Concepts Related to Constant Width

Abstract

A polytope P is circumscribed about a convex body $\phi \subset \mathbb {E}^n$ if $\phi \subset P$ and each facet of P intersects $\phi$; i.e., every facet of P is contained in a support hyperplane of $\phi$. A polytope P is inscribed in the convex body $\phi$ if $P\subset \phi$ and each of its vertices belongs to ${{\,\mathrm{\mathrm {bd}}\,}}\phi$.

Where there is matter, there is geometry!

Johannes Kepler

Appendices

Notes

Rotors in Polygons, Polytopes, and Related Topics

As already mentioned at the beginning of this section, the family of rotors in the sense described here is much larger than that of constant width sets, which are known as rotors in n-dimensional cubes or parallelotopes all whose pairs of parallel facets lie in hyperplanes at equal distance apart (in the plane yielding rhombi). Thus rotors can be regarded as a natural extension of constant width sets, and so we discuss here also the literature on rotors in a wider sense. Early results and references are summarized in [160, pp. 139–140]. We have also to mention the nice survey [431]; it is excellently complemented by the related material in the book [1204], also discussing rotors of regular triangles (see [1186], [1024] and below for this special case). In addition, we refer to [272, A 14].

It is well known that rotors are also used to drill holes with nonspherical (e.g., polygonal) cross sections. For further references on this topic see the Notes of Chapter 18. Surveys containing material about rotors as (extensions of) sets of constant width are [238], [527], [460], and [461]. Section 5.7 of Groemer’s book [464] is entirely devoted to rotors of 3- and higher dimensional polytopes, repeating results from [1034], but also going further, with methods based on Fourier series and spherical harmonics.

To start with the planar situation, we have to go back to Hurwitz [566], Meissner [815] and Fujiwara [379]. In [566] and [815] (see also [886]), with the help of Fourier series, constant width sets and the class of rotors of regular polygons are investigated; the latter is obtained as the intersection of two curve classes with related definitions. (Some of the results in [815] were rediscovered in [366] and [251].) In [379], the regular convex k-gons are fixed, which allow noncircular rotors; see also [605] and [1025]. Closer to constant width curves, Goldberg (see [431] and [425]) considered rotors composed of circular arcs, also of even numbers of arcs. His results simplify the discussion which started with [379]. In [425], symmetry properties of rotors play an essential role. For $k = 3$, Weissbach (see [1186] and [1187]) investigated the problem of finding rotors of regular k-gons with minimal area. Using an idea by Eggleston, he showed in [1186] that the intersection of two congruent circles, whose midpoints are separated by a distance of $\sqrt{3}$ times their radius $r>0$, is the rotor of minimal area in a regular triangle of side length r, nicely reproving an old result from [384] also presented in [1204]; see again [728] and of course Theorem 14.1.2. For arbitrary k this problem was also attacked by Focke [366] und Klötzler [636], in the latter paper with an application of Pontrjagin’s maximum principle. Further extremal problems for rotors of regular polygons have been settled in [367], where largest possible portions of the polygon boundary reached by the rotor, or smallest possible distance to the vertices of the polygon, are taken into consideration (see also [368]).

