Let $p_1,\dots,p_n$ be a collection of points in the plane $\mathbb{R}^2$ and let $a$ be a positive number such that $a<1$. Is there a good numerical algorithm to find points $x_1,\dots,x_n$ in the plane such that $$\|x_i - x_{i+1}\| = a(\|x_i - p_i\|+\|p_i-x_{i+1}\|)$$ for all $i$, where we use $n+1\equiv 1$ to simplify notation? A picture for $a=3$ is attached, in which the black dots indicate the $p_i$'s, the blue segments indicate the segments from $x_i$ to $x_{i+1}$, and the red segments indicate the $\|x_i - p_i\|+\|p_i-x_{i+1}\|$.

Let $p_1,\dots,p_n$ be a collection of points in the plane $\mathbb{R}^2$ and let $a$ be a positive number such that $a<1$. Is there a good numerical algorithm to find points $x_1,\dots,x_n$ in the plane such that $$\|x_i - x_{i+1}\| = a(\|x_i - p_i\|+\|p_i-x_{i+1}\|)$$ for all $i$, where we use $n+1\equiv 1$ to simplify notation? A picture for $a=1/3$ is attached, in which the black dots indicate the $p_i$'s, the blue segments indicate the segments from $x_i$ to $x_{i+1}$, and the red segments indicate the $\|x_i - p_i\|+\|p_i-x_{i+1}\|$.

Let $p_1,\dots,p_n$ be a collection of points in the plane $\mathbb{R}^2$ and let $a$ be a positive number such that $a<1$. Is there a good numerical algorithm to find points $x_1,\dots,x_n$ in the plane such that $$\|x_i - x_{i+1}\| = a(\|x_i - p_i\|+\|p_i-x_{i+1}\|)$$ for all $i$, where we use $n+1\equiv 1$ to simplify notation?

Let $p_1,\dots,p_n$ be a collection of points in the plane $\mathbb{R}^2$ and let $a$ be a positive number such that $a<1$. Is there a good numerical algorithm to find points $x_1,\dots,x_n$ in the plane such that $$\|x_i - x_{i+1}\| = a(\|x_i - p_i\|+\|p_i-x_{i+1}\|)$$ for all $i$, where we use $n+1\equiv 1$ to simplify notation? A picture for $a=3$ is attached, in which the black dots indicate the $p_i$'s, the blue segments indicate the segments from $x_i$ to $x_{i+1}$, and the red segments indicate the $\|x_i - p_i\|+\|p_i-x_{i+1}\|$.