Modeling of geometric correspondences and transformations via geometry constructions language

Abstract

Geometric correspondences and transformations can be applied in solutions to numerous problems of geometric modeling, mainly in order to create models of curves and surfaces. Among the tools of CAD systems, the geometric transformations are presented in the form of a limited set of editing tools, array tools, tools providing formation of 3D solids and surfaces through motion. In practice, this set of tools restricts application of the wide capabilities of geometric transformations and correspondences. In this regard, it is proposed to model the geometric transformations and correspondences via the geometry construction language. The elements of the geometric transformation theory are analyzed. The geometric transformations are proposed to be created on par with other objects of the geometric model. The commands of the geometry constructions language enabling construction of the transformation objects are considered, their application is discussed. Solutions to practical problems of surface modeling via the geometry constructions language with the use of geometric transformation are demonstrated on examples. The proposed approach allows us to facilitate modeling of curves and surfaces in CAD systems, if said curves and surfaces can be acquired through geometric transformation or correspondence that can be defined via a certain constructive algorithm.