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

Solution 1:

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:

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.

Solution 2:

As far as I understand the questions, you mean by the "Reuleaux-method" that you take a polygon and add circular arcs to obtain a convex body of constant width. Assuming this, the answer to question 1 should be negative:

In the book "How round is your circle" there is another method to construct convex bodies of constant width: See on the How round is your circle webpage, the example titled "Half a convex shape". In this example, half an ellipse is completed to a convex body. So for curvature reasons this body can not be generated by the "Reuleaux-method".

Solution 3:

It depends on what you mean by the number of sides of a curve.

If $k$ is an odd integer, and $p(t) = a\cos^2(kt/2) + b$ then $$ \begin{cases} x(t) = p(t)\cos t - p'(t)\sin t \\ y(t) = p(t)\sin t + p'(t)\cos t \end{cases} $$ where $0 \le t \le 2\pi$ is a curve of constant width. Choosing $k=3$, $a=3$, $b=1$ you end up with

How many sides does that curve have? More details can be found in this document

Solution 4:

Concerning your first question: It is easy to produce curves of constant width without corners. See here, in particular the upper half of page 107:

http://www.math.ethz.ch/~blatter/Konstante_Breite.pdf

This paper is in German. The relevant statement on page 107 is the following:

Put $${\bf u}(\phi):=(\cos\phi,\sin\phi),\qquad {\bf u}'(\phi):=(-\sin\phi,\cos\phi)$$ and consider the parametric representation $$\gamma:\quad {\bf z}(\phi):= p(\phi){\bf u}(\phi)+p'(\phi){\bf u}'(\phi)\qquad(\phi\in{\mathbb R}/(2\pi))\ ,\tag{1}$$ where the function $p\in C^2$ fulfills the following conditions: $$p(\phi)+p''(\phi)>0,\qquad p(\phi)+p(\phi+\pi)\equiv b\ .\tag{2}$$ Then $\gamma$ is a closed convex curve of constant width $b$, and $(1)$ is a regular parametrization of $\gamma$. In fact from $(1)$ one computes $${\bf z}'(\phi)=\bigl(p(\phi)+p''(\phi)\bigr){\bf u}'(\phi)\ ,$$ so that $$|{\bf z}'(\phi)|=p(\phi)+p''(\phi)>0,\qquad\arg\bigl({\bf z}'(\phi)\bigr)=\phi+{\pi\over2}\ .$$ It follows that the radius of curvature $\rho$ along $\gamma$ is given by $$\rho(\phi)=p(\phi)+p''(\phi)\ .$$ When $p$ is sufficiently smooth, then so is $\rho$, whereas in the case of curves obtained by the "Reuleaux method" the radius of curvature is piecewise constant and has jump discontinuities at the hinges of the construction.

On the other hand the curves obtained by the "Reuleaux method" have a representation of the form $(1)$ as well. The function $p$ is then only continuous and piecewise of the form $$p(\phi)=\rho_i+a_i\cos\phi+b_i\sin(\phi)\qquad(\alpha_{i-1}<\phi<\alpha_i)\ .$$

$C^\infty$-functions $p$ fulfilling the conditions $(2)$ are, e.g., the trigonometric polynomials $$p(\phi):={b\over2}+\sum_{j=1}^N\biggl(a_j\cos\bigl(2j+1)\phi\bigr)+b_j\sin\bigl((2j+1)\phi\bigr)\biggr)\ ,$$ where the $|a_j|$ and $|b_j|$ are sufficiently small.