The Reuleaux polygons are analogs of the regular polygons, except that the "sides" are composed of circle arcs instead of lines. It is known that for an odd number of sides, e.g. the Reuleaux triangle, the polygon has constant width.
After reading the paper Roads and Wheels by Stan Wagon and Leon Hall, I got curious on how one might construct the appropriate "road" for Reuleaux wheels; i.e., finding the curve such that when a Reuleaux polygon rolls on it, the axle at the centroid of the polygon experiences no vertical displacement.
My problem is that it does not seem straightforward, at least to me, how to construct the corresponding differential equation for the road, as presented in the paper. Since circles roll on horizontal lines, and equiangular spirals roll on inclined lines, I would suppose that the road needed for a rolling Reuleaux would not be piecewise linear. This demonstrates that the "road" cannot be a horizontal line for a Reuleaux triangle, as the axle does not remain level when the curve is rolling.
So, how does one construct the road? A solution for just the Reuleaux triangle would be fine, but a general solution is much better.