THEORY OF SURFACES IN FOUR-DIMENSIONAL GALILEAN SPACE

By means of a special choice of coordinate lines of the surface in four-dimensional Galilean space, the first and second quadratic shape of the surface is defined. It has been proved that the second-order surface equation in three-dimensional space can be converted to a canonical form by means of a special transformation, which is the rotation of the coordinate axes of three-dimensional Galilean space. Furthermore, the transformation matrix is an element of the Heisenberg group that is neither symmetric nor orthogonal. In four-dimensional space R41 - the concept of a surface indicator is introduced and the main curvature of the surface is defined.

On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability

It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential equation of third order, for the surface representing the stratum. In particular, the approach adopted here leads to natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature with one family of coordinate lines representing the fibres. Integrable cases are isolated by requiring that the Gauss equation be compatible with another third-order hyperbolic differential equation. In particular, a variant of the integrable Tzitz\'eica equation is derived which encodes orthogonal coordinate systems on pseudospherical surfaces. This third-order equation is related to the Tzitz\'eica equation by an analogue of the Miura transformation for the (modified) Korteweg-de Vries equation. Finally, the formalism developed in this paper is illustrated by focussing on the simplest ``fluid sheets'' of constant Gaussian curvature, namely the plane, sphere and pseudosphere.

4-dimensional lattice models for the quantum range

Some finite subspace models L are presented for quantum structures which replace the use of countable infinite Hilbert space H dimensions. A maximal Boolean sublattice, called block, is 24, where its four atoms directly above 0εL, base vectors of H in 24 are drawn as four points on an interval. Blocks can overlap in one or two atoms. Different kinds of operators can map one block onto another and interpretations are given such that subspaces can carry on their base vector tuple real, complex or quaternionic numbers, energies, symmetries and generate coordinate lines. Describing states of physical systems is done using L and its applications for dynamical modelling. They don‘t need the infinte dimensional vectors of H. L has in the first model 11 blocks and 24 atoms (figure 1). They correspond to the 24 elements of the tetrahedral S4 symmetry. S4 arises from a spin-line rgb-graviton whirl operator with center at the tip of a tetrahedron and a nucleon triangle base with three quarks as vertices. The triangles factor group D3 of S4 is due to the CPT Klein normal subgroup Z2 x Z2 of S4 . It has a strong interaction SI rotor for the nucleons inner dynamics which is used for integrating functions, exchanging energies of nucleon with its environment and setting barycentrical coordinates in the triangle. At their intersection B as barycenter sets a Higgs boson or field the rescaled quark mass of a nucleon. Each factor class of one element from D3 assigns to it a color charge, a coordinate, an energy vector and a symmetry. Symmetries attached can be different according to interactions involved. Every atom of L has then a specific character with different properties.Three characters are added to octonian base vectors, listed by their indices as n = 0,1,…,7, and named for the atoms of L as na, nb, nc. The structure and element attributes of the finite subspace lattices L are desribed in many examples and models which technical constructed run macroscopically. Several models are described below. Example, the color charge whirl as rgb-graviton projection operator maps the block 2c3b5a6a to 0a1a2a3a. The symmetries change dimension from 3x3- to 2x2-matrices. From SU(3) are λ1 on 3b mapped to the SU(2) x-coordinate Pauli matrix σ1, from λ2 on 5a to σ2 y-coordinate and from λ3 on 6a to σ3 z-coordinate of real Euclidean space R³. The SU(3) matrices have complex w3 = z +ict, w2 = (iy,f), w1 = (x,m) coordinates. In figure 3 is shown how a rotation of two proton tetrahedrons for fusion changes the two linearly independent wj vectors to the 1-dimensional x,y,z base vectors. In deuteron then on one coordinate line sit with Cooper paire u-d-quarks at the ends the Heisenberg coupled energy or space vector rays 15 (x,m), m mass measured in kg, x in meter, 23 (iy,E(rot)), E(rot) rotational energy measured in Joule J, y in meter, 46 (ict,f), t time measured in seconds, f = 1/∆t frequency s inverse time interval measured in Hz. The six color charges are red r on +x as octonian coordinate 1, green g on +y as 2 , blue b on -z as 6, turquoise on -x as 5, magenta on -y as 3, yellow on +z as 4..

