Re: Surface Normal

• To: mathgroup at smc.vnet.net
• Subject: [mg55195] Re: Surface Normal
• From: "Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk>
• Date: Wed, 16 Mar 2005 05:36:22 -0500 (EST)
• References: <d15s2g\$9k2\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```You need to compute the cross product of the derivatives of your parametric
surface w.r.t. the parameters.

Define your parametric surface (a torus).

f[u_, v_] := {(10 + 5*Cos[v])*Cos[u], (10 + 5*Cos[v])*Sin[u], 5*Sin[v]};

Compute the derivatives w.r.t. parameters and form the cross product.

df[u_, v_] = Simplify[Cross[D[f[u, v], u], D[f[u, v], v]]]

{25*Cos[u]*Cos[v]*(2 + Cos[v]), 25*Cos[v]*(2 + Cos[v])*Sin[u], 25*(2 +
Cos[v])*Sin[v]}

Plot the parametric surface.

g1 = ParametricPlot3D[f[u, v], {u, 0, 2*Pi}, {v, 0, 2*Pi}];

Plot a field of surface normals.

g2 = Show[Graphics3D[Table[Line[{f[u, v], f[u, v] + 0.1*df[u, v]}], {u, 0,
2*Pi, (2*Pi)/20},
{v, 0, 2*Pi, (2*Pi)/20}]]];

Display the above two plots together.

Show[g1,g2];

Steve Luttrell

<gouqizi.lvcha at gmail.com> wrote in message news:d15s2g\$9k2\$1 at smc.vnet.net...
> Hi, All:
>
> If I have a surface in parametric form
>
> For example,
> x = (10 + 5cosv)cosu
> y = (10 + 5cosv)sinu
> z = 5sinv
>
> How can I quickly calculate its normal for any (u,v) by mathematica
>
> Rick
>

```

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