## Around the wheel (*)

### Introduction

In a vehicle with four circular wheels, each pair of wheels is joined by an axle passing through their centers. The shape of the wheel ensures that all points on its edge are at the same distance from its center, so that the axle moves without oscillations. But if we want to transport a heavy object, the wheel-axle system may not be sufficiently robust and therefore rollers are often used: the objects are transported on a platform that rolls on cylinders with the same section. In this way the transport also occurs without oscillations, but now the property responsible for this smooth movement is the fact that the circumference has constant width. The essential difference is that in the first case (usually cars) the car's structure rests on the axles, while in the second it rests on the edges of the wheels (or rollers).

What does constant width mean?

Given a planar and closed curve $$C$$, the width of $$C$$ in a fixed direction r is the length of the line segment which is obtained by projecting, on r, each point of $$C$$ perpendicular to the line $$r$$. It is said that the width of the curve is constant if that length is the same for all directions in the plane.

For example, a circumference of diameter $$d$$ has constant width $$d$$. But this is not a property that characterizes it since there are infinitely many other curves that satisfy it. The simplest one is the Reuleaux triangle, which is obtained from an equilateral triangle of side $$L$$, if we draw three arcs of circumference with radius $$L$$ and center in each of the vertices of the triangle. Let us verify that the curve obtained in this way has constant width $$L$$.

A supporting line of a curve is a line that contains at least a point of the curve and is such that the curve lies completely on one of the sides of the line. Note that a supporting line may not be a tangent line; and that a closed curve has exactly two supporting lines in each direction: they can be found by placing the curve between two lines parallel to that direction and sliding them, keeping the parallelism, until they touch the curve.

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