Calculating radii allowing for circular placement of polygonal linkage's joints

Question

Given a planar polygonal linkage defined by a sequence of $n$ hinge joints $(j_0,\,\cdots,\,j_{n-1},j_n = j_0)$ with links of fixed lengths $\lbrace\|j_{k+1}-j_k\|=d_k\ |\ 0\le k\lt n\rbrace$ between adjacent joints,

how can the set of radii be calculated that allow for placing all joints of the linkage on a circle with such a radius, while preserving the lengths of its links?

In view of the answers I want to stress that all joint shall lie on the same circle and, that the linkage is cyclically closed, i.e. its joints and links resemble a connected 2-regular graph.
There is already a question that is related to polygonal linkages: Is the area of a polygonal linkage maximized by having all vertices on a circle? which however was not concerned with determining the radius of the circle on which all joints lie when the area enclosed by the linkage is maximal.

Being able to calculate the smallest and the largest of those radii would be sufficient for my purposes, which I will state in a later edit.

In reply to the answers I'd like to add the following findings:

• a sharp lower bound for the radius on which all joints of a linkage can simultaneously lie, can be found by differentiating the radius of the circumcircle of a triangle w.r.t. one of its side-lengths: $$\frac{\partial\left(\frac{x\,y\,z}{\sqrt{2(x^2y^2+y^2z^2+z^2x^2)-(x^4+y^4+z^4)}}\right)}{\partial x}=0\iff x=\sqrt{z^2-y^2} $$
  which indicates that the circumcircle of a pair of adjacent links is minimal if the longer of the two is the hypotenuse of a right triangle and that the lower bound for entire linkage equals the maximum of all those minima.
• if one divides the link-lengths by the by the sought radius, the resulting values are of the form $2\sin\left(\frac{\varphi_{ij}}{2}\right)$ and function for determining the radius can set up via the trigonometric addition theorems; that function can in turn be converted to rational one by utilizig $$\sin\left(\frac{\varphi_{ij}}{2}\right)=\frac{2t}{1+t^2}\quad \text{and}\quad \cos\left(\frac{\varphi_{ij}}{2}\right)=\frac{1-t^2}{1+t^2}$$

Answer 1

This problem has an algebraic-number solution (set up a system of quadratic equalities between the squared link lengths and the squared distances between hinge points) but with unsolvable Galois groups, so there is not going to be a nice closed formula solution. See: Varfolomeev, V.V.: Galois groups of the Heron–Sabitov polynomials for inscribed pentagons. Mat. Sb. 195 (2004) 3–16 Translation in Sb. Math. 195: 149–162, 2004.

Answer 2

Let the radius be $r$. Of course we need $r \ge d_k/2$ for all $k$. If $\theta_k$ is the angular coordinate of joint $j_k$, with say $\theta_0 = 0$, we have $$ \theta_{k+1} - \theta_k = \pm 2 \arcsin(d_k/(2r))$$ and we need $\theta_{n} = 2 \pi m$ for some integer $m$. By symmetry we may assume the first $\pm$ is $+$, so there are potentially $2^{n-1}$ choices there.

I suspect that for moderately large $n$ it may be best to handle this numerically, though in principle we could use a system of polynomials.

Answer 3

Some simple reductions.

For n=2 and 3 (rigid linkages) , there is a unique answer. Scaling the problem so that the longest link has length one, this results in radii between $1/2$ and $1/s$ where $s^2=3$. Depending on parity, these are also the absolute minimum values of radii possible.

If one is wiling to settle for a near optimum, one can do a "bicycle chain" argument (assuming a really accommodating chain and appropriate pipe, but hey, such assumptions are part of the benefits package of a theoretician), place the chain in a pipe of appropriate parity (or hopeful diameter) and start placing the chain in the pipe so that many links are flexed to touch the inner diameter of the pipe. One may get many joints on the inner surface, and one may be able to manage all but one joint this way on the pipe. The reason for this argument is that one can attempt to predict the discrepancy, and then adjust the pipe toward a more accommodating diameter. In particular, if one can do this while keeping one joint outside of the pipe after each intermediate placement, one can by continuity estimate an exact radius, which for large n should not differ much from the trial radius.

For lengths all the same, one can do stellar patterns by tracing a pattern on a regular ngon (for k relatively prime to n, place link j on point jk mod n). The desired radii should be easily calculated and are probably in the literature (for some interesting value of 'literature'). This gives for this special case a starting set of radii, and one wonders if between two such radii there is an intermediate radius that has a (most likely nonregular) link placement.

A correction to Robert's post: The set of radii can be infinite for some cases with even n: just use an arc. I suspect the set of radii is finite for odd n.

Gerhard "Building A Circle Of Ideas" Paseman, 2020.01.01.