On the detection of parallel curves: Models and representations

In this paper, we study three parallel models for curves. Based on their common properties we develop an algorithm for the detection of parallelism among curves. Furthermore, to speed up processing, we developed theorems for fast verification at critical points. False alarm is prevented by an enhancement in tangent representation, thus no postprocessing is required. The new tangent representation is called direction-dependent tangent (DDT). It incorporates concavity information into the tangent representation and prohibits false matching. Based on the theorems and the tangent representation, we show that parallelism detection can be formulated as a correspondence problem. The algorithm handles also partial parallelism, i.e., parallelism in certain ranges along the curves but not lasting throughout the whole curves spans. By "curve" we mean any planar curve with C³ continuity. The technique treats both curves and lines alike and is valid for both open and closed curves.