Closely related and suitably using methods from convex optimization, the authors of [369] investigate such rotors with sharpest supporting cones. In [1188] it is proved that the “best” region of constant width for drilling out the largest hole in a rhombus is the Reuleaux triangle. In [418], it is investigated which constant width sets minimize (according to different measures) the region left close to a particular vertex of a specified rhombus. Besides the Reuleaux triangle, other constant width sets will also reach the same extremal effect. More recent references related to this topic of extremal rotors are [84] and [989], using, for instance, semi-infinite simplex algorithms. In [83] optimal control theory is used to prove that rotors in regular polygons having minimal area consist of a finite union of arcs of circles. Moreover, the radii of these arcs are fixed, and based on this a constructive characterization of an optimal rotor with minimal area is given. It is shown that such a minimizing rotor has itself to satisfy a regularity property, which proves a long-standing conjecture formulated in [428]. Perimeter optimization combined with rotors was investigated in [839], and Lillington [728] derived extremal properties of rotors of regular hexagons and studied regular triangles in a related way. Fujiwara [381] investigated rotors in triangles which are bounded by circular arcs. Schaal [1024] showed that if any angle of a triangle is irrational, then it cannot have a noncircular rotor; and if all angles are rational, then there are infinitely many such curves, all having the same length as the circular rotor. Croft [271] gave two nice characterizations of closed convex curves in which regular triangles can revolve. In [251], special convex curves are characterized via constancy of the sum of radii of curvatures at touching points with circumscribed regular k-gons; this is done in terms of the Fourier coefficients of the radii of curvature functions, and the case $k=3$ characterizes rotors of regular triangles which are of constant width, see also Theorem 17.1.2. And in [421], the orbit of the center of mass of Reuleaux triangles rotating in squares and analogous situations for Reuleaux k-gons in $(k+1)$-gons are investigated. It is shown that for $k>3$ a Reuleaux k-gon cannot revolve in a regular $(k+1)$-gon P such that it has one point in common with each side of the polygon P. A compact convex set C is called an (m|k)-orbiform if each circumscribed and equal-angled n-gon has the same perimeter and the rotation symmetry of order k / m (m being not larger than k). (With $m=1$ this definition yields Euler’s classical definition of orbiforms; see [323].) Weissbach [1189] presents constructions of (m|k)-orbiforms. It turns out that the case of odd k is more difficult than the even case. In [1025], Fourier series are used to describe the curve of the midpoint of regular polygons rotating around their fixed rotor, thus getting support functions and parametrizations of such generalizations of constant width curves (see also [874]). Technical applications of such curves, such as practical roundness tests, are discussed in [1026]. We mention here that the rotor of the famous Wankel engine (see Chapter 18) does not have to be a rotor as considered here, and therefore not exactly a Reuleaux triangle. We refer to [556], [875], [876], [877], and [889] for discussions of the kinematics of the Wankel motor.

A general topological result on rotors in convex polygons is derived in [88]: it is proved there that the hyperspace of all rotors (respectively, all smooth rotors) in a regular polygon is homeomorphic to the Hilbert cube (respectively, the separable Hilbert space), see also Section 16.1. For further concepts related to planar rotors, we refer to [361] (about curves that have polygons as rotors), [425] (about special curves, such as hypocycloids of three cusps and Ribaucour curves, considered as rotors in regular triangles and squares), [429] (rotors of arrangements of circles whose vertices are those of the fixed polygon), [430] (rotors in connection with closed sets of minimum area containing at least one point of a square lattice in any position), and [607] (characterizing the superset of constant width curves all of whose representatives have the property that all circumscribed rectangles are squares).

The study of rotors in not necessarily regular polygons started with [379]; more recent presentations and results are given in [1024], [517], [605], and [366]. We also refer to the discussions and references in [238], [1034], and [527], even taking unbounded polygonal domains into consideration (see especially under 2.3 in the survey [461]). Going into another direction, Nakajima [886] considered the problem of characterizing strictly convex domains with the property that under rotation in a wedge (to which it osculates in any position) the length of the inner boundary curve between the two contact points is constant. He also studied (higher dimensional) versions of this, and a special case can be found in [659]. Another related paper is [1105].

Rotors have also been considered in non-Euclidean frameworks. Having spherical polygons in mind, Goldberg (see [426] and [424]) presented a kinematical construction of spherical rotors. Without giving a general formula for them, he described a method of successive approximation which would suffice for practical applications. For generalized Minkowski planes (having convex unit balls not necessarily symmetric about the origin, also called gauges), in [414] the self-perimeters and other quantities of rotors of triangles, their polars and constant width sets are estimated.

