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A sphere, a surface of constant radius and thus diameter, is a surface of constant width. Contrary to common belief the Reuleaux tetrahedron is not a surface of constant width. However, there are two different ways of smoothing subsets of the edges of the Reuleaux tetrahedron to form Meissner tetrahedra, surfaces of constant
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 contours and plane sections of the sphere are circles. This property defines the sphere uniquely. The sphere has constant width and constant girth. The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the Meissner body.
In particular, the unit sphere has surface area , while the surface of revolution of a Reuleaux triangle with the same constant width has surface area . [ 5 ] Instead, Barbier's theorem generalizes to bodies of constant brightness , three-dimensional convex sets for which every two-dimensional projection has the same area.
where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
The girth of a sphere in any direction equals the circumference of its equator, or of any of its great circles.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.
In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), (also Cauchy-Crofton formula) is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it.
The first mathematician to discover the existence of curves of constant width, and to observe that the Reuleaux triangle has constant width, may have been Leonhard Euler. [5] In a paper that he presented in 1771 and published in 1781 entitled De curvis triangularibus , Euler studied curvilinear triangles as well as the curves of constant width ...