Curve of Constant Width

In geometry, a curve of constant width is a convex planar shape whose width (defined as the perpendicular distance between two distinct parallel lines each having at least one point in common with the shape's boundary but none with the shape's interior) is the same regardless of the orientation of the curve.

More generally, any compact convex planar body D has one pair of parallel supporting lines in any given direction. A supporting line is a line that has at least one point in common with the boundary of D but no points in common with the interior of D. The width of the body is defined as before. If the width of D is the same in all directions, the body is said to have constant width and its boundary is a curve of constant width; the planar body itself is called an orbiform.

The width of a circle is constant: its diameter. On the other hand, the width of a square varies between the length of a side and that of a diagonal, in the ratio 1:√2. Thus the question arises: if a given shape's width is constant in all directions, is it necessarily a circle? The surprising answer is that there are many non-circular shapes of constant width. A nontrivial example is the Reuleaux triangle. To construct this, take an equilateral triangle with vertices ABC and draw the arc BC on the circle centered at A, the arc CA on the circle centered at B, and the arc AB on the circle centered at C. The resulting figure is of constant width.

The Reuleaux triangle lacks tangent continuity at three points, but constant-width curves can also be constructed without such discontinuities (as shown in the second illustration at right). Curves of constant width can be generated by joining circular arcs centered on the vertices of a regular or irregular convex polygon with an odd number of sides (triangle, pentagon, heptagon, etc.).