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MathGroup Archive 2005

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Re: Normal Disappear Problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg55340] Re: [mg55302] Normal Disappear Problem
  • From: DrBob <drbob at bigfoot.com>
  • Date: Sat, 19 Mar 2005 04:46:41 -0500 (EST)
  • References: <200503181035.FAA14770@smc.vnet.net>
  • Reply-to: drbob at bigfoot.com
  • Sender: owner-wri-mathgroup at wolfram.com

Of course the sphere has a normal everywhere, but that doesn't mean a given method of computing the normal WORKS everywhere.

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[u_, v_] = {Cos[u]Sin[v], Sin[u]Sin[v], Cos[v]};
normVec[f][u, v]

{(-Cos[u])*Sin[v]^2,
   (-Sin[u])*Sin[v]^2,
   (-Cos[u]^2)*Cos[v]*Sin[v] -
    Cos[v]*Sin[u]^2*Sin[v]}

(I haven't normalized, but neither did you.)

If you substitute u = v = 0, that's certainly the zero vector:

normVec[f][0,0]

{0,0,0}

Now consider the vectors used in the cross product:

showCross[f_][u__] := Block[{vars = Table[Unique["x"], {
     Length@{u}}],
         result}, result = cross @@ Transpose[D[f @@ vars, {vars}]] /. Thread[
       vars -> {u}];
     Remove @@ vars;
     result]
showCross[f][0,0]

cross[{0,0,0},{1,0,0}]

Clearly, that cross product is zero. But why is the first vector all zeroes?

Well, it's simply because:

D[f[u, v], {u}]
% /. v -> 0

{-Sin[u] Sin[v],Cos[u] Sin[v],0}

{0,0,0}

And that's for ALL values of u, so:

normVec[f][u,0]

{0,0,0}

That doesn't look good, but think about it. You started with a "parameterization" of the unit sphere. But it's not a one-to-one mapping:

f[u,0]

{0,0,1}

If the parameterization f is no good at a point, the normal can't be computed using f at that point.

Normalizing reveals the problem in a different way:

#/#.# &@normVec[f][u, v] // TrigExpand
% /. v -> 0

{-Cos[u],-Sin[u],-Cot[v]}

{-Cos[u],-Sin[u],ComplexInfinity}

Everything fails when v = 0.

So.... What's the simpler, better method?

Normalizing actually helps:

#.# &@normVec[f][u, v] // TrigExpand // FullSimplify

Sin[v]^2

normVec[f][u, v]/Sin[v] // TrigExpand

{-Cos[u] Sin[v], -Sin[u] Sin[v], -Cos[v]}

We'd probably reverse the sign, and get

f[u,v]==-%

True

In other words, the (outward-pointing) normal at f[u,v] is just f[u,v].

Bobby

On Fri, 18 Mar 2005 05:35:18 -0500 (EST), gouqizi.lvcha at gmail.com <gouqizi.lvcha at gmail.com> wrote:

> Hi, All:
>
> I have the following parametric equation for an unit sphere:
>
> x = cos(u)sin(v)
> y = sin(u)sin(v)
> z = cos(v)
>
> 0<=u<2*Pi ; 0<=v<=Pi
>
> Then I use
>
> normal = (Dx/Du, Dy/Du, Dz/Du) CROSS (Dx/Dv, Dy/Dv, Dz/Dv) to get the
> normal vector.
>
> I get the follwoing after calculation (with normalization):
>
> normal =  [sin(v) ^2 cos(u), sin(v)^2  sin(u), cos(u)^2  cos(v) sin(v)
> +  sin(u)^2  cos(v) sin(v)]
>
> Now when u=0, v=0 , Normal = (0,0,0)! How can it be? We know the fact
> that a sphere should have normal everywhere.
>
> Rick
>
>
>
>



-- 
DrBob at bigfoot.com


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