APPLICATION OF CIRCLES FOR CONJUGATION OF FLAT CONTOURS OF THE FIRST ORDER OF SMOOTHNESS

In geometric modeling of contours, especially for conjugation of sections of flat contours of the first order of smoothness, arcs of circles can be applied. The article proposes ways to determine the equations of a circle for two ways of its problem: the problem of a circle with a point and two tangents, none of which contains a given point, and the problem of a circle with three tangents. The equations of the circles were determined in both cases using a projective coordinate system. In the first case, when a circle is given by a point and two tangents, neither of which contains this point, the center of the conjugation circle is defined as the point of intersection of two locus of points - the bisector of the angle between the tangents and the parabola, the focus of which is a given point. given tangents. In the general case, there are 2 conjugation circles for which canonical equations are defined. Parametric equations of conjugate circles, the parameters of which are equal to 0 and ∞ on tangents and equal to one at a given point, with the help of affine and projective coordinates of points of contact are determined first in the projective coordinate system, and then translated into affine system. For the second case, when specifying a circle using three tangent lines, the equation of the second-order curve tangent to these lines is first determined in the projective coordinate system. The tangent lines are taken as the coordinate lines of the projective coordinate system. The unit point of the projective coordinate system is selected in the metacenter of the thus obtained base triangle. The equation of the tangent to the base lines of the second order contains two unknown variables, positive or negative values ​​which determine the location of four possible tangents of the second order. After writing the vector-parametric equation of the tangent curve of the second order in the affine coordinate system, the equation is written to determine the parameters of cyclic points. In order for the equation of the tangent curve of the second order obtained in the projective plane to be an equation of a circle, it must satisfy the coordinates of the cyclic points of the plane, which allows to write the second equation to determine the parameters of cyclic points. By solving a system of two equations, we obtain the required equations of circles tangent to three given lines.

The mapping of raster images on plane isometric grid

Drawing images on curvilinear shapes with the least distortion takes place in many design tasks. In most ways, build a grid, each elementary cell of which is painted a given color. In this problem it is necessary to solve two main problems: the first - to carry out the formation of a given curvilinear grid with elementary cells in the form of squares, which are called isometric (or isothermal); the second is to paint each cell of the curved area with the corresponding pixel color of the original raster The aim of the study is to reveal the way of displaying raster images on flat curvilinear areas represented by isometric grids, and with the help of a computer model in the Maple symbolic algebra to analyze the influence of isometric grid parameters on the position and size of displayed raster images. The mapping of images onto curvilinear forms with minimal distortion takes place in many design tasks. A method of conformal mapping of arbitrary raster images onto plane curvilinear region is proposed, which are represented by isometric (also called isothermal) grids. The essence of the proposed method is as follows. Any raster image, for example, digital photography in jpg format, is characterized by the dimensions N×M - the number of pixels in width and height. In addition, each pixel has a color and brightness, which are arranged in rows and columns. To apply a raster image to a curvilinear region, it is also necessary to divide the curvilinear domain into N×M, the number of elementary squares, each of which is assigned the corresponding color from the raster. The influence and arguments of the various isometric grids constructed on the sizes and positions of an arbitrary raster image are investigated in the article. It is shown how the isometric grid, depending on and localizes the raster image - it can be located both within the limits of the isometric grid coordinate lines and beyond it, can also be oriented in different directions with respect to the and coordinate lines. It is shown the possibility of scaling a raster image that can be performed relative to the relative dimensions of an isometric grid. Since there is a correspondence between the pixel matrix of the original raster image and the - cells of the isometric grid, the rotation of the image will affect its position in the isometric grid. For example, rotating the original bitmap image at an angle 90 degrees will change its location on a plane isometric grids – from along the coordinate lines to along the coordinate lines. Note that, the curvilinear cells of the constructed isometric grids differ somewhat from the shape of the squares because the values and of the corresponding arguments and of their coordinate lines were taken somewhat too large. Otherwise, cells would degenerate into points and the corresponding grid image would not be so clear.

Compressible Euler equations on a sphere and elliptic–hyperbolic property

In this work, we systematically derive the governing equations of supersonic conical flow by projecting the 3D Euler equations onto the unit sphere. These equations result from taking the assumption of conical invariance on the 3D flow field. Under this assumption, the compressible Euler equations reduce to a system defined on the surface of the unit sphere. This compressible flow problem has been successfully used to study the steady supersonic flow past cones of arbitrary cross section by reducing the number of spatial dimensions from three down to two while still capturing many of the relevant 3D effects. In this paper, the powerful machinery of tensor calculus is utilized to avoid reference to any particular coordinate system. With the flexibility to use any coordinate system on the surface of a sphere, the equations can be more readily solved numerically when a structured mesh is used by defining the mesh lines to be the coordinate lines. The type of the system of partial differential equations would be hyperbolic or elliptic based on whether the crossflow Mach number is supersonic or subsonic.

