Global theorems for symmetry sets of smooth curves and polygons in the plane

Abstract

SynopsisThe symmetry set of a plane curve y is the locus of centres of circles which either (i) are tangent to y at at least two distinct points, or (ii) have at least 4-point (A3) contact with y somewhere. The points (ii) also lie at cusps of the evolute of γ and the symmetry set, together with the evolute, can be studied by regarding their union as a full bifurcation set. For information on symmetry sets, see [2, 4, 5, 7, 8].Here we shall use a theorem of Ozawa [9] to deduce global formulae relating three features of symmetry sets. The transitions which occur on the symmetry set of a generic one-parameter family of plane curves have been found [4], and more insight into the global formulae can be gained by tracing them through the transitions and verifying that they are invariant (see Section 2). We also briefly introduce symmetry sets of plane polygons in Section 3, and show that, contrary to first impressions, the same global formulae hold there.