Supportedness and tameness differentialless geometry of plane curves

摘要

We introduce a class of planar arcs and curves, called tame arcs, which is general enough to describe (parts of) the boundaries of planar real objects. A tame arc can have smooth parts as well as sharp (non-differentiable) corners; thus a polygonal arc is tame. On the other hand, this class of arcs is restrictive enough to rule out pathological arcs which have infinitely many inflections or which turn infinitely often: A tame arc can have only finitely many inflections, and its total absolute turn must be finite.In order to relate boundary properties of discrete objects obtained by segmenting digital images to the corresponding properties of their continuous originals, the theory of tame arcs is based on concepts that can be directly transferred from the continuous to the discrete domain. A tame arc is composed of a finite number of supported arcs. We define supported digital arcs and motivate their definition by the fact that they can be obtained by digitizing continuous supported arcs. Every digital arc is tame, since it contains a finite number of points, and therefore it can be decomposed into a finite number of supported digital arcs.