Science Fair Project Encyclopedia
Evolute
In the differential geometry of curves, the evolute of a curve is the set of all its centers of curvature. It is equivalent to the envelope of the normals.
If r is the curve parametrised by arc length (i.e. | r'(s) | = 1; see natural parametrization) then the center of curvature at s is
 |
Such parametrisation is usually between difficult and impossible, but it's still feasible to access r". If x is any (reasonably differentiable) parametrisation, and s gives arc length over the same parameter, then the desired r would give r(s(t)) = x(t) which if differentiated twice gives
- r'(s(t))s'(t) = x'(t)
- r''(s(t))s'(t)2 + r'(s(t))s''(t) = x''(t)
which we rearrange to
Recognising that
- s'(t) = | x'(t) |
eliminates the need to know s itself, thus eliminating the integration in which the analytic impossibilities lie.
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details