Re: Urgent Help needed
- To: mathgroup at smc.vnet.net
- Subject: [mg20523] Re: [mg20390] Urgent Help needed
- From: "Richard I. Pelletier" <bitbucket at home.com>
- Date: Sat, 30 Oct 1999 00:13:46 -0400
- Organization: @Home Network
- References: <7v3bei$5t8@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
> Vladimir Tsyrlin wrote: > > > > Given the implcit form of a curve, i.e. F(x,y,z) = 0, do you know how to > > find the curvature of F at a point in 3D space? All the references I have > > assume F is in parametric form and take the standard differential geometry > > approach. > > > > -- > > ************************************************** > > *************Vladimir Tsyrlin ******************* > > vtsyrlin at cs.rmit.edu.au vtsyrlin at ozemail.com.au > > ****************************************************** I have seen one post suggesting that you _meant_ the curvature of a curve in 3-space, and therefore, need 2 equations. Let me assume you _meant_ one equation, and therefore, the curvatures of a surface in 3-space. And rather than try to show you the gory details, let me give you the name of the answer, and one reference. What you want is called _the shape operator_. It is the covariant derivative of a unit normal along a tangent to the surface. The nice thing about the equation F(x,y,z)=0 is that the gradient of F _is_ normal to the surface, so you just need to compute a covariant derivative. Once you have the shape operator, in this case a 2x2 matrix, the principal curvatures are its eigenvalues, so the mean and Gaussian curvatures are their average and product resp. Chapter V of Barrett O'Neill Elementary Differential Geometry (1966 ed.) is entitled _Shape Operators_, and pp. 216-219 work out the details for precisely this problem. It does seem very common to focus on the case where you have a parametric representation, but you don't need one. Vale, Rip -- Multiplication is not commutative before breakfast. Richard I. Pelletier NB eddress: r i p 1 [at] h o m e [dot] c o m