrizations of Canal Surfaces." J. Symbolic Computation, vol. 23, 255–266, 1997.

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