Finding a Curve of Space by its Cyclographic Image

Abstract

The present paper explores the questions of development and practical application of the constructive method of geometric modeling. In particular, the paper justifies possibility of solution to the inverse problem of cyclographic mapping of a curve of space R³, i.e. reconstruction of a spatial curve given its cyclographic projection. It is proven that knowing either orthogonal and one of the two branches of the cyclographic projection of a curve of space R³ in plane z = 0, or both of its branches, is sufficient to determine a curve of space. The spatial curve, its orthogonal and cyclographic projections have common parameterization, which allows one to establish point-to-point bijection between these three elements and perform solutions of the direct and the inverse problems of cyclographic modeling of a spatial curve. The paper formulates and establishes theses justifying the possibility of analytic solution of practical tasks of cyclographic modeling, for example, cutting tool trajectory calculation for high-precision pocket machining of machine-building products on NC units. The algorithm for solution to the inverse problem is demonstrated on examples.