Vyatkin S.I. Conversion of functionally defined forms

Abstract: The author studies geometrical transformation of functionally defined three-dimensional shapes. The paper suggests description of geometrical objects using functions and implementing the methods of transformation of the describing function for geometric operations such as projection, offsetting, set-theoretic and functions of metamorphosis including morphing nonhomeomorphic objects as well as more complex geometric operations: sweeping by moving solid object and twisting of objects. Of all existing methods the functional representation is the most accurate way of describing object geometry, needs less space for storing data required. Functional representation provides compactness and flexibility in setting surfaces and objects obtained as a result of logical operations on volumes. Using functional representation of objects makes it possible to implement new effects on objects due to the introduction of operations on functions. It can be useful in modelling some complex movements of object and particles in scientific applications and games. The method of the research is based on the use of systematic and targeted approach in the evaluation of algorithmic solutions, theory of sets and analytic geometry, interpolation theory and matrix theory, mathematic modeling and theory of computing systems. The main conclusions of the study are: the possibility of implementing complex geometric operations (metamorphosis, projections, offsetting, twisting, sweeping) on objects; proposed method of describing threedimensional scene objects using reference surfaces and functions of the perturbation has a more compact description in comparison with known methods of specifying functionally defined objects; in comparison with known algorithms the rendering algorithm determines the point on the surface of functionally defined objects in less time due to the smaller number of calculations; the proposed functional description of objects simplifies the implementation of the mentioned above operations on geometric functions of the perturbation.

Keywords: