Given a curve $X$ in the complex plane $\mathbb{C}^{2}$ and a smooth point $p\in X$ , an osculating conic to $X$ at $p$ generalizes the notion of a tangent line, and of an osculating circle. Beyond degree two, one may consider more general osculating spaces of $X$ at $p$ , where the degree and order of tangency are prescribed. This paper lays out a computational framework for computing such spaces. We work in local coordinates $y=f(x)$ for $X$ near $p$ and assume that $p$ is the origin. Then, the computation of an osculating space amounts to solving a linear system of equations in the derivatives of $f(x)$ at $x=0$ . We showcase our work by producing a formula for the osculating conic of a generic analytic curve and exhibit several specific examples with interesting properties.
