Abstract
We study the double-cross matrix descriptions of polylines in the two-dimensional plane. The double-cross matrix is a qualitative description of polylines in which exact, quantitative information is given up in favour of directional information. First, we give an algebraic characterization of the double-cross matrix of a polyline and derive some properties of double-cross matrices from this characterisation. Next, we give a geometric characterization of double-cross similarity of two polylines, using the technique of local carrier orders of polylines. We also identify the transformations of the plane that leave the double-cross matrix of all polylines in the two-dimensional plane invariant.
Highlights
Introduction and Summary of ResultsPolylines arise in Geographical Information Science (GIS) in a multitude of ways
We identify the transformations of the plane that leave the double-cross matrix of all polylines invariant
We have studied the double-cross matrix descriptions of polylines in the two-dimensional plane from an algebraic and geometrical point of view
Summary
Polylines arise in Geographical Information Science (GIS) in a multitude of ways. One recent example comes from the collection of moving object data, where trajectories of moving persons (or animals), that carry GPS-equipped devices, are collected in the form of time-space points that are registered at certain (ir) regular moments in time. The double-cross formalism is used, for instance, in the qualitative trajectory calculus, which, in turn, has been used to test polyline similarity with applications to query-by-sketch, indexing and classification [27]. The similarities of the plane are the translations, rotations and homotecies (scalings) of the plane This result allows us, for instance, to prove any statement about double-cross matrices of a polyline, only for polylines start in the origin of the two-dimensional plane and have a unit length first line segment. We identify the transformations of the plane that leave the double-cross matrix of all polylines invariant
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