Curvature of a curve PROOF help?

Question

I need help proving how the curvature of a curve given by the vector function r is k(t) = |r'(t) X r''(t)| / |r'(t)|^3.

The proof starts out like this... (btw, T = unit tangent vector)

Since T = r' / |r'| and |r'| = ds/dt
r' = |r'|T = (ds/dt) T
Apply product rule you get:

r'' = (d^2s/dt^2)T + (ds/dt)T'

I don't know what to do after. Any help?

Answer 1

Okay, we have:

||r'(t)×r''(t)|| / ||r'(t)||³

First note that r'(t) = T(t) * ||r'(t)||, so we have:

||(||r'(t)||*T(t))×r''(t)|| / ||r'(t)||³

Note that the cross product is linear in both arguments, so this is simply:

|| ||r'(t)||*(T(t)×r''(t))|| / ||r'(t)||³

And since ||cv|| = |c|*||v||, we can pull the scalar out, to obtain:

||T(t)×r''(t)||*||r'(t)||/ ||r'(t)||³
||T(t)×r''(t)|| / ||r'(t)||²

Now, consider that as you found, r'(t) = s'(t)*T(t), so we have that r''(t) = s''(t)*T(t) + s'(t)*T'(t). Thus we have:

||T(t)×(s''(t)*T(t) + s'(t)*T'(t))|| / ||r'(t)||²

Using the bilinearity of the cross product, we have:

||s''(t)*(T(t)×T(t)) + s'(t)*(T(t)×T'(t))|| / ||r'(t)||²

Of course, the cross product of any vector with itself is zero, so in particular, T(t)×T(t) = 0. Thus we have:

||s'(t)*(T(t)×T'(t))|| / ||r'(t)||²

And pulling the scalar out:

||T(t)×T'(t)|| * |s'(t)| / ||r'(t)||²
||T(t)×T'(t)|| * ||r'(t)|| / ||r'(t)||²
||T(t)×T'(t)|| / ||r'(t)||

Now we just look at ||T(t)×T'(t)||. This is just ||T(t)|| * ||T'(t)|| * sin θ, where θ is the angle between T'(t) and T(t). Of course, T'(t) is perpendicular to T(t), so sin θ = 1. Similarly, since T(t) is defined to be the _unit_ tangent vector, ||T(t)|| = 1. Thus ||T(t)×T'(t)|| = ||T'(t)||, so we have:

||T'(t)|| / ||r'(t)||

Which is the definition of curvature. So we are done.

Answer 2

First draw a diagram of a curve and a element P on the curve at which the tangent makes an attitude phi with the x axis. Then circulate a distance ds alongside the curve to he element Q. The tangent at element Q makes an attitude phi + dphi with the x axis. Now draw a element O on the concave factor of the curve. O is the centre of curvature. OPQ is a sector of a circle whose radius is r, and the arc length PQ=ds. the attitude POQ is dphi. subsequently we've r dphi = ds! or dphi/ds= a million/r that's the curvature, defined by using fact the reciprocal of the radius of curvature.