SUPPORTEDNESS AND TAMENESS DIFFERENTIALLESS GEOMETRY OF PLANE-CURVES

We introduce a class of planar arcs and curves, called tame arcs, whic h is general enough to describe (parts of) the boundaries of planar re al objects. A tame are can have smooth parts as well as sharp (non-dif ferentiable) corners; thus a polygonal are is tame. On the other hand, this class of arcs is restrictive enough to rule out pathological arc s which have infinitely many inflections or which turn infinitely ofte n: A tame are can have only finitely many inflections, and its total a bsolute turn must be finite. In order to relate boundary properties of discrete objects obtained by segmenting digital images to the corresp onding properties of their continuous originals, the theory of tame ar cs is based on concepts that can be directly transferred from the cont inuous to the discrete domain. A tame are is composed of a finite numb er of supported arcs. We define supported digital arcs and motivate th eir definition by the fact that they can be obtained by digitizing con tinuous supported arcs. Every digital are is tame, since it contains a finite number of points, and therefore it can be decomposed into a fi nite number of supported digital arcs. (C) 1998 Pattern Recognition So ciety. Published by Elsevier Science Ltd. All rights reserved.