Synthesis of Equations For Ruled Surfaces With Two Curvilinear And One Rectangular Directrixes

Ruled surfaces have long been known and are widely used in construction, architecture, design and engineering. And if from the technical point of view the developable surfaces are more attractive, then architecture and design successfully experiment with non-developable ones. In this paper are considered non-developable ruled surfaces with three generators, two of which are curvilinear ones. According to classification, such surfaces are called twice oblique cylindroids. In this paper has been proposed an approach for obtaining of twice oblique cylindroids by immersing a curve in a line congruence of hyperbolic type. Real directrixes of such congruence are a straight line and a curve. It has been proposed to use helical lines (cylindrical and conical ones) as a curvilinear directrix, and a helical line’s axis as the straight one. Then the congruence’s rectilinear ray will simultaneously intersect the helical line and its axis. Congruence parameters are the line’s pitch and the guide cylinder or cone’s radius. The choice of the curvilinear directrix is justified by the fact that the helical lines have found a wide application in engineering and architecture. Accordingly, the helical lines based surfaces can have a great potential. In this paper have been presented parametric equations of the considered congruences. The congruence equations have been considered from the point of view related to introducing a new curvilinear coordinate system. The obtained system’s coordinate surfaces and coordinate lines have been also studied in the paper. To extract the surface, it is necessary to immerse the curve in the congruence. To synthesize the equations has been used a constructive-parametric method based on the substitution of the immersed line’s parametric equations in the congruence equations according to a special algorithm. In the paper have been presented 5 examples for the synthesis of ruled surfaces equations such as the twice oblique cylindroid and their visualization. The method is universal and algorithmic, and therefore easily adaptable for the automated construction of surfaces with variable parameters of both the congruence and the immersed line.

Seasonal range test run with Global Eta Framework

Global Eta Framework (GEF) is a global atmospheric model developed in general curvilinear coordinates and capable of running on arbitrary rectangular quasi-uniform spherical grids, using stepwise (Eta) representation of the terrain. In this study, the model is run on a cubed-sphere grid topology, in a version with uniform Jacobians (UJ), which provides equal-area grid cells, and a smooth transition of coordinate lines across the edges of the cubed-sphere. Within a project at the Brazilian Center for Weather Forecasts and Climate Studies (CPTEC), a nonhydrostatic version of this model is under development and will be applied for seasonal prediction studies. This note describes preliminary tests with the GEF on the UJ cubed-sphere in which model performance is evaluated in seasonal simulations at a horizontal resolution of approximately 25 km, running in the hydrostatic mode. Comparison of these simulations with the ERA-Interim reanalyses shows that the 850 hPa temperature is underestimated, while precipitation pattern is mostly underestimated in tropical continental regions and overestimated in tropical oceanic regions. Nevertheless, the model is still able to well capture the main seasonal climate characteristics. These results will be used as a control run in further tests with the nonhydrostatic version of the model.

General Method for Modeling Slope Discontinuities and T-Sections Using ANCF Gradient Deficient Finite Elements

Slope discontinuities and T-sections can be modeled in a straight forward manner using fully parameterized absolute nodal coordinate formulation (ANCF) finite elements that have a complete set of gradient vectors. Linear transformations that define the element connectivity can always be obtained and used to preserve ANCF desirable features that include constant mass matrix and zero Coriolis and centrifugal forces in the case of spinning structures. The objective of this paper is to develop a general method that allows for modeling slope discontinuities and T-sections using gradient deficient ANCF finite elements that do not have a complete set of coordinate lines and gradient vectors. Linear connectivity conditions that preserve all the ANCF desirable features including the constant mass matrix are developed at the nodes of slope discontinuities. At these nodes of discontinuity, one can always define a complete set of independent coordinate lines that lie on the structure. These coordinate lines can be used to define a complete set of independent gradient vectors at these nodes. Since the proposed method is based on linear coordinate transformations, the method can be implemented in a preprocessor computer program. The application of the proposed general method is demonstrated using ANCF gradient deficient beam element example.