       Re: Re: Surface Normal

• To: mathgroup at smc.vnet.net
• Subject: [mg55246] Re: [mg55198] Re: Surface Normal
• From: DrBob <drbob at bigfoot.com>
• Date: Thu, 17 Mar 2005 03:30:23 -0500 (EST)
• References: <d15s2g\$9k2\$1@smc.vnet.net> <200503161036.FAA23857@smc.vnet.net> <opsnqkkgzgiz9bcq@monster.ma.dl.cox.net> <4238A0B5.70600@arcor.de>
• Sender: owner-wri-mathgroup at wolfram.com

```My intention was to get something more generally applicable, and I obviously failed!

This may be useful, though:

ClearAll[normVec]
normVec[f_][u__] := Block[{vars = Table[Unique["x"], {Length@{u}}], result},
result = Cross @@ Transpose[D[f @@ vars, {vars}]] /. Thread[vars -> {u}];
Remove @@ vars;
result
]
f[x_, y_] := {x*y, x, y};
normVec[f][2, 3]
normVec[f][u, v]

{1, -3, -2}
{1, -v, -u}

Bobby

On Wed, 16 Mar 2005 22:10:13 +0100, Peter Pein <petsie at arcor.de> wrote:

>  DrBob wrote:
>
>> This may be easier to type and remember (when f is vector valued):
>>
>> normVec[f_][u_, v_] := Cross @@ Transpose@D[f[u, v], {{u, v}}]
>>
>> Bobby
>>
>> On Wed, 16 Mar 2005 05:36:34 -0500 (EST), Peter Pein <petsie at arcor.de>
>> wrote:
>>
>>> gouqizi.lvcha at gmail.com wrote:
>>>
>>>> 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
>>>>
>>> The same way you would do with pencil & paper:
>>>
>>> In:= normVec[f_][u_,v_]:=
>>>   Cross[Derivative[1,0][f][u,v],Derivative[0,1][f][u,v]]
>>> In:= f[u_,v_]:={5 Cos[u](Cos[v]+2),5 Sin[u](Cos[v]+2),5 Sin[v]};
>>> In:= nv = FullSimplify[normVec[f][u, v]]
>>> Out=
>>>   {25*Cos[u]*Cos[v]*(2 + Cos[v]),
>>>    25*Cos[v]*(2 + Cos[v])*Sin[u],
>>>    25*(2 + Cos[v])*Sin[v]}
>>>
>>> If you need it normalized; divide by
>>> In:= absnv = Simplify[Sqrt[#1 . #1]&[nv], v \[Element] Reals]
>>> Out= 25*(2 + Cos[v])
>>> or insert "(#/Sqrt[#.#])&@" (without quotes) just before "Cross" in the
>>> above definition.
>>>
>>> Peter
>>>
> More readable and easier to remember - OK. But:
>
> In:=
>   yourNormVec[f_][u_, v_] := Cross @@ Transpose[D[f[u, v], {{u, v}}]]
>   myNormVec[f_][u_, v_] :=
>     Cross[Derivative[1, 0][f][u, v], Derivative[0, 1][f][u, v]]
> In:=
>   f[x_, y_] := {x*y, x, y};
> In:=
>   StringJoin[ToString[#1], ": ", ToString[Evaluate[#1[f][2, 3]]]] &
>     {yourNormVec, myNormVec}
> From In:=
>   <<error msg omitted>>
> Out=
>   "yourNormVec: Cross[D[{6, 2, 3}, {{2, 3}}]]"
>   "myNormVec: {1, -3, -2}"
>
> _This_ was the reason for using Derivative[1,0][f] instead of D[f[u,v],u].
>
> Peter
>
>
>

--
DrBob at bigfoot.com

```

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