On a property of plane curves
Abstract
Description
Let $γ: [0,1] \to [0,1]^2$ be a continuous curve such that $γ(0)=(0,0)$, $γ(1)=(1,1)$, and $γ(t) \in (0,1)^2$ for all $t\in (0,1)$. We prove that, for each $n \in \mathbb{N}$, there exists a sequence of points $A_i$, $0\leq i \leq n+1$, on $γ$ such that $A_0=(0,0)$, $A_{n+1}=(1,1)$, and the sequences $π_1(\overrightarrow{A_iA_{i+1}})$ and $π_2(\overrightarrow{A_iA_{i+1}})$, $0\leq i \leq n$, are positive and the same up to order, where $π_1,π_2$ are projections on the axes.
8 pages
8 pages