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>
- Reply-to: drbob at bigfoot.com
- 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[1]:= normVec[f_][u_,v_]:= >>> Cross[Derivative[1,0][f][u,v],Derivative[0,1][f][u,v]] >>> In[2]:= f[u_,v_]:={5 Cos[u](Cos[v]+2),5 Sin[u](Cos[v]+2),5 Sin[v]}; >>> In[3]:= nv = FullSimplify[normVec[f][u, v]] >>> Out[6]= >>> {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[4]:= absnv = Simplify[Sqrt[#1 . #1]&[nv], v \[Element] Reals] >>> Out[4]= 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[1]:= > 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[3]:= > f[x_, y_] := {x*y, x, y}; > In[4]:= > StringJoin[ToString[#1], ": ", ToString[Evaluate[#1[f][2, 3]]]] & > {yourNormVec, myNormVec} > From In[4]:= > <<error msg omitted>> > Out[4]= > "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
- References:
- Re: Surface Normal
- From: Peter Pein <petsie@arcor.de>
- Re: Surface Normal