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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a curve of constant width is a
simple closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
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 Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
. Standard examples are the
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
and the Reuleaux triangle. These curves can also be constructed using circular arcs centered at crossings of an arrangement of lines, as the
involute In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the Locus (mathematics), locus of a point on a piece of taut string as the string is eith ...
s of certain curves, or by intersecting circles centered on a partial curve. Every body of constant width is a
convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
, 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 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. Curves of constant width have been generalized in several ways to higher dimensions and to
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
.


Definitions

Width, and constant width, are defined in terms of the supporting lines of curves; these are lines that touch a curve without crossing it. Every
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
curve in the plane has two supporting lines in any given direction, with the curve sandwiched between them. The
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
between these two lines is the ''width'' of the curve in that direction, and a curve has constant width if this distance is the same for all directions of lines. The width of a bounded
convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
can be defined in the same way as for curves, by the distance between pairs of parallel lines that touch the set without crossing it, and a convex set is a body of constant width when this distance is nonzero and does not depend on the direction of the lines. Every body of constant width has a curve of constant width as its boundary, and every curve of constant width has a body of constant width as its
convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
. Another equivalent way to define the width of a compact curve or of a convex set is by looking at its
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it we ...
onto a line. In both cases, the projection is a
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
, whose length equals the distance between support lines that are perpendicular to the line. So, a curve or a convex set has constant width when all of its orthogonal projections have the same length.


Examples

Circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
s have constant width, equal to their
diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
. 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 In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the 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, the
zero set In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or eq ...
of the
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
below forms a non-circular smooth
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
of constant width: :\begin 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 Its degree, eight, is the minimum possible degree for a polynomial that defines a non-circular curve of constant width.


Constructions

Every
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
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. For instance, this construction generates a Reuleaux triangle from an equilateral triangle. Some irregular polygons also generate Reuleaux polygons. In a closely related construction, called by
Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing magic, scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writin ...
the "crossed-lines method", an arrangement of lines in the plane (no two parallel but otherwise arbitrary) is sorted into cyclic order by the slopes of the lines. The lines are then connected by a curve formed from a sequence of circular arcs; each arc connects two consecutive lines in the sorted order, and is centered at their crossing. 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 In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...
it can form a Reuleaux polygon.
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
constructed curves of constant width from
involute In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the Locus (mathematics), locus of a point on a piece of taut string as the string is eith ...
s of curves with an odd number of cusp singularities, having only one
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
in each direction (that is, projective hedgehogs). An intuitive way to describe the involute construction is to roll a line segment around such a curve, keeping it tangent to the curve without sliding along it, until it returns to its starting point of tangency. The line segment must be long enough to reach the cusp points of the curve, so that it can roll past each cusp to the next part of the curve, and its starting position should be carefully chosen so that at the end of the rolling process it is in the same position it started from. When that happens, the curve traced out by the endpoints of the line segment is an involute that encloses the given curve without crossing it, with constant width equal to the length of the line segment. 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 In geometry, a deltoid curve, also known as a tricuspoid curve or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along the insi ...
, and the involutes of the deltoid that enclose it form smooth curves of constant width, not containing any circular arcs. Another construction chooses half of the curve of constant width, meeting certain requirements, and forms from it a body of constant width having the given curve as part of its boundary. The construction begins with a convex curved arc, whose endpoints are the intended width w apart. The two endpoints must touch parallel supporting lines at distance w from each other. Additionally, each supporting line that touches another point of the arc must be tangent at that point to a circle of radius w containing the entire arc; this requirement prevents the
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
of the arc from being less than that of the circle. The completed body of constant width is then the intersection of the interiors of an infinite family of circles, of two types: the ones tangent to the supporting lines, and more circles of the same radius centered at each point of the given arc. This construction is universal: all curves of constant width may be constructed in this way. Victor Puiseux, a 19th-century French mathematician, found curves of constant width containing elliptical arcs that can be constructed in this way from a
semi-ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
. To meet the curvature condition, the semi-ellipse should be bounded by the
semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longe ...
of its ellipse, and the ellipse should have
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
at most \tfrac\sqrt. Equivalently, the semi-major axis should be at most twice the semi-minor axis. Given any two bodies of constant width, their
Minkowski sum In geometry, the Minkowski sum of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...
forms another body of constant width. A generalization of Minkowski sums to the sums of support functions of hedgehogs produces a curve of constant width from the sum of a projective hedgehog and a circle, whenever the result is a convex curve. All curves of constant width can be decomposed into a sum of hedgehogs in this way.


