Curvature of Ellipse We all know that the curvature of a circle is defined by the equation
$$k=\frac{1}{r}$$

What about ellipses?

In terms of major axis $a$, minor axis $b$, $x$ and $y$, what is the curvature of an ellipse?

Thanks lots!
$$x=a\cos(t),y=b\sin(t)$$
$$\dot x=-a\sin(t),\dot y=b\cos(t)$$
$$\ddot x=-a\cos(t),\ddot y=-b\sin(t)$$

$$\kappa=\frac{\dot x\ddot y-\ddot x\dot y}{(\dot x^2+\dot y^2)^{3/2}}=\frac{ab}{(a^2\sin^2(t)+b^2\cos^2(t))^{3/2}}=\frac{ab}{((\frac ab y)^2+(\frac bax)^2)^{3/2}}$$
The second answer is the maximum principal curvature, as explained on pages 71-76 of "Vector and Tensor Analysis" by Harry Lass, McGraw-Hill (1950).  The denominator is G^{3/2}, where G is the third term in the first fundamental form for the Ellipse.  The minimal principal curvature is c / ( a \sqrt{G}).