On a property of plane curves

TL;DR

The paper proves that for any continuous curve connecting (0,0) to (1,1) within the unit square, one can find a sequence of points along it with ordered projections on axes that are permutations of each other, for any number of segments.

Contribution

It establishes a new property of plane curves relating to the existence of point sequences with ordered and permuted projections, extending understanding of curve structure.

Findings

Existence of point sequences with ordered projections for any number of segments.

Projections on axes are permutations of each other.

Applicable to all continuous curves connecting (0,0) and (1,1).

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 $A_i$, $0\leq i \leq n+1$, on $\gamma$ such that $A_0=(0,0)$, $A_{n+1}=(1,1)$, and the sequences $\pi_1(\overrightarrow{A_iA_{i+1}})$ and $\pi_2(\overrightarrow{A_iA_{i+1}})$, $0\leq i \leq n$, are positive and the same up to order, where $\pi_1,\pi_2$ are projections on the axes.