Radius of Curvature

**CrashDummy11** · April 11th 2010, 12:12 PM

How can I calculate the radius of curvature of a 3D curve that is parameterized in the form:
x(t), y(t), z(t)?

**maddas** · April 11th 2010, 12:17 PM

It is the reciprocal of the curvature, whose formula can be found at Curvature - Wikipedia, the free encyclopedia

Essentially, at a point p, one takes the limiting value of the radius of the circle passing through p-dp, p, and p+dp.

**Plato** · April 11th 2010, 12:22 PM

Originally Posted by CrashDummy11

How can I calculate the radius of curvature of a 3D curve that is parameterized in the form:
x(t), y(t), z(t)?

If $r(t)=<x(t),y(t),z(t)>$ then the curvature is: $\kappa = \left\| {\frac{{dT}}<br /> {{ds}}} \right\| = \frac{{\left\| {r' \times r''} \right\|}}<br /> {{\left\| {r'} \right\|^3}}$

**CrashDummy11** · April 11th 2010, 12:26 PM

Thats a really ugly formula but it was what I was looking for. Thanks!

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