Re: Surface Normal
- To: mathgroup at smc.vnet.net
- Subject: [mg55183] Re: Surface Normal
- From: Roland Franzius <roland.franzius at uos.de>
- Date: Wed, 16 Mar 2005 05:35:58 -0500 (EST)
- Organization: Universitaet Hannover
- References: <d15s2g$9k2$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
gouqizi.lvcha at gmail.com schrieb:
> 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
Vector of torus surface point
X =
{(10 + 5Cos[v])Cos[u],
(10 + 5Cos[v])Sin[u],
5Sin[v]}
ParametricPlot3D[X, {u, 0, 1.5 Pi}, {v, 0, 1.5 Pi}]
Local tangent vectors
{du,dv}=D[X,#]&/@{u,v}//FullSimplify
{{-5 (2 + Cos[v]) Sin[u], 5 Cos[u] (2 + Cos[v]), 0},
{-5 Cos[u] Sin[v], -5 Sin[u] Sin[v], 5 Cos[v]}}
local normal vector (provided du!=0, dv!=0)
dn=Cross[du,dv]//FullSimplify
{25 Cos[u] Cos[v] (2 + Cos[v]), 25 Cos[v] (2 + Cos[v]) Sin[u],
25 (2 + Cos[v]) Sin[v]}
{du.dv,dn.du,dn.dv }
{0,0,0}
Scale factors
{du.du, dv.dv, dn.dn} // FullSimplify
{25((2 + Cos[v]))^2, 25, 625 ((2 + Cos[v]))^2})
Metric has no singularities in this parametrisation.
--
Roland Franzius