Using a general Green function formulation, we re-derive, both, (i) Spitzer and his followers results for the winding angle distribution of the planar Brownian motion, and (ii) Edwards-Prager-Frisch results on the statistical mechanics of a ring polymer entangled with a straight bar. In the statistical mechanics part, we consider both cases of quenched and annealed topology. Among new results, we compute exactly the (expectation value of) the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory (= ring polymer). We also consider the generalizations of the problem for the finite diameter (disc-like) obstacle and winding within a cavity.