Help on getting started with this exercise
Another approach to the curvature of a unit-speed plane curve $\gamma$ at a point $\gamma(s_0)$ is to look for the `best approximating circle' at this point. We can then define the curvature of $\gamma$ to be the reciprocal of the radius of this circle.
Carry out this programme by showing that the centre of the circle which passes through three nearby points $\gamma(s_0)$ and $\gamma(s_0 \pm \delta s)$ on $\gamma$ approaches the point $$\boldsymbol \epsilon(s_0) = \gamma(s_0) + \frac{1}{\kappa_s(s_0)} \mathbf n_s(s_0)$$ as $\delta$s tends to zero. The circle $\mathcal C$ with centre $\epsilon(s_0)$ passing through $\gamma(s_0)$ is called the osculating circle to $\gamma$ at the point $\gamma(s_0)$, and $\boldsymbol \epsilon(s_0)$ is called the centre of curvature of $\gamma$ at $\gamma(s_0)$. The radius of $\mathcal C$ is $1/|\kappa_s(s_0)| = 1/\kappa(s_0)$, where $\kappa$ is the curvature of $\gamma$ $-$ this is called the radius of curvature of $\gamma$ at $\gamma(s_0)$.