The geometry of a smooth line is characterized locally by its curvature and torsion, or globally by its writhe. In many situations of physical interest the line is, however, not smooth so that the classical Frenet description of the geometry breaks down everywhere. One example is a thermalized stiff polymer such as DNA, where the shape of the molecule is the integral of a Brownian process. In such systems a natural frame is defined by parallel transport. In order to calculate the writhe of such non-smooth lines we study the area distributions of random walks on a sphere. A novel transposition of these results occurs in multiple light scattering where the writhe of the light path gives rise to a Berry phase recently observed in scattering experiments in colloidal suspensions.