This is similar to Parameterizing equilateral polygons but for 3 dimensions.
I have $n$ points in $\mathbb{R}^3$ ($n \geq 3$), labeled $1$, $2$, ..., $n$. Point $1$ is unit distance from point $2$, which is unit distance from point $3$, ..., and point $n$ is unit distance from both point $n-1$ and point $1$. WLOG, I set point $1$ to be the origin and point $n$ to be $(1, 0, 0)$ by translating and rotating my set of points as necessary.
This gives me a set of $n - 1$ conditions (one for each pair of numbering-adjacent points other than $1$ and $n$) on a set of $3(n - 2)$ variables, namely the coordinates for the $n-2$ points $2$ through $n - 1$.
Based on what I know about how many variables a single equation can/will eliminate, this should require less than $3(n-2)$ parameters to specify a group of these points, but cannot possibly take less than $3(n - 2) - (n - 1) = 2n - 5$ parameters. How many parameters do I actually need (in other words, how many degrees of freedom does this system have) and how do I retrieve the points from a given set of parameters?
Improved upper or lower bounds on the number of degrees of freedom, or values for special cases which might otherwise deviate from a general pattern, would also be appreciated. So far I know the following values:
n | degrees of freedom
---+--------------------
3 | 1
4 | 3
5 | 5
6 | 7
but these each seem to have relatively independent ways of obtaining the points from a parameter set which I cannot see a way to generalize to more points.