Figure 17.14

An intermittent rotor

Now we turn to higher dimensions (referring again to Euclidean geometry). Using representations in terms of support functions, Meissner (see [816] and [819]) succeeded in determining all 3-dimensional regular polytopes with nonspherical rotors (these are the tetrahedron, the cube, and the octahedron); see also [431] and, weakly related, [886] as well as Section 17.1.3. Using spherical harmonics, Schneider [1034] was able to solve this question completely in $n > 2$ dimensions, also for the not necessarily bounded case. The results and proof details from [1034] are nicely represented in Section 5.7 of the monograph [464], where higher dimensional rotors are extensively discussed. The author propagates Schneider’s approach as one of the most sophisticated applications of spherical harmonics in geometry. Another application of spherical harmonics for the construction of special rotors is given in [349] (this paper refers to the existence of constant width bodies with certain symmetry properties). In [441], a stability result is proved as a far-reaching extension of Minkowski’s theorem on projections of bodies of constant width, and the equality case of a related inequality yields an extremal characterization of rotors. In [413], rotors of a regular simplex in n-dimensional Euclidean space and its respective polar bodies are taken into consideration. An inequality for the volume of these polars is proved, with equality if and only if the polar of the rotor is a ball centered at the centroid of the simplex.

Rotors in Mechanisms

The use of drills in the shape of rotors in polygons has been extended from the drilling of even polygons (the square - see Section 18.5 - and the hexagon) to the drilling of odd-gons (the triangle and the pentagon). There are many other problems closely related to rotors in polygons that have been investigated. One of them is the determination of the noncircular shapes of pivoted rotors which remain in contact with two straight arms of a pivoted rocker arm. The only possible angles between the arms are rational fractions of a circle as are the angles of a regular polygon. Examples of the ovals are described by

$$p_1(\theta )= \cos \big (m\pi /2n + k\sin (n\theta )\big ),$$

which is a rotor in a polygon. These ovals can be described as curves whose isoptic curves are circles; cf. [449]. They are the basis of a patent issued for a series of double-contact cam mechanisms (see [427]).

Another related mechanism is the intermittent rotor discussed in [430]. This makes contact with a series of fixed elements, but not always with all the elements. In the example shown in Figure 17.14 the rotor is restrained in its motion by contact with three of the four fixed points until all four of the fixed points are touched. The motion may then be continued with another set of three fixed points as constraints. As already mentioned, we refer also to Chapter 18 below where related topics are discussed.

Billiards

Billiards in connection with floating body problems and sets of constant width were already discussed in the Notes of Chapter 5. Thus, the reader is also referred to this chapter, and we cite here again the references [68], [108], [351], [352], [353], [499], [496], and [640] from there; see also the sections A 4 and A 6 of the problem book [272]. Additional themes and topics are also discussed in these papers, such as billiards and caustics (see, for example, [640] where also fractal caustics and the evolution of wave fronts on 3-dimensional billiards of equal thickness are discussed).

We continue here with some further results. Dynamical systems of billiards in a convex body are studied in [1068]. First it is shown that an n-dimensional smooth convex body is a ball iff each billiard orbit is contained in an affine 2-space. Furthermore, the double normal property is used to show that a smooth convex body in the plane is of constant width iff each billiard orbit consists of only right-handed reflections or of only left-handed reflections when hitting the boundary. In [1117], properties of planar billiard trajectories are investigated, where the theory of parallels and the parallel axiom are essential in the geometry of the configuration space; the main result is a characterization of the circle within the family of smooth constant width curves. Bezdek and Connelly [116] proved that the compact convex sets of constant width $\frac{1}{2}$ present the class of planar translation covers for every closed curve of length one or less having minimum perimeter, see Theorem 17.2.4 and Remark 17.2.1. For the proof they use Helly’s Theorem, Sperner’s Lemma and Fagnano’s Theorem about pedal triangles. They prove their covering problem with the help of billiard conditions for triangles, applying their results also to Reuleaux triangles. Finally we mention here the weakly related paper [1206].

Exercises

• 17.1. Prove that a convex body $\Phi$ is a rotor in the n-dimensional cube $Q^n$ if and only if $\Phi$ has constant width.

• 17.2. Prove Theorem 17.1.1 for an unbounded triangle T.

• 17.3. Prove that the biangle $\Delta$ is the intersection of two disks of radius h whose centers are at distance $\sqrt{3} h$ apart.

• 17.4. In the context of the proof that the biangle $\Delta$ is a rotor in the equilateral triangle, prove that the angle at A is one half of the sum of the arc $M^\prime N^\prime$ plus the arc $MKN$.

• 17.5. Prove that the area of the $\Delta$-biangle is $(\pi /3 - \sqrt{3}/2) h^2$, and that its perimeter is $2\pi h/3$.

