Abstract.A centrally symmetric plane curve has a point called it's centre of symmetry. We define (following Janeczko) a set which measures the central symmetry of an arbitrary strictly convex plane curve, or surface in R 3 . We investigate some of it's properties, and begin the study of non-convex cases.

The affine distance symmetry set (ADSS) of a plane curve is an affinely invariant analogue of the euclidean symmetry set (SS) [7], [6]. We list all transitions on the ADSS for generic 1-parameter families of plane curves. We show that for generic convex curves the possible transitions coincide with those for the SS but for generic non-convex curves, further transitions occur which are generic in 1-parameter families of bifurcation sets, but are impossible in the euclidean case. For a nonconvex curve there are also additional local forms and transitions which do not fit into the generic structure of bifurcation sets at all. We give computational and experimental details of these.