Curve of constant width

For closed convex planar bodies with boundary a smooth curve, one notes that there are exactly two parallel tangent lines to the boundary curve in any given direction. One defines the width of the curve in a given direction to be the perpendicular distance between the tangents perpendicular to that direction.

More generally, any compact convex planar body D has one pair of parallel supporting lines in a given direction. A supporting line is a line that intersects the boundary of D but not the interior of D. One defines the width of the body as before. If the width of D if the same in all directions, then one says that the body is of constant width and calls its boundary a curve of constant width.

Curves of constant width can be rotated between parallel line segments. To see this, simply note that one can rotate parallel line segments (supporting lines) around curves of constant width by definition. Consequently, a curve of constant width can be rotated in a square.

The circle is obviously a curve of constant width. A nontrivial example is the Reuleaux triangle. To construct this, take an equilateral triangle ABC and draw the arc BC on the circle with radius A, the arc CA on the circle with radius B, and the arc AB on the circle with radius C. The resulting figure is of constant width.

A basic result on curves of constant width is Barbier's theorem, which asserts that the perimeter of any curve of constant width a is equal to πa.

By the isoperimetric inequality and Barbier's theorem, the circle has the maximum area of any curve of given constant width. The Blaschke-Lebesgue theorem says that the Reuleaux triangle had the least area of any curve of given constant width.

Δ curves, or curves which can be rotated in the equilateral triangle, have many similar properties to curves of constant width. The generalization of the the definition of bodies of constant width to convex bodies in R³ and their boundaries leads to the concept of surface of constant width. There is also the concept of space curves of constant width, which are space curves whose normal planes intersect them at exactly two points and are normal to them at both points.