The visualization of vector fields has attracted much attention over the last decade due to the vast variety of applications in science and engineering. Topological methods have been used intensively for global structure extraction and analysis. Recently, there has been a growing interest in local structure analysis due to its connection to automatic feature extraction and speed. We present an algorithm that extracts local topological structure of arbitrary regions in a two-dimensional vector field. It is based on a mathematical analysis of the topological vector field structure in these regions. The algorithm deals with piecewise linear vector fields and arbitrary polygonal regions. We have tested the algorithm for well known analytic vector fields and data sets resulting from computational fluid dynamics.