Barbier's theorem, a basic result on curves of constant width

Curve of constant width

In geometry, a curve of constant width is a convex planar shape whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines.More generally, any compact...

 first proved by Joseph Emile Barbier, states that the perimeter of any curve of constant width w is πw.

The most familiar examples of curves of constant width are the circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

 and the Reuleaux triangle

Reuleaux triangle

A Reuleaux triangle is, apart from the trivial case of the circle, the simplest and best known Reuleaux polygon, a curve of constant width. The separation of two parallel lines tangent to the curve is independent of their orientation...

. A circle of width (diameter

Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

) w has perimeter

Perimeter

A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 πw. A Reuleaux triangle of width w consists of three arc

Arc (geometry)

In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle...

s of circles of radius

Radius

In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

 w. Each of these arcs has central angle

Central angle

A central angle is an angle which vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an arc between those two points whose angle is equal to the central angle itself...

 π/3, so the perimeter of the Reuleaux triangle of width w is equal to ½ the perimeter of a circle of radius w and therefore is equal to πw. A similar analysis of other simple examples such as Reuleaux polygons gives the same answer.

The analogue of Barbier's theorem for surfaces of constant width

Surface of constant width

In geometry, a surface of constant width is a convex form whose width, measured by the distance between two opposite parallelplanes touching its boundary, is the same regardless of the direction of those two parallel planes. One defines the width of the surface in a given direction to be the...

 is false.

External links

• The Theorem of Barbier (Java) at cut-the-knot
  Cut-the-knot
  Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...