Editing Curve of constant width (section)

==Examples==
[[File:A curve of constant width defined by 8th-degree polynomial.png|thumb|A curve of constant width defined by an 8th-degree polynomial]]
[[Circle]]s have constant width, equal to their [[diameter]]. On the other hand, squares do not: supporting lines parallel to two opposite sides of the square are closer together than supporting lines parallel to a diagonal. More generally, no [[polygon]] can have constant width. However, there are other shapes of constant width.  A standard example is the [[Reuleaux triangle]], the intersection of three circles, each centered where the other two circles cross.{{r|gardner}} Its boundary curve consists of three arcs of these circles, meeting at 120° angles, so it is not [[Smooth function#Smoothness of curves and surfaces|smooth]], and in fact these angles are the sharpest possible for any curve of constant width.{{r|rt}}

Other curves of constant width can be smooth but non-circular, not even having any circular arcs in their boundary.
For instance, the [[zero set]] of the [[polynomial]] below forms a non-circular smooth [[algebraic curve]] of constant width:{{r|rabinowitz}}
:<math display="block">\begin{align}
f(x,y)={}&(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(x^2 - 3y^2)\left(16(x^2 + y^2)^2 - 5544(x^2 + y^2) + 266382\right) - 720^3.
\end{align}</math>
Its [[Degree of a polynomial|degree]], eight, is the minimum possible degree for a polynomial that defines a non-circular curve of constant width.{{r|bb}}