Measuring the width of a Reuleaux triangle as the distance between parallel supporting lines. Because this distance does not depend on the direction of the lines, the Reuleaux triangle is a curve of constant width.
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] Standard examples are the circle and the Reuleaux triangle. These curves can also be constructed using circular arcs centered at crossings of an arrangement of lines, as the involutes of certain curves, or by intersecting circles centered on a partial curve.
Every body of constant width is a convex set, its boundary crossed at most twice by any line, and if the line crosses perpendicularly it does so at both crossings, separated by the width. By Barbier's theorem, the body's perimeter is exactly π times its width, but its area depends on its shape, with the Reuleaux triangle having the smallest possible area for its width and the circle the largest. Every superset of a body of constant width includes pairs of points that are farther apart than the width, and every curve of constant width includes at least six points of extreme curvature. Although the Reuleaux triangle is not smooth, curves of constant width can always be approximated arbitrarily closely by smooth curves of the same constant width.
Cylinders with constant-width cross-section can be used as rollers to support a level surface. Another application of curves of constant width is for coinage shapes, where regular Reuleaux polygons are a common choice. The possibility that curves other than circles can have constant width makes it more complicated to check the roundness of an object.
Curves of constant width have been generalized in several ways to higher dimensions and to non-Euclidean geometry.
^Cite error: The named reference euler was invoked but never defined (see the help page).
and 23 Related for: Curve of constant width information
the width is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers. For a curveof constant...
In geometry, a Reuleaux polygon is a curveofconstantwidth made up of circular arcs ofconstant radius. These shapes are named after their prototypical...
ofconstantwidth is the three-dimensional analogue of a curveofconstantwidth, a two-dimensional shape with a constant distance between pairs of parallel...
circles, there are other curvesofconstantwidth. By Barbier's theorem, every curveofconstantwidth has perimeter π times its width. The Reuleaux triangle...
and named by analogy to the Reuleaux triangle, a two-dimensional curveofconstantwidth; both shapes are named after Franz Reuleaux, a 19th-century German...
Bodies ofconstant brightness are a generalization ofcurvesofconstantwidth, but are not the same as another generalization, the surfaces ofconstant width...
coins using a non-circular curveofconstantwidth include the 7-sided British twenty pence and fifty pence coins (the latter of which has similar size and...
width, this is a very useful result. Curveofconstantwidth Jiazu, Zhou; Deshuo, Jiang (2008), "On mean curvatures of a parallel convex body", Acta Mathematica...
important domains of science and knowledge. Today, he may be best remembered for the Reuleaux triangle, a curveofconstantwidth that he helped develop...
side coinciding with part of the triangle's longest side. A Reuleaux triangle, or more generally any curveofconstantwidth, can be inscribed with any...
{\displaystyle K} is (the interior of) a curveofconstantwidth, then the Minkowski sum of K {\displaystyle K} and of its 180 ∘ {\displaystyle 180^{\circ...
fixed points is a constant. When the two fixed points coincide, a circle results. A curveofconstantwidth is a figure whose width, defined as the perpendicular...
{\displaystyle C} is a curveofconstantwidth. Let C be a simple closed curve on a sphere of radius 1. Denote by L the length of C and by A the area enclosed...
planar region with legs of unit width. The area thus obtained is referred to as the sofa constant. The exact value of the sofa constant is an open problem...
Gaussian RMS width) controls the widthof the "bell". Gaussian functions are often used to represent the probability density function of a normally distributed...
Global Information