## The curvature and torsion: how to distinguish the shape of a curve

### Osculating circle

The osculating circle of a curve $$C$$ at point $$P=\left(x\left(t_{0}\right),y\left(t_{0}\right)\right)$$ is the tangent circle to the curve at $$P$$ that best approximates the curve in the neighbourhood of $$P$$; more precisely, it is the circle that has the same tangent in $$P$$ as $$C$$ as well as the same curvature. Therefore, the radius of such circle is equal to the inverse of the curvature at $$P$$, that is, $$r=\left|\frac{1}{k(t)}\right|$$.