Constant Breadth Solid Roller

I came across ‘Curves of Constant Breadth’ in an old book – ‘Riddles In Mathematics: a book of paradoxes’ by Eugene P. Northrop. Northrop introduces the idea of Curves of Constant Breadth by questioning whether a roller used under a sledge for moving heavy objects must be round. Surprisingly, the answer is “No.” Northrop produces the example of the Reuleaux triangle as a Curve of Constant Breadth, though he notes that the Reuleaux triangle makes a terrible roller due to its relatively sharp points. However, Northrop notes the points can be smoothed by prolonging the sides of the triangle some distance (s) and drawing a curve of radius (s) between the two extended sides. Where the breadth of the Reuleaux triangle is (L) (i.e. the length of each side of the underlying equilateral triangle), the breadth of the smoothed Curve of Constant Breadth is (2s+L).

https://archive.org/details/riddles-in-mathematics-a-book-of-paradoxes-eugene-p-northrop/page/58/mode/2up

A bit of math obscurit