Geometry – Must a Curve of Constant Width Be Generated with an Odd Number of Sides?

geometry

From what I have seen (and to some extend read), a curve of constant width generated from a polygon with an even number of sides is not possible.
Wikipedia cites an Oxford University paper when it says

Curves of constant width can be generated by joining circular arcs centered on the vertices of a regular or irregular convex polygon with an odd number of sides (triangle, pentagon, heptagon, etc.)

It says curves of width can be generated with polygons with an odd number of sides, it does not explicitly rule out even numbers of sides. I suppose this can be asked in two questions:

Question 1 Are all curves of constant width generateable with the "Reaulaux Method"?

It seems that every document I read mentions curves of constant width generated with circular arcs, which does make sense. Is this the only possible method? Examples can be seen on the Wikipedia Article. I'm going to say that a circle can be generated with this method as well. What is the justification for the correct answer?

Question 2 Must a curve of constant width be generated with an odd number of sides?

This question is harder to answer if the answer to question 1 is no, so if that is the case this can be left out (unless it is still easy to answer, that is). Mrf's answer provided insight that could be lead to a proof that regular polygons used to generate with the "Reulaux Method" must have an odd number of sides, but is this also true for irregular polygons? I'm guessing yes but what I need is a proof

Best Answer

Just for fun, I made a little animation of the half-convex example linked to by @Alexander Schmeding. The upper half is an ellipse with minor-to-major axis ratio ranging from $b = 1/2$ to $b = \sqrt{2}$. The curve can be parametrized as $$f_b(\theta) = \begin{cases} (\cos \theta, b \sin \theta), & 0 \le \theta \le \pi, \\ {\displaystyle \frac{2 (b \cos \theta, \sin \theta)}{\sqrt{b^2 \cos^2 \theta + \sin^2 \theta}}} - (\cos \theta, b \sin \theta), & \pi < \theta \le 2\pi. \end{cases}$$ In Mathematica, we can generate an interactive plot with

F[t_, b_] := Piecewise[{{{Cos[t], b Sin[t]}, 0 <= t < Pi},
             {2 {b Cos[t], Sin[t]}/Sqrt[(b Cos[t])^2 + Sin[t]^2]
               - {Cos[t], b Sin[t]}, Pi <= t <= 2 Pi}}]

Manipulate[Show[ParametricPlot[F[t, b], {t, 0, 2 Pi}, 
                PlotRange -> {{-1, 1}, {-1.5, 1.5}}, PlotStyle -> Black], 
           Graphics[Flatten[{Opacity[0.5], {Hue[#/Pi], 
           Line[{F[#, b], F[# + Pi, b]}]} & /@ (Range[n] Pi/n)}]]]
       {{b, 1}, 1/2, Sqrt[2]}, {n, 1, 75, 1}]

And we get this:

enter image description here

What I found really interesting is how the curve looks like it's the same near the extremes of the animation, but it obviously isn't so from the definition of $f$ itself. Bonus points if you can parametrize the envelope of normals for $b \in [1/2, \sqrt{2}]$. And more bonus points if you can compute the enclosed area.

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