Measuring Linearity of Connected Configurations of a Finite Number of $$2D$$ 2 D and $$3D$$ 3 D Curves

Abstract

We define a new linearity measure for a wide class of objects consisting of a set of of curves, in both \(2D\) and \(3D\). After initially observing closed curves, which can be represented in a parametric form, we extended the method to connected compound curves—i.e. to connected configurations of a number of curves representable in a parametric form. In all cases, the measured linearities range over the interval \((0,1],\) and do not change under translation, rotation and scaling transformations of the considered curve. We prove that the linearity is equal to \(1\) if and only if the measured curve consists of two straight line overlapping segments. The new linearity measure is theoretically well founded and all related statements are supported with rigorous mathematical proofs. The behavior and applicability of the new linearity measure are explained and illustrated by a number of experiments.