Constant width curve

The Ryelo triangle is a curve of constant width. The sides of the square are support lines: each side touches the triangle, but does not intersect it. The Ryo triangle can be rotated, and at the same time it will always touch each side of the square; thus the width of the triangle (the distance between the two support lines) is constant.

Constant width curve $w$ ${\ displaystyle w}$ $w$ Is a flat convex curve , the length of the orthogonal projection of which onto any line is $w$ ${\ displaystyle w}$ $w$ .

In other words, a curve of constant width is called a flat convex curve , the distance between any two parallel support lines of which is constant and equal to $w$ ${\ displaystyle w}$ $w$ - the width of the curve.

Content

1 Related Definitions
2 Examples
3 Functional Representation
4 Properties
5 Applications
6 Variations and generalizations
7 Notes
8 Literature

Related Definitions

A figure of constant width is a figure whose boundary is a curve of constant width.

Examples

Röhl polygons

A smooth curve of constant width, built on the basis of a triangle and made up of fragments of six conjugate circles. Width w = a + b - c + 2y, where a, b, c are sides of the triangle (a, b> c, y> 0).

Figures of constant width, in particular, are the Rölö circle and polygons (a special case of the latter is the Rölö triangle ). Röhlo polygons are composed of fragments of circles and are not smooth curves. From conjugate fragments of circles, you can build a smooth curve of constant width (figure on the right), but a further increase in the smoothness of the curve along this path is impossible

Functional View

Unlike the simplest examples given above, curves of constant width may not coincide with a circle on any finite segment and be arbitrarily smooth everywhere. In general, a figure of constant width $w$ ${\ displaystyle w}$ $w$ with reference function $p(t)$ ${\ displaystyle p (t)}$ defined by parametric equations ^[1]

$x=p(t)cos(t)-p'(t)sin(t)$ ${\ displaystyle x = p (t) cos (t) -p '(t) sin (t)}$

$y=p(t)sin(t)+p'(t)cos(t)$ ${\ displaystyle y = p (t) sin (t) + p '(t) cos (t)}$ ,

under conditions

$w=p(t)+p(t+\pi )$ ${\ displaystyle w = p (t) + p (t + \ pi)}$ ,
the resulting curve is convex.

According to elementary trigonometry, the first condition is satisfied by the Fourier series of the following form:

p(t)={\frac {w}{2}}+\sum \limits _{k=2}^{+\infty }a_{k}\cos \left((2k-1)t+\theta _{k}\right)

{\ displaystyle p (t) = {\ frac {w} {2}} + \ sum \ limits _ {k = 2} ^ {+ \ infty} a_ {k} \ cos \ left ((2k-1) t + \ theta _ {k} \ right)}

^[2]

If the coefficients of the series decrease quickly enough, then the resulting curve will be convex (without self-intersections).

In particular, the support function $p(t)=9+cos(3t)$ ${\ displaystyle p (t) = 9 + cos (3t)}$ generates a constant-width curve for which an implicit representation is found in the form of an equation for an 8th degree polynomial ^[3]

(x^{2}+y^{2})^{4}-45(x^{2}+y^{2})^{3}-41283(x^{2}+y^{2})^{2}+7950960(x^{2}+y^{2})+16(x^{2}-3y^{2})^{3}

{\ displaystyle (x ^ {2} + y ^ {2}) ^ {4} -45 (x ^ {2} + y ^ {2}) ^ {3} -41283 (x ^ {2} + y ^ {2}) ^ {2} +7950960 (x ^ {2} + y ^ {2}) + 16 (x ^ {2} -3y ^ {2}) ^ {3}}

+48(x^{2}+y^{2})(x^{2}-3y^{2})^{2}+(x^{2}-3y^{2})x[16(x^{2}+y^{2})^{2}-5544(x^{2}+y^{2})+266382]-720^{3}=0

{\ displaystyle +48 (x ^ {2} + y ^ {2}) (x ^ {2} -3y ^ {2}) ^ {2} + (x ^ {2} -3y ^ {2}) x [16 (x ^ {2} + y ^ {2}) ^ {2} -5544 (x ^ {2} + y ^ {2}) + 266382] -720 ^ {3} = 0}

This curve is an analytic function in a neighborhood of any point of either x or y and does not coincide with a circle in any neighborhood.

