Re: Approx. Curvature from Three points in space

Golabi Doon wrote:

Hello Folks,

I tried to look up information about curvature of curves (i.e. 1-
dimensional manifolds) in n-dimensional spaces but did not find any
relevant information on wikipedia. I would appreciate your help.

Consider a vector valued function F(t) which maps [0,1] to R^n. F(t)
is nothing but representation of of a curve in n-dimensional space (n
might be larger than 3) parameterized by t.

I would like to approximate the curvature of the curve at some
intermediate point t0. We are given three consequative samples of the
curve at F(t0), F(t0-delta), F(t0+delta) where delta is a very small
number. We do not have access to any information about F except these
three successive vectors.

How can I approximate the curvature of F at t0?

Your help would be highly appreciated

Golabi
One may use the Frenet formula

dv1 / ds = k v2 (1)

where v1 and v2 denote unit tangential and normal vectors and s the arc
length. So one deterines in a first step from

dF / dt = ds /dt v1

the arc length s and the tangential unit vector at two positions t0 -
delta/2 and t0 + delta/2 (appoximating the differential quotients by
difference quotients). Similarly one obtains from (1) the unit normal
vector and the curvature k.
.

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