curvature of a curve

hi all,

it might be a naive question, but i’m trying to find the curvature of a curve at each point on the curve.

i’m not sure i know how to find it. i looked online at wikipedia/mathworld, but it didnt make too much sense.

can someone please tell me how to find the curvature? thanks.


Here is an idea which may not work:
For each point, fit a circle to the point and two neighbor points. The curvature will be 1 divided by the radius of the fitted circle.

I like the circle idea. That’s related to the circle of curvature, where circle and curve share a common tangent

r = 1 / k,


k = 1 / r.

But to use this you need to know the radius of curvature. For an sphere/ellipse this would be a great way to go I think.

Another way to express curvature from calculus is

k = ||V x A|| / ||V||^3,

where V and A are the velocity and acceleration vectors along the curve at some instant in time. For a set of points along the curve, you could use the definition of the derivative

r’(t) = V
~ (r(t+h) - r(t)) / h
~ (Pi+1 - Pi) / h,

r’’(t) = A
~ (r’(t+h) - r’(t)) / h
~ (r(t+2h) - 2 r(t+h) + r(t)) / h^2
~ (Pi+2 - 2 Pi+1 + Pi) / h^2

for consecutive points Pi, Pi+1, Pi+2 along the curve.

Of course if this is for a curve that can be differentiated directly, then you should probably do so.