Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 BinormalSurface

Source Notebook

Compute the binormal surface to a curve

Contributed by: Alfred Gray

ResourceFunction["BinormalSurface"][c,t,{u,v}]

computes the binormal surface of a curve c with parameter t using u and v to parametrize the result.

Details and Options

The binormal surface associated to a curve is generated by its binormal vector field.
The binormal surface is effectively a ruled surface.

Examples

Basic Examples (4) 

Define and plot a helix curve:

In[1]:=
helix = Entity["SpaceCurve", "Helix"]["ParametricEquations"][1, 2][t]
Out[1]=
In[2]:=
vp = ParametricPlot3D[Evaluate[helix], {t, 0, 2 \[Pi]}, PlotStyle -> Thickness[.02]]
Out[2]=

Compute the normal surface of the helix curve with the resource function NormalSurface and plot the result:

In[3]:=
ns = ResourceFunction["NormalSurface"][helix, t, {u, v}] // Simplify
Out[3]=
In[4]:=
nsp = ParametricPlot3D[Evaluate[ns], {u, 0, 2 \[Pi]}, {v, 0, 5}, PlotPoints -> {20, 10}]
Out[4]=

Compute and plot the binormal surface of the helix curve:

In[5]:=
ResourceFunction["BinormalSurface"][helix, t, {u, v}]
Out[5]=
In[6]:=
bsp = ParametricPlot3D[ Evaluate[ResourceFunction["BinormalSurface"][helix, t, {u, v}]], {u, 0, 2 \[Pi]}, {v, 0, 5}, PlotPoints -> {20, 10}]
Out[6]=

Show both surfaces along with the helix curve:

In[7]:=
Show[vp, nsp, bsp]
Out[7]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 24 February 2020

Source Metadata

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

License Information