Curve of constant width

Measuring the width of a Reuleaux triangle as the distance between parallel supporting lines. Because this distance does not depend on the direction of the lines, the Reuleaux triangle is a curve of constant width.

In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. Standard examples are the circle and the Reuleaux triangle. These curves can also be constructed using circular arcs centered at crossings of an arrangement of lines, as the involutes of certain curves, or by intersecting circles centered on a partial curve.

Every body of constant width is a convex set, its boundary crossed at most twice by any line, and if the line crosses perpendicularly it does so at both crossings, separated by the width. By Barbier's theorem, the body's perimeter is exactly π times its width, but its area depends on its shape, with the Reuleaux triangle having the smallest possible area for its width and the circle the largest. Every superset of a body of constant width includes pairs of points that are farther apart than the width, and every curve of constant width includes at least six points of extreme curvature. Although the Reuleaux triangle is not smooth, curves of constant width can always be approximated arbitrarily closely by smooth curves of the same constant width.

Cylinders with constant-width cross-section can be used as rollers to support a level surface. Another application of curves of constant width is for coinage shapes, where regular Reuleaux polygons are a common choice. The possibility that curves other than circles can have constant width makes it more complicated to check the roundness of an object.

These curves have been generalized in several ways to higher dimensions and to non-Euclidean geometry.

Definitions

Every compact set in the plane has one pair of parallel supporting lines in any given direction. A supporting line is a line that has at least one point in common with the boundary of the set but does not separate any two points from the set. The width of the set in that direction is the Euclidean distance between these two lines. Equivalently, the convex hull of the orthogonal projection perpendicular to that direction is a line segment, and the width in that direction is the length of this line segment. If the width is the same in all directions, the boundary of the set is a curve of constant width and its convex hull is a body of constant width.

Examples

A curve of constant width defined by an 8th-degree polynomial

Circles have constant width, equal to their diameter. On the other hand, squares do not: supporting lines parallel to two opposite sides of the square are closer together than supporting lines parallel to a diagonal. More generally, no polygon can have constant width. However, there are other shapes of constant width. A standard example is the Reuleaux triangle, the intersection of three circles, each centered where the other two circles cross. Its boundary curve consists of three arcs of these circles, meeting at 120° angles, so it is not smooth, and in fact these angles are the sharpest possible for any curve of constant width.

Other curves of constant width can be smooth but non-circular, not even having any circular arcs in their boundary. For instance, for the polynomial $f(x,y)$ in the formula below, the zero set (the points $(x,y)$ for which $f(x,y)=0$) forms a non-circular smooth algebraic curve of constant width:

$${\begin{aligned}f(x,y)={}&(x^{2}+y^{2})^{4}-45(x^{2}+y^{2})^{3}-41283(x^{2}+y^{2})^{2}\\&+7950960(x^{2}+y^{2})+16(x^{2}-3y^{2})^{3}+48(x^{2}+y^{2})(x^{2}-3y^{2})^{2}\\&+x(x^{2}-3y^{2})\left(16(x^{2}+y^{2})^{2}-5544(x^{2}+y^{2})+266382\right)-720^{3}.\end{aligned}}$$

Its degree, eight, is the minimum possible degree for a polynomial that defines a non-circular curve of constant width.

Constructions

An irregular Reuleaux polygon

Applying the crossed-lines method to the sides of an irregular triangle

Body of constant width (yellow) formed by intersecting disks (blue) centered on a semi-ellipse (black). The eccentricity of the semi-ellipse, ${\tfrac {1}{2}}{\sqrt {3}}$, is the maximum possible while maintaining the property that every supporting line is tangent to a circle of radius equal to the width, containing the semi-ellipse; this tangent circle is shown in red for the point of minimum curvature of the semi-ellipse.

