Pavel Chalmoviansky

Comenius University Bratislava · Department of Algebra, Geometry and Didactics of Mathematics

We introduce the novel method for estimation of mean and Gaussian curvature and several related quantities for polygonal meshes. The algebraic quadric fitting curvature (AQFC) is based on local approximation of the mesh vertices and associated normals by a quadratic surface. The quadric is computed as an implicit surface, so it minimizes algebraic...

Voronoi diagrams belong to frequently used structures in computational geometry with application in many fields of science. The properties of Voronoi diagram are already studied in various metric spaces – Euclidean, Manhattan, Minkowski, Hausdorff, or Karlsruhe and also in the hyperbolic metric. In this paper, we focus on the Voronoi diagram and it...

Let O=(0,0) be the intersection point of two plane algebraic curves F and G. According to existing results, we know that their intersection multiplicity IO at O satisfies the inequality IO(F,G)≥mn+t, where m and n are the multiplicities of O on F and G respectively, and t is the number of their common tangents at O (counted with multiplicity). The...

The aim of point cloud upsampling is to generate dense and regular point sets from the sparse and usually irregular inputs. Numerous approaches based on deep learning have been recently introduced, since they are able to outperform the most of the optimization-based methods. In this paper, we introduce novel upsampling method PU-QF, which is based...

We prove that intersection multiplicity of two plane curves defined by Fulton's axioms is equivalent to the multiplicity computed using blowup. The algorithm based on the latter is presented and its complexity is estimated. We compute for polynomials over $\Q$ and its algebraic extensions.

The notion of a multi-valued function is frequent in complex analysis and related fields. A graph of such a function helps to inspect the function, however, the methods working with single-valued functions can not be applied directly. To visualize such a type of function, its Riemann surface is often used as a domain of the function. On such a surf...

Multifocal lemniscate is a set of points in E2, whose product of distances to a finite set of fixed points is a constant. In our work, we look for a set of foci and a corresponding radius value, so that the resulting lemniscate approximates the input data set sufficiently accurate. Our algorithm searches for approximating lemniscate by doubling and...

We classify mutual position of a quadratic Bézier curve and a regular quadric in three dimensional Euclidean space. For given first and last control point, we find the set of all quadratic Bézier curves having no common point with a regular quadric. This system of such quadratic Bézier curves is represented by the set of their admissible middle con...

The topology and structure of the ADE singularities in terms of their topological invariants are recalled. A representation of these curves as Riemann surfaces is used to propose a novel technique of visualization of multivalued complex functions. Here, not only the entire domain is displayed, but also the method of domain coloring is extended via...

We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals. By iterating the refinement procedure, we get a limit curve interpolating the data. We show that...

In this paper, a sequential coupling of two-dimensional (2D) optimal topology and shape design is proposed so that a coarsely discretized and optimized topology is the initial guess for the following shape optimization. In between, we approximate the optimized topology by piecewise Bézier shapes via least square fitting. For the topology optimizati...

We describe a method to approximate a segment of the inter- section curve of two implicitly dened surfaces by a rational parametric curve. Starting from an initial solution, the method applies predictor and corrector steps in order to obtain the result. Based on a preconditioning of the two given surfaces, the corrector step is formulated as an opt...

We report on approximate techniques for conversion between the implicit and the parametric representation of curves and surfaces, i.e., implicitization and parameterization. It is shown that these techniques are able to handle general free–form surfaces, and they can therefore be used to exploit the duality of implicit and para-metric representatio...

We describe a method for approximate parameterization of a planar algebraic curve by a rational Bézier (spline) curve. After briefly discussing exact methods for param-eterization and methods for rational interpolation, we de-scribe a new technique for rational parameterization. Our approach is based on the minimization of a suitable non– linear ob...

We develop methods for the variational design of algebraic curves. Our approach is based on truly geometric fairness criteria, such as the elastic bending energy. In addition, we take certain feasibility criteria for the algebraic curve segment into account. We describe a compu- tational technique for the variational design of algebraic curves, usi...

Laser scans of real objects produce data sets (point clouds) which may have holes, due to problems with visibility or with the optical properties of the surface. We describe a method for detecting and filling these holes. After detecting the boundary of the hole, we fit an algebraic surface patch to the neighbourhood and sample auxiliary points. Th...

The modeling of complex shapes usually requires a well-based space of splines. The aim of this work is to give the construction method of such spline space basis over the chosen class of triangulations. This basis has several useful properties | local minimal support, low degree of polynomials. We also present

ó We describe a new method for con- structing a sequence of rened polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from cir- cles which approximate adjacent points and the corresponding normals. By iterating the renement procedure, we get a limit curve interpolating the data. We show th...