Inverse problem of cyclographic modeling of spatial curve

Abstract

The objective of the present study is to justify the possibility of constructive and analytic solution to the inverse problem of cyclographic modeling of a curve of space R3 and development of a respective algorithm. The orthogonal projection and the two components of the cyclographic projection of a spatial curve form a triad of elements in plane z=0. These elements are the result of the direct problem solution and constitute the basis for the inverse problem solution. The direct problem consists in construction in plane z=0 of a cyclographic projection (a model) of a given spatial curve, while the inverse problem consists in determination of a spatial curve given its cyclographic projection. Insufficient knowledge on the inverse problem as well as its relevance in practical applications, e.g. in cutting tool trajectory calculation for pocket machining of mechanical engineering products on NC units, make urgent the definition and the solution of the inverse problem. In the present paper a simple convex closed curve is considered as the given cyclographic projection. It is proven that there exists a unique spatial curve, for which the given curve constitutes a cyclographic projection. The algorithm for the inverse problem solution is demonstrated on examples.

Keywords

• cyclographic mapping
• medial axis
• medial transformation axis
• inverse task
• α-shell
• vertex points of the curve