According to Longman Dictionary of Contemporary English the word shape means the outer form of something, that you see or feel. Reasoning about the shape is a common way of describing, representing and visualization of real and abstract objects in computer science, engineering, mathematics, biology, physics, chemistry, medicine, entertainment applications and even daily life. Shape modeling is an interdisciplinary area composing theoretical and experimental results from mathematics, physics, computer graphics, computer-aided design, computer animation, and others fields. Shape modeling and mathematical simulation stand side by side, and one upholds the other.
The heart of my work was solving applied problems of mathematical simulation. In general, a problem of mathematical simulation includes three main parts: mathematical model, numerical method and programming realization. In 1983, I had the opportunity to be involved in the international project of the comet Halley investigation. It is important to mention the results of investigation of the orbital motion of the comet Halley as an example of mathematical simulation. The study of comets motion put forward a problem of developing a mathematical model which could reproduce true motion of the comet with a high degree of accuracy. Development of a new mathematical model and numerical methods in a combination with computer graphics algorithms and methods led to the attainment of the best accuracy in the world for forecasting the appearance of Halley’s Comet.
As mathematical simulation becomes a tool of knowledge, such simulation is now connected with the progress in developing new computers and first of all with parallel and distributed processing. To find a common methodological approach for making distributed systems as easy for programming as conventional systems I have investigated a number of computing problems: gas-dynamics finite differences schemes, sheet forming modeling, solving of large systems of linear algebraic equations, image processing, bioengineering and geometric modeling. As a result of this activity a "serialization" approach in the design of parallel programs was proposed. Now we are able to answer for uneasy question: is it correct to say that a coach harnessed with a team of four horses would move faster that one harnessed with only two?
A challenging goal in CG and CAD is to provide powerful technique for modeling the shape of an object. Indeed, when we are using only generic properties such as position of a point of the object that is deformed the problem of constructing smooth surface satisfying certain constrains can be formulated as a mapping function from R3 to R3. Such space mapping technique based on radial based functions (RBFs) is a powerful tool, which offers simple and quite general control of simulated shapes. In fact, a model of extended space mapping is used and incorporates geometric space mappings and a function mapping from R3 to R. Constructive solid geometry (CSG) is usually used in many CAD applications. Traditionally, CSG modeling uses simple geometric objects for a base model. Actually, RBFs offer a mechanism to get extrapolated points of a surface for various parts of a reconstructed object that can be used as “CSG components” to design a model.
Volume modeling supports combinations of representational styles, including constructive geometry, sweeping, soft objects, voxel-based objects, deformable and other animated objects. A number of unique computer graphics techniques were introduced and reported in more than 30 publications while working on the project. Application examples of aesthetic design, physically based collision's simulation, animation of volumetric data were considered. I introduced an extended Bezier clipping operation for fast inverse mapping in the case of the changed domain. Problems of simulation and analysis of growing mammalian cell colony with the help of genetic algorithms, 3D reconstruction from medial axis transform have been studied. Also I have investigated questions of finite element approximation for visualizing scattered data and for function-to-volume conversion problem. Due to this work volume geometric modeling paradigm was introduced.
As result, was the recipient of the Best WWW Award of EUROGRAPHICS'96. Entries were encouraged from all areas of computer graphics and were judged on basis of technological innovation and contents.
In spite of a flurry of activity in the field of scattered data reconstruction and interpolation, this matter remains a difficult and computationally expensive problem. Despite their long history, RBFs have never really become a widely-used tool for surface generation in CAD. I can state that according to our experiments with various applications of RBFs, for instance an optical design, we have a good alliance of geometric modeling and optimization techniques to determine the reconstructed surface and assure overall smoothness. Another approach, which allows significantly increase size of 3D data sets, is to use compactly supported RBFs. Thus, RBFs seem ready-made for many applications in shape modeling and CG, even for interactive 3D modification, sculpting, image retouching, and real-time animation.