Intersection points of planar curves can be computed

Abstract

Consider two paths $\varphi ,\psi :[0;1]\to [0;1{]}^2$ in the unit square such that $\varphi (0)=(0,0)$, $\varphi (1)=(1,1)$, $\psi (0)=(0,1)$ and $\psi (1)=(1,0)$. By continuity of $\varphi$ and $\psi$ there is a point of intersection. We prove that from $\varphi$ and $\psi$ we can compute closed intervals $S_{\varphi },S_{\psi }\subseteq [0;1]$ such that $\varphi (S_{\varphi })=\psi (S_{\psi })$.