Every regular polygon with an odd number of sides gives rise to a curve of constant width, a Reuleaux polygon, formed from circular arcs centered at its vertices that pass through the two vertices farthest from the center; irregular Reuleaux polygons are also possible. This is a special case of a more general construction, called by Martin Gardner the "crossed-lines method", in which any arrangement of lines in the plane (no two parallel), sorted into cyclic order by their slopes, are connected by a smooth curve formed from circular arcs between pairs of consecutive lines in the sorted order, centered at the crossing of these two lines. The radius of the first arc must be chosen large enough to cause all successive arcs to end on the correct side of the next crossing point; however, all sufficiently-large radii work. For two lines, this forms a circle; for three lines on the sides of an equilateral triangle, with the minimum possible radius, it forms a Reuleaux triangle, and for the lines of a regular star polygon it can form a Reuleaux polygon.

Leonhard Euler constructed curves of constant width as the involutes of curves with an odd number of cusp singularities, having only one tangent line in each direction (that is, projective hedgehogs). If the starting curve is smooth (except at the cusps), the resulting curve of constant width will also be smooth. An example of a starting curve with the correct properties for this construction is the deltoid curve, and involutes of the deltoid form smooth curves of constant width, not formed from circular arcs. The same construction can also be obtained by rolling a line segment along the same starting curve, without sliding it, until it returns to its starting position. For any long enough line segment, there is a starting position tangent to one of the cusps of the curve for which it will return in this way, obtained by a calculation involving an alternating sum of the lengths of arcs of the starting curve between its cusps.

Another construction chooses half of the curve of constant width, meeting certain conditions, and then completes it to a full curve. The construction begins with a convex curved arc, connecting a pair of closest points on two parallel lines whose separation is the intended width $w$ of the curve. The arc must have the property (required of a curve of constant width) that each of its supporting lines is tangent to a circle of radius $w$ containing the entire arc; intuitively, this prevents its curvature from being smaller than that of a circle of radius $w$ at any point. As long as it meets this condition, it can be used in the construction. The next step is to intersect an infinite family of circular disks of radius $w$, both the ones tangent to the supporting lines and additional disks centered at each point of the arc. This intersection forms a body of constant width, with the given arc as part of its boundary. In a special case of this construction found by 19th-century French mathematician Victor Puiseux, it can be applied to the arc formed by half of an ellipse between the ends of its two semi-major axes, as long as its eccentricity is at most ${\tfrac {1}{2}}{\sqrt {3}}$, low enough to meet the curvature condition. (Equivalently, the semi-major axis should be at most twice the semi-minor axis.) This construction is universal: all curves of constant width may be constructed in this way.

Given any two bodies of constant width, their Minkowski sum forms another body of constant width.

Properties

The Reuleaux triangle rolling within a square while at all times touching all four sides

A curve of constant width can be rotated between two parallel lines separated by its width, while at all times during the rotation touching those lines. This sequence of rotations of the curve can be obtained by keeping the curve fixed in place and rotating two supporting lines around it, and then applying rotations of the whole plane that instead keep the lines in place and cause the curve to rotate between them. In the same way, a curve of constant width can be rotated between two pairs of parallel lines with the same separation. In particular, by choosing the lines through opposite sides of a square, any curve of constant width can be rotated within a square. Although it is not always possible to rotate such a curve within a regular hexagon, every curve of constant width can be drawn within a regular hexagon in such a way that it touches all six sides.

A curve has constant width if and only if, for every pair of parallel supporting lines, it touches those two lines at points whose distance equals the separation between the lines. In particular, this implies that it can only touch each supporting line at a single point. Equivalently, every line that crosses the curve perpendicularly crosses it at exactly two points of distance equal to the width. Therefore, a curve of constant width must be convex, for every non-convex simple closed curve has a supporting line that touches it at two or more points. Curves of constant width are examples of self-parallel or auto-parallel curves, curves traced by both endpoints of a line segment that moves in such a way that both endpoints move perpendicularly to the line segment. However, there exist other self-parallel curves, such as the infinite spiral formed by the involute of a circle, that do not have constant width.

