I have two software solutions. One from ‘Will Hunting’ in C++/Qt and one from Bruce Pearson in Python/WxWidgets. Both use heuristic approaches.

Bruce Pearson’s code does pretty well for up to N=27. But the circles overlap beyond that.

Will Hunting’s does better with circle placement over a wider range of N. It also had the advantage of being in C++, making it easier to incorporate into PerfectTablePlan. But there is a visible gap between circle 0 and circle N-1. if this error could be spread equally between all the circles I think it might be good enough.

GJ Peltenburg has sent me some quartic solver code and Christopher Wells has filled in more details on an analytic approach. But every time I try to write out the quartic equation for x2 in terms of x1 and y1 my brain starts to bleed. ;0)

I think a lot got real close to the solution but you are focusing on the wrong ellipse.

You first do what the original poster did and set C = N*D.

That gives you the ellipse of the chairs. Then use the equation of an ellipse and the distance equation for two points.

That is two equations and two unknowns (the new X and Y position).

You will need to pick the location of the first chair (circle origin), but then after that those two equations will give you two results, one in each direction around the ellipse.

After you have used that chair ellipse, shrink both sides by D and you have the table ellipse.

For small values of N (say 6 or less), the use of C = N*D does not work too well, so I would suggest scaling D by a factor of 2 or so in all calculations. That will allow the chairs to not have to touch.

You first do what the original poster did and set C = N*D.

That gives you the ellipse of the chairs. They use the equation of an ellipse and the distance equation for two points.

That is two equations and two unknowns (the new X and Y position).

You will need to pick the location of the first chair (circle origin), but then after that those two equations will give you two results, one in each direction around the ellipse.

After you have used that chair ellipse, shrink both sides by D and you have the table ellipse.

For small values of N (say 6 or less), the use of C = N*D does not work too well, so I would suggest D by a factor of 2 or so in all calculations. That will allow the chairs to not have to touch.

Thanks. I don’t remember seeing *that* in school.

> seat centers fall on the ellipse a+r/b+r where r is the radius of the seat circle

I think those seats will overlap the ellipse at all except the 4 cardinal points (and that this will be most visible when a is much smaller than b, and the seats are sized such that there are about 8 or 12 seats around the table).

I’ve been placing seat centers such that for a table with major/minor axes of a/b, the seat centers fall on the ellipse a+r/b+r where r is the radius of the seat circle. The seats seem to be tangent to the table ellipse, but maybe there’s some error I’m not seeing in the resulting graphics (I’m using Genius … it’s kind of like Matlab).

Here’s why:
http://planetmath.org/encyclopedia/QuarticFormula.html

NB: at one point I implemented that in code, but I suspect it requires complex numbers to work, because the results I got were completely wrong.

Also the arc length between each pair of chairs isn’t constant (the length pf the straight line between the centres of each pair of chairs is constant, but the shape’s curvature isn’t constant and the arc length is greater betwen pairs where the shape’s curvature is larger).