Tangents to curves and a dyadic parameterization
It is well-known that a rectifiable curve has a tangent almost everywhere with respect to arclength measure. The purpose of this paper is to characterize those plane curves which have a tangent almost everywhere with respect to Hausdorff linear measure. To this end, a method of parametrizing arbitrary Jordan curves is introduced. This parametrization is the limit of a sequence of approximating polygons which are generated by a bisection algorithm, with corners corresponding to dyadic points on the parametric domain \([0,1]\). The characterization of points at which the curve has a tangent is in terms of the sequence of line segments which converge to a given point \(z\). In particular, if the difference of arguments of a pair of successive segments is denoted by \(\theta_ n(z)\), then the tangent properties at \(z\) are almost everywhere characterized by the convergence of \(\sum^ \infty_{n= 1} \theta^ 2_ n(z)\). The connection to harmonic measure is made by way of a result of McMillan.
scientific article; zbMATH DE number 682541