Poncelet's porism

Abstract

Given two non-degenerate conics $C$ and $D$ in the complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$ , consider the following problem: constructing a closed polygon inscribed in $C$ and circumscribed about $D$. Assuming that the polygon may have self-intersections, a first approach to build such a polygon could be the next one. Take an arbitrary point $p_0 \in C$ and choose $l_0$ one of the two tangent lines to $D$ passing through $p_0$. If the line $l_0$ is not tangent to $C$ there exists a point $p_1 \in {C} \cap l_0 $ different from $p_0$. Then, take $l_1 \neq l_0$ the tangent line to $D$ through $p_1$. In a similar way, $l_1$ must intersect $C$ at a point $p_2 \neq p_1$.