**Research
Description**

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?

At the

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 **R**^{3} to **R**^{3}. 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 **R**^{3} 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.