Hello,
Right figure (ignore left)
I am having an issue solving this equation for even sided polygons (odd is easy as working out radius is easy)
I have a horizontal line (say 2100mm) (a1 from right image),
and a perpendicular measurement from the large horizontal line to the parallel/opposite side of the polygon (say 500mm) (ai from right image).
E.g.
6 sided polygon (need equation to be n dependent)
length n1 = 2100. length n2 = n3 = n4 = n5 = n6
Arc of n1n2 = n6n1
arc of n2n3 = n3n4 = n4n5 = n5n6
all intersecting point are on circumference of a circle.
How do I work out length n2,n3,n4,n5,n6
and arcs n2n3,n3n4,n4n5,n5n6 OR n1n2,n6n1
with constraints n1 = 2100, n1 & n4 are parallel and perpendicular distance = 500.
Any help would be greatly appreciated.
The length of a chord corresponding to the aperture angle $2\alpha$ is $r\cos\alpha$. So with your data, and denoting $m:=n-1$,
$$\begin{cases}a_1=r\cos(m\alpha),\\a_i=r\cos(\alpha).\end{cases}$$
So you have to solve the trigonometric equation
$$a_i\cos(m\alpha)=a_1\cos(\alpha)$$ also written
$$a_iT_m(\cos(\alpha))=a_1\cos(\alpha)$$ where $T_m$ is a Chebyshev polynomial.
Unfortunately, the equation is polynomial and for $n>4$ you can't spare a numerical resolution.
If you don't know $a_i$ but the distance between $a_1$ and $a_i$, the resolution will be similar, with a more complicated equation.