I have the following parameterization of an Ellipse:
$$x=a \cos (\omega t),\space \space \space y=b\sin(\omega t),\space \space \space \omega=const., \space \space \space a,b \in \Bbb R$$
I want to determine the curvature $\color{blue}{\kappa}$ and the radius of curvature $\color{red}{\rho}$
I know that the curvature is the absolute value of the unit-tangent vector $\hat{e}_t$ differentiated with respect to arc-length:
$$\color{blue}{\kappa}=\lvert \frac{d \hat{e}_t}{ds}\lvert$$
The radius of curvature is:
$$\color{red}{\rho}=\frac{1}{\color{blue}{\kappa}}$$
My questions:
Am I right in saying that curvature just tells me how much my curve bends at a certain point?
How do I calculate $\frac{d \hat{e}_t}{ds}$?
I know how to get $\hat{e}_t$:
$$\dot{\vec{r}}(t)=\begin{bmatrix}-a \omega \sin(\omega t) \\b \omega \cos(\omega t)\end{bmatrix} \implies \frac{\dot{\vec{r}}(t)}{\lvert \dot{\vec{r}}(t)\rvert}=\frac{1}{\sqrt{a^2\omega^2\sin^2(\omega t)+b^2\omega^2\cos(\omega t)}} \begin{bmatrix}-a \omega \sin(\omega t) \\b \omega \cos(\omega t)\end{bmatrix}=\hat{e}_t$$
However, how do I differentiate this with respect to arc-length $s$?
For your first question, the “right”ness depends on how precise you want to define “bends”. But if you look at the expression $\left|\frac{d\hat e_t}{ds}\right|$, you can interpret it as the speed at which the unit tangent vector rotates as you move uniformly along the curve.
For the second, the arc-length functions along ellipses are not integrable in elementary terms (they are the pathway to the study of elliptic functions). Instead, you might find one of the equivalent formulas for curvature more helpful: $$ \kappa(t) = \frac{\left|\dot{\hat e_t}(t)\right|}{\left|\dot{\vec r}(t)\right|} = \frac{\left|\dot{\vec r}(t) \times \ddot{\vec r}(t)\right|}{\left|\dot{\vec r}(t)\right|^3} $$