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[20] [21] Every curve of constant width can be approximated arbitrarily closely by a piecewise circular curve or by an analytic curve of the same constant width. [ 22 ] A vertex of a smooth curve is a point where its curvature is a local maximum or minimum; for a circular arc, all points are vertices, but non-circular curves may have a finite ...
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.
These Reuleaux polygons have constant width, and all have the same width; therefore by Barbier's theorem they also have equal perimeters. In geometry, Barbier's theorem states that every curve of constant width has perimeter π times its width, regardless of its precise shape. [1] This theorem was first published by Joseph-Émile Barbier in ...
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]
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 ...
where L and w are, respectively, the perimeter and the width of any curve of constant width. A = π r 2 {\displaystyle A=\pi r^{2}} where A is the area of a circle .
Curve of constant width; R. ... Surface of constant width This page was last edited on 9 November 2020, at 07:49 (UTC). Text is available under the ...