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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. [1]
The angles made by each pair of arcs at the corners of a Reuleaux triangle are all equal to 120°. This is the sharpest possible angle at any vertex of any curve of constant width. [9] Additionally, among the curves of constant width, the Reuleaux triangle is the one with both the largest and the smallest inscribed equilateral triangles. [15]
One defines the width of the surface in a given direction to be the perpendicular distance between the parallels perpendicular to that direction. Thus, a surface of constant width is the three-dimensional analogue of a curve of constant width, a two-dimensional shape with a constant distance between pairs of parallel tangent lines.
More generally, if S is a surface of constant width w, then every projection of S is a curve of constant width, with the same width w. All curves of constant width have the same perimeter, the same value πw as the circumference of a circle with that width (this is Barbier's theorem). Therefore, every surface of constant width is also a surface ...
The same theorem is also true in the hyperbolic plane. [11] For any convex distance function on the plane (a distance defined as the norm of the vector difference of points, for any norm), an analogous theorem holds true, according to which the minimum-area curve of constant width is an intersection of three metric disks, each centered on a boundary point of the other two.
Gambian dalasi coin, a Reuleaux heptagon. In geometry, a Reuleaux polygon is a curve of constant width made up of circular arcs of constant radius. [1] These shapes are named after their prototypical example, the Reuleaux triangle, which in turn is named after 19th-century German engineer Franz Reuleaux. [2]
The problem was posed by Henri Lebesgue in a letter to Gyula Pál in 1914. It was published in a paper by Pál in 1920 along with Pál's analysis. [1] He showed that a cover for all curves of constant width one is also a cover for all sets of diameter one and that a cover can be constructed by taking a regular hexagon with an inscribed circle of diameter one and removing two corners from the ...
The mean width is the average of this "width" over all ^ in . The definition of the "width" of body B in direction n ^ {\displaystyle {\hat {n}}} in 2 dimensions. More formally, define a compact body B as being equivalent to set of points in its interior plus the points on the boundary (here, points denote elements of R n {\displaystyle \mathbb ...