In real-time computer graphics, surfaces are typically represented by triangular meshes, as surfaces defined this way can be rendered quickly. However, since triangular meshes are not the most convenient choice for more complex surface operations or analysis, transitioning between representations may become necessary, which is often not a simple task. My research focuses on the geometry, topology, and computer representation of surfaces defined by triangular meshes. It encompasses the mathematical (geometrical and algebraic) theory of triangular meshes and the efficient computational implementation of their local and global operations (e.g., smoothing, subdivision, cutting, and slicing).

In my research, I apply various techniques to the approximation of parametric curves and surfaces. In the parametric case, even formulating the approximation-theoretic problem is not straightforward, and it is challenging to construct an approximation method that delivers acceptable results with a tolerable computational load. In the one-dimensional problem, I work with special parametric curves such as polylines, polygons, and splines; in the two-dimensional case, I focus on constructing approximation methods for surfaces defined on discrete grids and triangular meshes. The most typical applications of the above include simplification, repairing, smoothing, under- and oversampling, and compression of polylines/meshes.

For the computer simulation of the motion of objects represented by triangular meshes, in certain cases, the problem can be reduced to operations performed on triangular faces. In such cases, the task can be simplified so that only the calculation of integrals over triangular domains is necessary to determine the complete dynamic behavior. In this research, I focus on similar tasks and develop specific, numerically stable computational methods for the approximate or exact calculation of the arising integrals. In addition to the simulation of rigid bodies, my research is tangentially related to the physical simulation of deformable and breakable bodies.