Properties

A curve of constant width can rotate between two parallel lines separated by its width, while at all times touching those lines, which act as supporting lines for the rotated curve. In the same way, a curve of constant width can rotate within a rhombus or square, whose pairs of opposite sides are separated by the width and lie on parallel support lines. Not every curve of constant width can rotate within a regular
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is de ...
in the same way, because its supporting lines may form different irregular hexagons for different rotations rather than always forming a regular one. However, every curve of constant width can be enclosed by at least one regular hexagon with opposite sides on parallel supporting lines. 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, since 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 A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...
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 In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. '' Isoperimetric'' ...
and Barbier's theorem, the circle has the maximum area of any curve of given constant width. The
Blaschke–Lebesgue theorem In plane geometry the Blaschke–Lebesgue theorem states that the Reuleaux triangle has the least area of all curves of given constant width. In the form that every curve of a given width has area at least as large as the Reuleaux triangle, it ...
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 In geometry, the four-vertex theorem states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives ...
, 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 In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point (geometry), point is reflected across a designated inversion center, which remains Fixed p ...
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 together touch the curve in at least three pairs of opposite points, but these points are not necessarily vertices. A convex body has constant width if and only if the Minkowski sum of the body and its 180° rotation is a circular disk; if so, the width of the body is the radius of the disk.


Applications

Because of the ability of curves of constant width to roll between parallel lines, any
cylinder A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
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. Some coinage shapes are non-circular bodies of constant width. For instance the British 20p and 50p coins are Reuleaux heptagons, and the Canadian loonie is a Reuleaux 11-gon. These shapes allow automated coin machines to recognize these coins from their widths, regardless of the orientation of the coin in the machine. On the other hand, testing the width is inadequate to determine the roundness of an object, because such tests cannot distinguish circles from other curves of constant width. Overlooking this fact may have played a role in the
Space Shuttle Challenger disaster On January 28, 1986, the Space Shuttle Challenger, Space Shuttle ''Challenger'' broke apart 73 seconds into its flight, killing all seven crew members aboard. The spacecraft disintegrated above the Atlantic Ocean, off the coast of Cape Can ...
, as the roundness of sections of the rocket in that launch was tested only by measuring widths, and off-round shapes may cause unusually high stresses that could have been one of the factors causing the disaster.


Generalizations

The curves of constant width can be generalized to certain non-convex curves, the curves that have two tangent lines in each direction, with the same separation between these two lines regardless of their direction. As a limiting case, the projective hedgehogs (curves with one tangent line in each direction) have also been called "curves of zero width". One way to generalize these concepts to three dimensions is through the surfaces of constant width. The three-dimensional analog of a Reuleaux triangle, the Reuleaux tetrahedron, does not have constant width, but minor changes to it produce the Meissner bodies, which do. The curves of constant width may also be generalized to the bodies of constant brightness, three-dimensional shapes whose two-dimensional projections all have equal area; these shapes obey a generalization of Barbier's theorem. A different class of three-dimensional generalizations, the
space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s of constant width, are 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 In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
and for non-Euclidean
normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
s.


See also

*
Mean width In geometry, the mean width is a measure of the "size" of a body; see Hadwiger's theorem for more about the available measures of bodies. In n dimensions, one has to consider (n-1)-dimensional hyperplanes perpendicular to a given direction \hat in ...
, the width of a curve averaged over all possible directions * Zindler curve, a curve in which all perimeter-bisecting chords have the same length


References


External links


Interactive Applet
by Michael Borcherds showing an irregular shape of constant width (that you can change) made usin
GeoGebra
* * {{cite web, title=Shapes and Solids of Constant Width, url=http://www.numberphile.com/videos/shapes_constant.html, work=Numberphile, publisher=
Brady Haran Brady John Haran (born 18 June 1976) is an Australian-British independent filmmaker and video journalist who produces educational videos and documentary films for his YouTube channels, the most notable being ''Computerphile'' and ''Numberph ...
, author=Mould, Steve, access-date=2013-11-17, archive-url=https://web.archive.org/web/20160319140111/http://www.numberphile.com/videos/shapes_constant.html, archive-date=2016-03-19, url-status=dead
Shapes of constant width
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...
Curves Constant width