Spatial Geometric Cells — Quasipolyhedra

Tiling of three-dimensional space is a very interesting and not yet fully explored type of tiling. Tiling by convex polyhedra has been partially investigated, for example, works [1, 15, 20] are devoted to tiling by various tetrahedra, once tiling realized by Platonic, Archimedean and Catalan bodies. The use of tiling begins from ancient times, on the plane with the creation of parquet floors and ornaments, in space - with the construction of houses, but even now new and new areas of applications of tiling are opening up, for example, a recent cycle of work on the use of tiling for packaging information [17]. Until now, tiling in space has been considered almost always by faceted bodies. Bodies bounded by compartments of curved surfaces are poorly considered and by themselves, one can recall the osohedra [14], dihedra, oloids, biconuses, sphericon [21], the Steinmetz figure [22], quasipolyhedra bounded by compartments of hyperbolic paraboloids described in [3] the astroid ellipsoid and hyperbolic tetrahedra, cubes, octahedra mentioned in [6], and tiling bodies with bounded curved surfaces was practically not considered, except for the infinite three-dimensional Schwartz surfaces, but they were also considered as surfaces, not as bodies., although, of course, in each such surface, you can select an elementary cell and fill it with a body, resulting in a geometric cell. With this work, we tried to eliminate this gap and described approaches to identifying geometric cells bounded by compartments of curved surfaces. The concept of tightly packed frameworks is formulated and an approach for their identification are described. A graphical algorithm for identifying polyhedra and quasipolyhedra - geometric cells are described.

MODELING OF THE 2ND ORDER CURVES AND SURFACES OF ENGINEERING STRUCTURES SHELLS BASED ON THEIR BASIS

The paper describes an example of modeling an arc of a 2nd order curve using an engineering discriminant and its analytical description based on a graphical algorithm for constructing a curve in point calculus. Examples of modeling the surfaces of engineering structures shells on an elliptical and rectangular plan are given. Research methods include geometric algorithms: modeling of 2nd order curves passing through 3 predetermined points in advance and having tangents at the start and end points, and shell surfaces based on them; analytical definition of curves arcs and sections of surfaces using the mathematical apparatus point calculation in a given parametrization and taking into account all predetermined geometric conditions. This approach can be widely used in the practice of modeling the shells of engineering structures for various technical purposes. It allows the designer to choose the best curvature of the shell surface, which will have the necessary strength characteristics, technical aesthetics and artistic expressiveness. The possibility of dividing the surface of the shell into finite elements of a given amount is also provided for studying the stress-strain state of the shell under the action of various loads in the systems of finite element analysis.

Imaginary Points in Cartesian Coordinate System

Geometric models are considered that allow symbolic representation of imaginary points on a real Cartesian coordinate plane XY. The models are based on the fact that through every pair of imaginary conjugate points A~B with complex coordinates x = a ± jb, y = c ± jd one unique real line m passes. For the image of imaginary points, it is proposed to use the graphic symbol m{OL} consisting of the line m passing through the imaginary points, the center O of the elliptic involution σ with imaginary double points A~B on the line m, and the Laguerre point L, from which the corresponding points involutions σ are projected by an orthogonal pencil of lines. According to A.G. Hirsch, the symbol m{OL} is called the marker of imaginary conjugate points A~B. A theorem is proved that establishes a one-to-one correspondence between the real Cartesian coordinates of the points O, L of the marker, and the complex Cartesian coordinates of the pair of imaginary conjugate points represented by this marker. The proved theorem allows us to solve both the direct problem (the construction of a marker depicting these imaginary points) and the inverse problem (the determination of the Cartesian coordinates of imaginary points represented by the marker). A graphical algorithm for constructing a circle passing through a real point and through a pair of imaginary conjugate points is proposed. An example of the graph-analytical determination of the Cartesian coordinates of imaginary points of intersection of two conics that have no common real points is considered.

The Method of Substitutive Circles Family: Application in Catia Design Environment for Gear Shaped Tool Profiling

Tools which generate by enveloping using the rolling method may be profiled using various methods. The substitutive circles family method is a complementary method developed based a specifically theorem, in which is determined a family of circles associated with the blank’s centrode, family which envelop the profile to be generate. The method assumes the determination of the circles family, transposed in the rolling process between the blank and tool centrodes. In this paper is proposed an algorithm for curling surfaces in enveloping, associated with a pair of rolling circular centrodes. The graphical algorithm is based on the representation of the circles family enveloped the blank’s profile. It is generated the circles family transposed on the centrode associated with the gear shaped cutter and is determined a new position of contact points with the blank. The assembly of these points forms the profile of the gear shaped cutter. The numerical data proof the proposed method quality.

The Evolution of Design: SALTS New Sail Training Schooner Project

The Sail and Life Training Society is building a new purpose-designed 35m wooden sail-training schooner for unrestricted foreign-going operations. Working with an international team of consultants, SALTS has initiated an ambitious agenda of analytical and experimental investigations to support design, including a parametric study of hull form as it relates to stability at high angles of heel, the development of bespoke parametric design and analysis tools using the graphical algorithm editor Grasshopper, a towing tank campaign at the Wolfson Unit to investigate the behavior of three keel profiles, and a wind tunnel campaign at Politecnico di Milano to investigate the behavior of fifteen sail plans. Preliminary results from these studies will be presented, set in the context of the unfolding story of the evolution of the design of the new vessel.

The Robinson―Schensted Correspondence and $A_2$-webs

International audience The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the 3$n$th tensor power of $V^+$ correspond bijectively to $[n,n,n]$ standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov–Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan–Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson–Schensted correspondence.