Rectifiability Property for Plane Paths and Descent Curves

A family of plane, oriented and continuous paths depending on a fixed real positive number is considered. For any point on the path, the previous points lie out of any disk with same radius, having interior normal in a suitable tangent cone to the path. These paths are locally descent curves of a nested family sets of same positive reach. Avoiding any smoothness requirements, we get angle estimate and not intersection property. Afterwards, we are able to estimate the length and detour of this curve.