Fiber Strong Shape Theory for Topological Spaces

In the paper we construct and develop a fiber strong shape theory for arbitrary spaces over fixed metrizable space \Bo. Our approach is based on the method of Marde\v{s}i\'{c}-Lisica and instead of resolutions, introduced by Marde\v{s}i\'{c}, their fiber preserving analogues are used. The fiber strong shape theory yields the classification of spaces over \Bo which is coarser than the classification of spaces over \Bo induced by fiber homotopy theory, but is finer than the classification of spaces over \Bo given by usual fiber shape theory

Visualising the structure of architectural open spaces based on shape analysis

This paper proposes the application of some well known two-dimensional geometrical shape descriptors for the visualisation of the structure of architectural open spaces. The paper demonstrates the use of visibility measures such as distance to obstacles and amount of visible space to calculate shape descriptors such as convexity and skeleton of the open space. The aim of the paper is to indicate a simple, objective and quantifiable approach to understand the structure of open spaces otherwise impossible due to the complex construction of built structures.Comment: 10 pages, 9 figure

Shape analysis on Lie groups and homogeneous spaces

In this paper we are concerned with the approach to shape analysis based on the so called Square Root Velocity Transform (SRVT). We propose a generalisation of the SRVT from Euclidean spaces to shape spaces of curves on Lie groups and on homogeneous manifolds. The main idea behind our approach is to exploit the geometry of the natural Lie group actions on these spaces.Comment: 8 pages, Contribution to the conference "Geometric Science of Information '17

On products in the coarse shape categories

The paper is devoted to the study of coarse shape of Cartesian products of topological spaces. If the Cartesian product of two spaces $X$ and $Y$ admits an HPol-expansion, which is the Cartesian product of HPol-expansions of these spaces, then $X\times Y$ is a product in the coarse shape category. As a consequence, the Cartesian product of two compact Hausdorff spaces is a product in the coarse shape category. Finally, we show that the shape groups and the coarse shape groups commute with products under some conditions.Comment: 11 page