Intersection points of planar curves can be computed

TL;DR

The paper proves that for two continuous paths in a square with specific endpoints, one can algorithmically compute intervals where the paths intersect, providing a constructive approach to intersection points.

Contribution

It introduces a method to compute intervals of intersection for continuous paths with given boundary conditions in the unit square.

Findings

Intervals of intersection can be effectively computed.

Paths with specified endpoints necessarily intersect.

The method is constructive and algorithmic.

Abstract

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