ISSN 8756-6990, Optoelectronics, Instrumentation and Data Processing, 2015, Vol. 51, No. 6, pp. 593–600. c  Allerton Press, Inc., 2015. Original Russian Text c  S.I. Vyatkin, 2015, published in Avtometriya, 2015, Vol. 51, No. 6, pp. 70–78. MODELING IN PHYSICAL AND TECHNICAL RESEARCH Method of Interactive Modeling of Functionally Defined Objects without Preliminary Triangulation of the Surface S. I. Vyatkin Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences, pr. Akademika Koptyuga 1, Novosibirsk, 630090 Russia E-mail: sivser@mail.ru Received November 13, 2014 Abstract—The process of interactive modeling of geometric shapes defined by mathematical functions is studied. A method of interactive modeling of functionally defined objects without preliminary trian- gulation is proposed. This allows more accurate definition of three-dimensional shapes and simplifies the modeling system. A method adapted for graphic processing units is used for rapid visualization of objects. Keywords: interactive modeling, geometric objects, perturbation functions. DOI: 10.3103/S8756699015060114 INTRODUCTION The possibility of interactive generation of shapes of three-dimensional objects was considered in [1–3]. Before those publications, no reliable methods for modeling functional objects were developed. The objects were described manually in a text form. This operation was extremely labor-consuming. Therefore, it became very important to have a suitable source environment for editing three-dimensional objects. Systems for modeling polygonal objects were first developed. The AutoCAD computer-aided design system (version R11) offered a possibility of three-dimensional simulations. As an alternative for this system of computer-aided engineering, a bCAD system was developed, which provided both two-dimensional and three-dimensional design technologies and built-in visualization tools. More recent developments include various commercial systems, such as SoftImage, Alias Wavefront (Maya), 3D Studio Max, LightWave 3D, Modo 3D software system for three-dimensional modeling and rendering, etc. There are interactive systems for modeling on the basis of polynomials, implicit surfaces [4], triangular grids [5], images [6], volumes [7], and functionally defined models [8, 9]. For polygonal models, it is difficult to ensure transformations of geometric objects with the use of oper- ations over functions, which is necessary to perform unary and binary geometric operations and to detect collisions between the objects. In the case of polygonal definition of objects, it is rather difficult to compute surface deformation because all vertices of the triangular grid have to be subjected to geometric calculations. Today’s realistic graphics requires a compact definition of a scene with an accurate description of three- dimensional objects, which ensures effective implementation of various geometric operations with object models in simulations of the behavior of interacting objects. Thus, the low level of realism of polygonal models and limited functional capabilities in terms of formation of graphical scenes give rise to an academic and simultaneously applied problem. Solving this problem requires the development of theoretical grounds for modeling and visualization of three-dimensional objects of a virtual medium, which would eliminate the drawbacks typical for polygonal definition of object models. Three basic types of functions are usually used for modeling object shapes: parametric, explicit, and implicit functions. Parametric functions are included into many modeling systems, in particular, the NURBS 593