Symmetry sets of piecewise-circular curves

Synopsis Piecewise-circular (PC) curves are made up of circular arcs and segments of straight lines, joined so that the (undirected) tangent line turns continuously. PC curves have arisen in various applications where they are used to approximate smooth curves. In a previous paper, the authors introduced some of their geometrical properties. In this paper they investigate the ‘symmetry sets’ of PC curves and one-parameter families of such curves. The symmetry set has also arisen in applications (this time to shape recognition) and its mathematical properties for smooth curves have been investigated by Bruce, Giblin and Gibson. It turns out that the symmetry sets of general one-parameter families of plane curves are mirrored remarkably faithfully by the symmetry sets arising from the much simpler class of PC curves.