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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.
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 of constant girth: its girth in all directions is the same number πw. Hermann Minkowski proved, conversely, that every convex surface ...
On the Sphere and Cylinder (Greek: Περὶ σφαίρας καὶ κυλίνδρου) is a treatise that was published by Archimedes in two volumes c. 225 BCE. [1] It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder , and was the first to do so.
Suppose that the distance between points λ 1 (t) and λ 2 (t) are constant for each t ∈ R and that the curve defined by the midpoints between λ 1 and λ 2 is such that its tangent vector at the point t is parallel to the segment from λ 1 (t) to λ 2 (t) for each t.
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