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Re: Normal vector on a surface

• To: mathgroup at smc.vnet.net
• Subject: [mg29314] Re: [mg29269] Normal vector on a surface
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Wed, 13 Jun 2001 03:10:46 -0400 (EDT)
• References: <200106090708.DAA29328@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Matthias.Bode at oppenheim.de wrote:
> 
> Dear Colleagues,
> 
> I have a function in the variables x1 and y1:
> 
> Out[27]=
> 19.74211746962547 - 61.78321746073334*
>    x1 + 70.84823523445556*x1^2 -
>   34.64681309362152*x1^3 +
>   5.822595947190386*x1^4 -
>   61.783217460733795*y1 +
>   188.56171712734522*x1*y1 -
>   208.7457484391798*x1^2*y1 +
>   99.21114279328117*x1^3*y1 -
>   16.223098505477388*x1^4*y1 +
>   70.8482352344551*y1^2 -
>   208.7457484391805*x1*y1^2 +
>   225.08774661852397*x1^2*y1^2 -
>   103.5151716236312*x1^3*y1^2 +
>   16.351931921608763*x1^4*y1^2 -
>   34.64681309362163*y1^3 +
>   99.21114279328117*x1*y1^3 -
>   103.51517162363109*x1^2*y1^3 +
>   45.654124756950296*x1^3*y1^3 -
>   6.928857192755963*x1^4*y1^3 +
>   5.822595947190411*y1^4 -
>   16.22309850547743*x1*y1^4 +
>   16.351931921608763*x1^2*y1^4 -
>   6.928857192755952*x1^3*y1^4 +
>   1.0137658500940734*x1^4*y1^4
> 
> This function yields a surface very similar to Sin[x1*y1] for 1<x1<3 and
> 1<y1<3.
> 
> Now I want to calculate (how?) and draw (how?) several "Normalenvektors"
> (sorry, I do not know the English termini technici) which should sit smugly
> - like palisades - on the plane tangential to the surface.
> 
> The "Normalenvektor" N in point P - according to Bronstein-Semendjajew - is
> a unity vector perpendicular to the tangential plane; its accompanying
> vectors e1 and e2 on the plane form a "right-handed system". N, e1 and e2
> are referred to as the "accompanying tripod". - I understand the words but
> not their meaning.
> 
> My attempts with Calculus`VectorAnalysis` and PlotVectorField3D &c. failed
> dismally.
> 
> Thank you for your assistance,
> 
> Matthias Bode
> Sal. Oppenheim jr. & Cie. KGaA
> Koenigsberger Strasse 29
> D-60487 Frankfurt am Main
> GERMANY
> Tel.: +49(0)69 71 34 53 80
> Mobile: +49(0)172 6 74 95 77
> Fax: +49(0)69 71 34 6380
> E-mail: matthias.bode at oppenheim.de
> Internet: http://www.oppenheim.de

Your surface is given in the form z = f[x,y] so you can take

-{D[f[x,y],x], Df[x,y],y], -1}

as an upward-pointing normal vector, and then normalize to get unit
length. The code below will do this.

normalvector[x_,y_]:=  With[
  {vec = -{D[f[x1,y1],x1],D[f[x1,y1],y1],-1} /. {x1->x,y1->y}},
  vec / Sqrt[vec.vec]]


Daniel Lichtblau
Wolfram Research

• References:
  • Normal vector on a surface
    • From: Matthias.Bode@oppenheim.de