A Minimalist Approach to Rolling Wheels

Referee: [Derivation of road equations for non-differentiable wheels] The manuscript must explicitly define the contact map and prove single-valuedness plus arc-length preservation for nowhere-differentiable rectifiable curves; the step from the no-slip assumption to the integral formula for the road appears to require additional justification (e.g., via approximation or measure-theoretic arguments) that is not visible in the provided abstract and may not follow directly from continuity alone.

Authors: We agree that an explicit definition of the contact map and a self-contained proof of its properties are needed for full rigor. In the revised version we will add a dedicated preliminary section that defines the contact map as the correspondence between arc-length parameters on the wheel and road induced by the no-slip condition. For any rectifiable curve (including nowhere-differentiable ones) we parametrize by arc length; the no-slip assumption then equates the arc-length increments directly, which immediately yields single-valuedness and arc-length preservation because the arc-length function is strictly increasing and continuous. The integral formula for the road is obtained by integrating the appropriately rotated unit tangent of the wheel; this construction is valid in the absolutely continuous sense for rectifiable curves. To justify the passage from the classical differentiable case to the general rectifiable case we will insert a short approximation argument: any rectifiable curve is the uniform limit of smooth curves with the same total length, the corresponding roads converge uniformly, and the limiting road satisfies the integral formula by continuity of the integral. These additions will be placed before the main derivation so that the abstract claim is fully supported in the body of the paper. revision: yes