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. 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 pi 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. Width, and constant width, are defined in terms of the supporting lines of curves; these are lines that touch a curve without crossing it. Every compact curve in the plane has two supporting lines in any given direction, with the curve sandwiched between them.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (5)

Related people (2)

Related units (1)

Related concepts (5)

Related courses (2)

Related lectures (27)

Reuleaux triangle

A Reuleaux triangle ʁœlo is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the other two. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation.

Barbier's theorem

In geometry, Barbier's theorem states that every curve of constant width has perimeter pi times its width, regardless of its precise shape. This theorem was first published by Joseph-Émile Barbier in 1860. The most familiar examples of curves of constant width are the circle and the Reuleaux triangle. For a circle, the width is the same as the diameter; a circle of width w has perimeter piw. A Reuleaux triangle of width w consists of three arcs of circles of radius w.

Surface of constant width

In geometry, a surface of constant width is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes. 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.

CH-353: Introduction to electronic structure methods

Repetition of the basic concepts of quantum mechanics and main numerical algorithms used for practical implementions. Basic principles of electronic structure methods:Hartree-Fock, many body perturbat

MATH-126: Geometry for architects II

Ce cours traite des 3 sujets suivants : la perspective, la géométrie descriptive, et une initiation à la géométrie projective.

Linear Response and Complex Diffusivity

Explores martingale-based linear response, complex diffusivity, and Nyquist relation in stochastic systems with time-dependent perturbation.

Basis Sets I CH-353: Introduction to electronic structure methods

Explores the solution of the Schrödinger equation for many-electron systems using basis sets and the concept of basis functions.

Convex Bodies: Properties and Theorems

Covers properties and theorems related to convex bodies and their transformations.

Adaptive Stochastic Variance Reduction for Non-convex Finite-Sum Minimization

Volkan Cevher, Kimon Antonakopoulos, Efstratios Panteleimon Skoulakis, Leello Tadesse Dadi, Ali Kavis

We propose an adaptive variance-reduction method, called AdaSpider, for minimization of L-smooth, non-convex functions with a finite-sum structure. In essence, AdaSpider combines an AdaGrad-inspired [Duchi et al., 2011, McMahan & Streeter, 2010], but a fai ...

2022

Order Types of Convex Bodies

We prove a Hadwiger transversal-type result, characterizing convex position on a family of non-crossing convex bodies in the plane. This theorem suggests a definition for the order type of a family of convex bodies, generalizing the usual definition of ord ...

2011

Phosphocreatine line with changes in localised 31P MRS of exercising muscle at 7T

Rolf Gruetter, Christine Nabuurs

The aim was to investigate changes in the line width of high-energy phosphates solely originating from the intramyocellular compartment by temporally resolved localised 31P-MRS after aerobic calf muscle exercise. Semi-Laser [2] localised 31P-MRS of gastroc ...

2011