Barbier's theorem asserts that the perimeter of any curve of constant width is equal to the width multiplied by $\pi$. As a special case, this formula agrees with the standard formula $\pi d$ for the perimeter of a circle given its diameter. By the isoperimetric inequality and Barbier's theorem, the circle has the maximum area of any curve of given constant width. The Blaschke–Lebesgue theorem says that the Reuleaux triangle has the least area of any convex curve of given constant width. Every proper superset of a body of constant width has strictly greater diameter, and every Euclidean set with this property is a body of constant width. In particular, it is not possible for one body of constant width to be a subset of a different body with the same constant width. Every curve of constant width can be approximated arbitrarily closely by a piecewise circular curve or by an analytic curve of the same constant width.

A vertex of a smooth curve is a point where its curvature is a local maximum or minimum; for a circular arc, all points are vertices, but non-circular curves may have a finite discrete set of vertices. For a curve that is not smooth, the points where it is not smooth can also be considered as vertices, of infinite curvature. For a curve of constant width, each vertex of locally minimum curvature is paired with a vertex of locally maximum curvature, opposite it on a diameter of the curve, and there must be at least six vertices. This stands in contrast to the four-vertex theorem, according to which every simple closed smooth curve in the plane has at least four vertices. Some curves, such as ellipses, have exactly four vertices, but this is not possible for a curve of constant width. Because local minima of curvature are opposite local maxima of curvature, the only curves of constant width with central symmetry are the circles, for which the curvature is the same at all points. For every curve of constant width, the minimum enclosing circle of the curve and the largest circle that it contains are concentric, and the average of their diameters is the width of the curve. These two circles again touch the curve in at least three pairs of opposite points, but these touching points might not be vertices.

A convex body has constant width if and only if the Minkowski sum of the body and its central reflection is a circular disk; if so, the width of the body is the radius of the disk.

Applications

Rollers of constant width

Because of the ability of curves of constant width to roll between parallel lines, any cylinder with a curve of constant width as its cross-section can act as a "roller", supporting a level plane and keeping it flat as it rolls along any level surface. However, the center of the roller moves up and down as it rolls, so this construction would not work for wheels in this shape attached to fixed axles.

Several countries have coins shaped as non-circular curves of constant width; examples include the British 20p and 50p coins. Their heptagonal shape with curved sides means that the currency detector in an automated coin machine will always measure the same width, no matter which angle it takes its measurement from. The same is true of the 11-sided loonie (Canadian dollar coin).

Because of the existence of non-circular curves of constant width, checking the roundness of an object requires more complex measurements than its width. Overlooking this fact may have played a role in the Space Shuttle Challenger disaster, as the roundness of sections of the rocket in that launch was tested only by measuring different diameters, and off-round shapes may cause unusually high stresses that could have been one of the factors causing the disaster.

Generalizations

The generalization of the definition of bodies of constant width to convex bodies in $\mathbb {R} ^{3}$ and their boundaries leads to the concept of surface of constant width (in the case of a Reuleaux triangle, this does not lead to a Reuleaux tetrahedron, but to Meissner bodies). There is also a concept of space curves of constant width, defined by the properties that each plane that crosses the curve perpendicularly intersects it at exactly one other point, where it is also perpendicular, and that all pairs of points intersected by perpendicular planes are the same distance apart.

Curves and bodies of constant width have also been studied in non-Euclidean geometry and for non-Euclidean normed vector spaces.

See also

• Mean width, the width of a curve averaged over all possible directions
• Zindler curve, a curve in which all perimeter-bisecting chords have the same length

External links

• Interactive Applet by Michael Borcherds showing an irregular shape of constant width (that you can change) made using GeoGebra.
• Weisstein, Eric W. "Curve of Constant Width". MathWorld.
• Mould, Steve. "Shapes and Solids of Constant Width". Numberphile. Brady Haran. Archived from the original on 2016-03-19. Retrieved 2013-11-17.
• Shapes of constant width at cut-the-knot