Properties

Constant width curve length $w$ ${\ displaystyle w}$ is equal to $\pi w$ ${\ displaystyle \ pi w}$ ( Barbier theorem ).
The centers of the inscribed and circled circles in the curve of constant width coincide, and the sum of their radii is equal to the width of the curve.
Constant width figure $w$ ${\ displaystyle w}$ can rotate squared with side $w$ ${\ displaystyle w}$ touching each side all the time.
Among all the figures of a given constant width, the Ryelo triangle has the smallest area, and the circle has the largest.
Any flat figure of diameter $w$ ${\ displaystyle w}$ can be covered with a figure of constant width $w$ ${\ displaystyle w}$ .

Applications

A drill made on the basis of the Röhlo triangle allows ^{[4] to} drill almost square holes (with an inaccuracy of about 2% of the square area).
British coins in denominations of 20 ^[5] and 50 pennies have the shape of a figure of constant width, built on a heptagon.
The Wankel engine uses ^[5] as a piston the Roelot triangle rotating inside the chamber, which allows you to immediately receive rotational motion.
The clamshell mechanism responsible for the “discrete” pulling of the tape in the Luch-2 film projector uses the Ryelo triangle rotating inside the moving square ^[5] .

Variations and generalizations

Lenticular Δ-gon rotating inside an equilateral triangle

Shapes of constant width can be defined as convex figures that can rotate inside a square, while touching all its sides. You can also consider shapes that can rotate by touching all sides of a certain $n$ {\ displaystyle n} -gon, for example, of the right $n$ {\ displaystyle n} -gon. Such figures are called rotors ^[6] .
- For example, a rectangle formed by the intersection of two identical circles with a vertex angle equal to $\pi /3$ ${\ displaystyle \ pi / 3}$ , is the rotor of an equilateral triangle. With a drill of this shape, in principle, triangular holes could be drilled without smoothed corners.
For figures of constant width, there are multidimensional analogues; see Body of constant width .

Notes

↑ Heinrich W. Guggenheimer, Differential Geometry. Dover. New York: 1977.
↑ The coefficient with the number k = 1 can be zeroed, since this term is responsible only for the position of the figure on the plane.
↑ Rabinowitz, Stanley. A Polynomial Curve of Constant Width (neopr.) // Missouri Journal of Mathematical Sciences. - 1997 .-- T. 9 . - S. 23-27 . Archived on June 17, 2009. Archived June 17, 2009 on the Wayback Machine
↑ “ Drilling square holes ” / Mathematical studies
↑ ¹ ² ³ “ Round Relo triangle ” / Mathematical studies
↑ Helmut Groemer, Geometric Applications of Fourier Series and Spherical Harmonics

Literature

I. M. Yaglom , V. G. Boltyansky , Convex figures , issue 4 of the series “Library of the Mathematical Circle” M.-L., GTTI, 1951.-343 p.

[1] Heinrich W. Guggenheimer, Differential Geometry. Dover. New York: 1977.

[2] The coefficient with the number k = 1 can be zeroed, since this term is responsible only for the position of the figure on the plane.

[3] Rabinowitz, Stanley. A Polynomial Curve of Constant Width (neopr.) // Missouri Journal of Mathematical Sciences. - 1997 .-- T. 9 . - S. 23-27 . Archived on June 17, 2009. Archived June 17, 2009 on the Wayback Machine

[4] “ Drilling square holes ” / Mathematical studies

[Reuleaux-5] ¹ ² ³ “ Round Relo triangle ” / Mathematical studies

[6] Helmut Groemer, Geometric Applications of Fourier Series and Spherical Harmonics