Osculating Curves

An osculating curve to $f(x)$ at $x_0$ is tangent at that point and has the same Curvature. It therefore satisfies

$$y^{(k)}(x_0)=f^{(k)}(x_0)$$

for $k=0$, 1, 2. The point of tangency is called a Tacnode. The simplest example of osculating curves are $x^2$ and $x^4$, which osculate at the point $x_0=0$.

See also Tacnode