arXiv subjects

Jakub Řada

Publications and source records attributed to Jakub Řada.

3 recordsLinked to original sources

arXiv2026

4-D Visualization of Minkowski Quaternionic Point Set Operations

The contribution emphasizes the geometric modeling point of view on Minkowski point set operations. In this paper, the Minkowski product is specified as the quaternionic product. Selected point sets are visualized using double orthogonal projection and perspective projection from four-dimensional to three-dimensional space. In particular, we demonstrate the generation of sets containing circles (Clifford torus, 3-sphere), lines (quadratic cone), or both.

math.GM↗

arXiv2024

Geometric Illumination of Implicit Surfaces

Illumination of scenes is usually generated in computer graphics using polygonal meshes. In this paper, we present a geometric method using projections. Starting from an implicit polynomial equation of a surface in 3-D or a curve in 2-D, we provide a semi-algebraic representation of each part of the construction. To solve polynomial condition systems and find constrained regions, we apply algebraic computational algorithms for computing the Gr{\" o}bner basis and cylindrical algebraic decomposition. The final selection of illuminated and self-shaded components for polynomial surfaces of a degree higher than three is discussed. The text is accompanied by visualizations of illumination of surfaces up to degree eight.

cs.CG↗

arXiv2024

3-D Shadows of 4-D Algebraic Hypersurfaces in a 4-D Perspective

The paper is focused on the four-dimensional visualization of hypersurfaces represented by implicit equations without their parametrization. We describe a general method to find shadow boundaries in an arbitrary dimension and apply it in a three- and four-dimensional space. Furthermore, we design a system of polynomial equations to construct occluding contours of algebraic surfaces in a 4-D perspective. The method is presented on a composed 3-D scene and three 4-D cases with gradual complexity. In general, our goal is to improve the understanding of spatial properties in a four-dimensional space.

math.GM↗