Let $r(t), t\in [0,1]$ be a continuous piecewise $C^1$ curve on the plane where $r(0)=(0,0)$ and $r(1)=(1,0)$. The distance $|r(t)|$ is a non-decreasing function and the distance $|r(t)-(1,0)|$ is a non-increasing function. $r\_m(t), t\in [0,1]$ defined below (a semi vesica piscis) is such a curve.
$$
\begin{equation}
r\_m(t) := \begin{cases}
\left(1-\cos\Big(\frac{2\pi}{3}t\Big),\sin\Big(\frac{2\pi}{3}t\Big)\right), & \forall t\in \Big[0,\frac{1}{2}\Big]; \\
\left(\cos\Big(\frac{2\pi}{3}(1-t)\Big),\sin\Big(\frac{2\pi}{3}(1-t)\Big)\right), & \forall t\in \Big[\frac{1}{2},1\Big].
\end{cases}
\end{equation}
$$


Question 1: Does curve $r\_m(t)$ maximize the length of all admissible curves $r(t)$?
Question 1 has been answered by Pietro Major below in the negative.


Question 2: Suppose $r(t)$ consists of $n$ straight line segments, what is the maximal length curve $r(t)$?



Is vesica piscis a maximal length curve constrained to two points?
The length of these curves is unbounded. For a positive integer $n$ consider a [triangular wave](https://en.wikipedia.org/wiki/Triangle_wave) 
$f\_n:[0,1]\to\mathbb{R}$ with support on $[1/3,2/3]$, making $n$ (isosceles) triangular impulses on $[1/3,2/3]$ with $\|f\_n(x)\|\_\infty=\frac{1}{6\sqrt n}$. The graph of $f\_n$ is a curve satisfying the monotonicity constraint, with length larger than $\sqrt n/3$.


