Dupin Cyclides as a Subspace of Darboux Cyclides

Dupin cyclides are interesting algebraic surfaces used in geometric design and architecture to join canal surfaces smoothly and construct model surfaces. Dupin cyclides are special cases of Darboux cyclides, which in turn are rather general surfaces in R^3 of degree 3 or 4. This article derives the algebraic conditions (on the coefficients of the implicit equation) for recognition of Dupin cyclides among the general implicit form of Darboux cyclides. We aim at practicable sets of algebraic equations describing complete intersections inside the parameter space.Comment: 20 pages, 1 figur

Darboux cyclides and webs from circles

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Moebius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure

Affine equivalences of surfaces of translation and minimal surfaces, and applications to symmetry detection and design

We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces, and therefore, in particular, the symmetries of a surface of translation or a minimal surface of the considered types. Additionally, we apply our results to designing surfaces of translation and minimal surfaces with symmetries, and to computing the symmetries of the higher-order Enneper surfaces.publishedVersio

On organizing principles of Discrete Differential Geometry. Geometry of spheres

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse

Foliations 2016. Paweł Walczak’s 50 Years in Mathematics. Program & Abstracts. Będlewo, July 11–17, 2016, 7th European Congress of Mathematics Satellite Conference

The international conference Foliations 2016 is in the series of conferences on foliations organized in Poland (1990, 1995, 2000, 2005, 2012). This time we celebrate one of the most recognizable person in the field Professor Paweł Walczak from Uniwersytet Łódzki who coorganized all the previous events. The conference is hosted by the Research and Conference Centre in Będlewo, Poland – a part of Mathematical Institute which belongs to the Polish Academy of Sciences, and takes place in the period of July 11-17, 2016. Foliations 2016 is a satellite of the 7th European Congress of Mathematics (July 18-22, 2016, Berlin, Germany). We wish you a pleasant stay in Będlewo.Stefan Banach Intermational Mathematical Center, Warsaw Center of Mathematics and Computer Science, Polish Academy of Science, Faculty of Mathematics and Computer Science, University of Łódź