• 17.6. Prove that the perimeter of the rotor $\Upsilon$ is equal to the perimeter of the biangle $\Delta$, but its area is larger. Indeed, prove that the area bounded by $\Upsilon$ is $\big ( \pi /3+1-\sqrt{3}\big )h^2$.

• 17.7. Prove that $\Phi$ is a rotor in the regular hexagon if and only if $\Phi$ is a rotor in an equilateral triangle and has constant width.

• 17.8*. Let $\rho$ be a rotation of $120^{\circ }$. Suppose that $\Phi + \rho \Phi + \rho ^2 \Phi$ is a disk. Prove that the convex figure $\Phi$ is a rotor in an equilateral triangle.

• 17.9*. Suppose $\Phi _1$ is a rotor in the polytope $P_1$ and $\Phi _2$ is a rotor in the polytope $P_2$. Is it true that $\Phi _1+\Phi _2$ is a rotor in the polytope $P_1+P_2$?

• 17.10*. Assume the polygon $P_1$ circumscribes the convex figure $\phi _1$ and the polygon $P_2$ circumscribes the convex figure $\phi _2$. Is it true that the polygon $P_1+P_2$ circumscribes the convex figure $\phi _1+\phi _2$?

• 17.11**. In the context of the proof of Theorem 17.1.4, use a continuity argument to show the existence of hexagons H and $H^\prime$ covering $\Phi$ and $\rho \Phi$, respectively, with $V(H, H^\prime )\le \sqrt{3}/4$.

• 17.12*. Let $\rho$ be a rotation of $90^{\circ }$ and let $\Phi$ be a convex figure in the plane. Denote by $\mathcal{P}(\theta )$ its support function, where

  $$\begin{aligned} \mathcal{P}(\theta ) \sim a_0 + \sum _1^\infty \big (a_k \cos (k\theta ) + b_k \sin (k\theta )\big ). \end{aligned}$$

  Prove that $\Phi + \rho \Phi + \rho ^2 \Phi +\rho ^3 \Phi$ is a disk if and only if $a_{4k}=b_{4k}=0$, for every $k>0$.

• 17.13*. Let $\rho$ be a rotation of $60^{\circ }$. Prove that if $\Phi$ is a rotor in the regular hexagon, then $\Phi + \rho \Phi +\dots +\rho ^5 \Phi$ is a disk. Construct a convex figure $\Phi$ for which $\Phi + \rho \Phi +\dots +\rho ^5 \Phi$ is a disk, but $\Phi$ is not a rotor in the regular hexagon.

• 17.14*. Prove that the four altitudes of a tetrahedron belong to one ruling of a quadratic surface.

• 17.15. Suppose $\mathcal{C}$ is a curve that cannot be translated inside a convex body $\phi$. Prove that there is a polygonal curve P of length $l(P)<l(\mathcal{C})$ that cannot be translated into the interior of $\phi$.

• 17.16*. Prove that for an acute triangle the pedal triangle formed by the feet of altitudes is a generalized billiard trajectory. Furthermore, prove that it is the unique generalized billiard trajectory of period 3.

• 17.17*. Prove that for a Reuleaux triangle of width 1 there is only one generalized billiard trajectory of period greater than 2. Prove that this billiard trajectory has period 3 and length 3.

• 17.18*. If $\Phi$ is a convex body and $S(\Phi )$ a Steiner symmetrization of $\Phi$, prove that $M(\Phi )\le M(S (\Phi ))$, where the Mahler volume $M(\Phi )$ of the body is defined as the product of the volumes of $\Phi$ and its polar body. That is, $M(\Phi ) := \hbox {vol}(\Phi ) \hbox {vol}(\Phi ^\circ )$.

• 17.19**. If $\Phi _1 \subset \mathbb {E}^{n-1}, \Phi _2 \subset \mathbb {E}^{n-2}$ are convex symmetric bodies, prove that

  $$\begin{aligned} M(\Phi _1 \times \Phi _2) = \frac{M(\Phi _1) M(\Phi _2) }{ \left( {\begin{array}{c}n_1+n_2\\ n_1\end{array}}\right) }. \end{aligned}$$