Re: Normal Disappear Problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg55335] Re: [mg55302] Normal Disappear Problem*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Sat, 19 Mar 2005 04:46:17 -0500 (EST)*References*: <200503181035.FAA14770@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.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 http://mathworld.wolfram.com/HairyBallTheorem.html Your tangent vectors are: In[2]:= {x,y,z} = {Cos[u]*Sin[v], Sin[u]*Sin[v], Cos[v]}; In[3]:= tanvex = {{D[x,u],D[y,u],D[z,u]},{D[x,v],D[y,v],D[z,v]}} Out[3]= {{-(Sin[u] Sin[v]), Cos[u] Sin[v], 0}, {Cos[u] Cos[v], Cos[v] Sin[u], -Sin[v]}} The theorem indicates each must vanish somewhere, hence their cross product must vanish. Upshot: the sphere has everywhere a unit normal, but it cannot be obtained as a cross product from an everywhere smooth tangential frame. Daniel Lichtblau Wolfram Research

**References**:**Normal Disappear Problem***From:*"gouqizi.lvcha@gmail.com" <gouqizi.lvcha@gmail.com>