Analytic properties of plane offset curves

Abstract

We survey the principal geometric and topological features of plane offset curves. With appropriate sign conventions, the irregular points of the offset at distance d from a regular generator curve arise where the generator has curvature κ=− 1 d . Usually, this induces a cusp on the offset, but if κ is also a local extremum, we observe instead a tangent-continuous extraordinary point of infinite curvature. Such irregular points are intimately related to the evolute, or locus of centers of curvature, of the generator. Certain special regular points are then identified: those of horizontal or vertical tangent, and those where the curvature or its derivative vanish. A one-to-one correspondence (with due allowance for irregularities) is established between such characteristic points on the generator and its offsets at each distance d. In the absence of irregular points, simple relations between certain global properties of the generator and offset curves, such as their arc length, the area they bound, and their mean square curvature or “smoothness” may be derived. The self-intersections of offset curves, and the trimming of certain extraneous loops they delineate, are also addressed.