Curvature of Ellipse

Question

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!

Answer 1

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

$$\kappa=\frac{\dot x\ddot y-\ddot x\dot y}{(\dot x^2+\dot y^2)^{3/2}}=\frac{ab}{(a^2\sin^2(t)+b^2\cos^2(t))^{3/2}}=\frac{ab}{((\frac ab y)^2+(\frac bax)^2)^{3/2}}$$

Answer 2

You can find the curvature in parametric form here, tag 11.