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 obscurity, Curves of Constant Breadth are the counter example to the oft asked question “Why are manhole covers round?” The traditional answer is “because only a round cover won’t fall in the hole.” The Reuleaux triangle is a counter example that shows that any Curve of Constant Breadth will make a serviceable manhole cover . . . although circular covers are probably easier to make and are certainly much more common.

In contrast to stamped manhole covers, the beauty of 3-D printing is that objects with “complex” shapes are just as easy to slice and print as simpler ones, so with a little math and OpenSCAD we have an entire family of rollers made of Curves of Constant Breadth based off of an equilateral triangle.

The OpenSCAD code/module included here takes the concept of a smoothed Reuleaux triangle a little further by introducing the concept of roundedness to quantify how much the Reuleaux triangle has been smoothed. An unsmoothed Curve of Constant Breadth would have a roundedness of 0 and in the OpenSCAD code/module would be a Reuleaux triangle, a fully smoothed Curve of Constant Breadth has a roundedness of 1 and would be a circle.

Breadth (Diameter) =2s+L where L is the length of the side of the base equilateral triangle and the roundedness (Round) = 2s/D.

Inputs for the module / customizer are:

Diameter - or the Breadth of the solid

Round - the roundedness of the roller, a number from 0 to 1. (Note: rollers with less roundness than .4 or .5 tend not to roll to well)