A Reuleaux triangle is a curve of constant width. A Reuleaux polygon is a Curve of constant width - that is a curve in which all diameters are the same length Opposite sides on each square are parallel supporting lines, since each touches the curve but does not intersect the interior. The fact that the Reuleaux triangle can be rotated whilst still just intersecting the boundary shows that width (separation between parallel supporting lines) is constant in all directions.

In geometry, a curve of constant width is a convex planar shape whose width, measured by the distance between two opposite parallel lines touching its boundary, is the same regardless of the direction of those two parallel lines. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined One defines the width of the curve in a given direction to be the perpendicular distance between the parallels perpendicular to that direction. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent

More generally, any compact convex planar body D has one pair of parallel supporting lines in a given direction. A supporting line is a line that intersects the boundary of D but not the interior of D. One defines the width of the body as before. If the width of D if the same in all directions, then one says that the body is of constant width and calls its boundary a curve of constant width, and the planar body itself is referred to as an orbiform.

Curves of constant width can be rotated between parallel line segments. To see this, simply note that one can rotate parallel line segments (supporting lines) around curves of constant width by definition. Consequently, a curve of constant width can be rotated in a square. Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides

The circle is obviously a curve of constant width. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the A nontrivial example is the Reuleaux triangle. A Reuleaux polygon is a Curve of constant width - that is a curve in which all diameters are the same length To construct this, take an equilateral triangle ABC and draw the arc BC on the circle centered at A, the arc CA on the circle centered at B, and the arc AB on the circle centered at C. Properties The area of an equilateral triangle with sides of length a\\! The resulting figure is of constant width.

A basic result on curves of constant width is Barbier's theorem, which asserts that the perimeter of any curve of constant width is equal to the width (diameter) multiplied by π. Barbier's theorem is a basic result on curves of constant width first proved by Joseph Emile Barbier. The perimeter is the distance around a given two-dimensional object 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 A simple example of this would be a circle with width (diameter) d having a perimeter of πd. 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

By the isoperimetric inequality and Barbier's theorem, the circle has the maximum area of any curve of given constant width. The isoperimetric inequality is a geometric Inequality involving the square of the Circumference of a Closed curve in the plane and the The Blaschke-Lebesgue theorem says that the Reuleaux triangle had the least area of any curve of given constant width.

Δ curves, or curves which can be rotated in the equilateral triangle, have many similar properties to curves of constant width. The generalization of the definition of bodies of constant width to convex bodies in R³ 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 a Meissner body). In Geometry, a surface of constant width is a convex Form whose width measured by the distance between two opposite parallelplanes touching its boundary A Reuleaux polygon is a Curve of constant width - that is a curve in which all diameters are the same length The Reuleaux tetrahedron is the intersection of four spheres of Radius s centered at the vertices of a regular Tetrahedron with side length There is also the concept of space curves of constant width, which are space curves whose normal planes intersect them at exactly two points and are normal to them at both points.

Famous examples of a curve of constant width are 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 correct diameter through the middle, no matter which angle it takes its measurement from. A currency detector is a device that determines if a piece of currency is or is not counterfeit

A curve of constant width is also used in the Wankel engine, which uses a Reuleaux triangle with slightly flattened corners as a rotor within an epitrochoidal combustion chamber. The Wankel engine is a type of Internal combustion engine which uses a rotary design to convert pressure into a rotating motion instead of using reciprocating A Reuleaux polygon is a Curve of constant width - that is a curve in which all diameters are the same length An epitrochoid (ɛpɨˈtrɒkɔɪd -ˈtroʊ- is a roulette traced by a point attached to a Circle of Radius r rolling around the outside The property of constant width allows the rotor to pass eccentrically through the chamber, maintaining seals against the walls while compressing and expanding volumes of a fluid. In practice, the seals have proven very difficult to maintain.

There exist polynomial f(x,y) of degree 8, which graphic (e. g. set of points in R², for which f(x,y) = 0) is curve of constant width [1].

External links

• Star Construction of Shapes of Constant Width at cut-the-knot