On a property of plane curves

Abstract

Let $\gamma :[0,1]\to [0,1{]}^2$ be a continuous curve such that $\gamma (0)=(0,0)$, $\gamma (1)=(1,1)$, and $\gamma (t)\in (0,1{)}^2$ for all $t\in (0,1)$. We prove that, for each $n\in \mathbb{N}$, there exists a sequence of points $A_i$, $0\le i\le n+1$, on $\gamma$ such that $A_0=(0,0)$, $A_{n+1}=(1,1)$, and the sequences ${\pi }_1(\overrightarrow{A_i{A}_{i+1}})$ and ${\pi }_2(\overrightarrow{A_i{A}_{i+1}})$, $0\le i\le n$, are positive and the same up to order, where ${\pi }_1,{\pi }_2$ are projections on the axes.

8 pages