Convex Quadrilaterals Research Articles

An evolutionary design of weighted minimum networks for four points in the three-dimensional Euclidean space

Abstract We find the equations of the two interior nodes (weighted Fermat–Torricelli points) with respect to the weighted Steiner problem for four points determining a tetrahedron in R 3 \mathbb{R}^{3} . Furthermore, by applying the solution with respect to the weighted Steiner problem for a boundary tetrahedron, we calculate the twist angle between the two weighted Steiner planes formed by one edge and the line defined by the two weighted Fermat–Torricelli points and a non-neighboring edge and the line defined by the two weighted Fermat–Torricelli points. By applying the plasticity principle of quadrilaterals starting from a weighted Fermat–Torricelli tree for a boundary triangle (monad) in the sense of Leibniz, established in [A. N. Zachos, A plasticity principle of convex quadrilaterals on a convex surface of bounded specific curvature, Acta Appl. Math. 129 (2014), 81–134], we develop an evolutionary scheme of a weighted network for a boundary tetrahedron in R 3 \mathbb{R}^{3} . By introducing the inverse weighted Steiner network with two interior nodes built by two different quantities of the subconscious (remaining weights) for boundary tetrahedra, we describe the evolution of a weighted network with two nodes that have inherited a subconscious. The cancellation of the dynamic plasticity of these weighted networks may be applied to the creation of evolutionary scenarios, in order to prevent the development of a quadrilateral or tetrahedral virus (a monad that has got a subconscious) and the cancerogenesis of quadrilateral cells. We continue by giving the plasticity equations for a generalized weighted minimum network with two nodes that have got a subconscious whose vertices are replaced by Euclidean spheres. This evolutionary approach may be applied to the determination of the bond strengths of molecular structures between atoms in the sense of Pauling. We obtain the analytical solutions of the weighted Fermat–Torricelli problem for the case of pairs of equal weights or one pair of equal weights. We consider as a DNA-like chain a sequence of tetrahedra whose vertices possess some symmetrical weights. By calculating the twist angles of each sequence and by applying the weighted Fermat–Torricelli tree structures with symmetrical weights or weighted Steiner tree structures, we may approximate the curve axis of a DNA-like tree chain. Finally, we construct a minimum tree, which is not a minimal Steiner tree for some boundary symmetric tetrahedra in R 3 \mathbb{R}^{3} , which has two interior nodes with equal weights having the property that the common perpendicular of some two opposite edges passes through their midpoints. We prove that the length of this minimum tree may have length less than the length of the full Steiner tree for the same boundary symmetric tetrahedra, under some angular conditions.

Sequences of ratios of a convex quadrilateral

We introduce the concept of the sequence of the ratios of convex quadrilaterals, identify some properties of these sequences and use them to provide new characterizations for some classic quadrilateral families. The research involves aspects of geometry, arithmetic and mathematical analysis, which converge to produce the results.

Generalized Regularity and the Symmetry of Branches of ''Botanological'' Networks

We derive the generalized regularity of convex quadrilaterals in R^2, which gives a new evolutionary class of convex quadrilaterals that we call generalized regular quadrilaterals in R^2. The property of generalized regularity states that the Simpson line defined by the two Steiner points passes through the corresponding Fermat-Torricelli point of the same convex quadrilateral. We prove that a class of generalized regular convex quadrilaterals consists of convex quadrilaterals, such that their two opposite sides are parallel. We solve the problem of vertical evolution of a ''botanological'' thumb (a two way communication weighted network) w.r to a boundary rectangle in R^2 having two roots,two branches and without having a main branch, by applying the property of generalized regularity of weighted rectangles. We show that the two branches have equal weights and the two roots have equal weights, if the thumb inherits a symmetry w.r to the midperpendicular line of the two opposite sides of the rectangle, which is perpendicular to the ground (equal branches and equal roots). The geometric, rotational and dynamic plasticity of weighted networks for boundary generalized regular tetrahedra and weighted regular tetrahedra lead to the creation of ''botanological'' thumbs and ''botanological'' networks (with a main branch) having symmetrical branches

3D Cartesian vertically boundary-adaptive grids construction for coastal hydrophysics problems

Coastal systems are characterized by coastline complex shape, the presence of far protruding spits, bays of complex shape. The rectangular grids using for modeling hydrodynamic processes in the coastal zone when constructing discrete models leads to significant errors. In this regard, when considering such processes, the initial geometry of the region is often specified in the surface grid form, and disturbances in the system in the form of boundary conditions. Particular attention is paid to the construction of a high-quality computational grid near the boundaries. The paper presents a technology for constructing 3D Cartesian vertically boundary-adaptive grids based on surface 2D grids, which are created using the procedure of minimizing the generalized Dirichlet functional. Previously, this approach has shown its effectiveness in constructing 2D non-degenerate regular grids containing the minimum number of cells (convex quadrangles) for test areas of the Z-shape, such as the «Maltese cross» and others, as well as in the numerical solution of 2D coastal hydrophysics problems.

The Mathematics of the Viereckschanzen of the La Tène Culture

The development of script-using ancient civilizations and their achievements in sciences such as astronomy and mathematics have been well researched. Illiterate cultures were able to attain an adequate level of knowledge and transferred this knowledge to succeeding generations via oral tradition and mnemonic artefacts. The constructions of the square ditched enclosures (Viereckschanzen) of the last phase of the La Tene period (LT D, 150-1 BC) are documentations of the mathematical knowledge of this culture and served as a kind of mnemonic artefact. The constructions comprise the application of Pythagorean theorem (in terms of Pythagorean triangles) and square-root approximation triangles. Via these types of triangles, convex quadrangles are constructed (i.e., trapezium, parallelogram, rectangle, kite, lozenge, and square). The constructions were laid out in the terrain on the basis of a consistent ‘Babylonian/Egyptian’ metrology.

Robust nonconforming polynomial finite elements over quadrilaterals

This work proposes two nonconforming polynomial finite elements over general convex quadrilaterals. The first one is designed for fourth order elliptic singular perturbation problems, and the other works for Brinkman problems, approximating the velocity with piecewise constant pressure. We show the robustness of these methods, namely, the discrete solution converges uniformly in the given parameters of the corresponding model problem. Numerical examples are also provided.

Additive two-dimensional splitting schemes for solving 3D suspension transport problems on optimal boundary-adaptive grids with uniform spacing’s in the vertical direction

The article considers3D matter transport model in dissolved and suspended forms (impurities) in coastal marine systems. The initial boundary value problem numerical solution is carried out on the basis of local2D splitting schemes. In this case, special attention is paid to the hig-hquality computational grid construction. As a rule, in the proposed methods, Cartesian grids are used, a variety of which are grids with boundary adaptation. The technology of constructing 3D vertically uniform boundary-adaptive grids on the basis of surface 2D grids, which is created using the procedure of minimizing the generalized Dirichlet functional, is presented. Previously, this approach has shown its effectiveness in constructing 2D non-degenerate regular grids containing the minimum number of cells (convex quadrangles) for test Z-shape regions, such as the “Maltese cross” and others, as well as in the 2Dhydrophysicsproblems numerical solution of coastal systems.

On Almost Empty Monochromatic Triangles and Convex Quadrilaterals in Colored Point Sets

Let S be a set of N points on the plane in general position, colored arbitrarily with c colors ($$c\in {\mathbb {N}}$$). A subset of points is said to be monochromatic if all its points have the same color. In this paper we show that if N is sufficiently large, then S contains a monochromatic triangle with vertices in S and containing at most $$c-3$$ points in its interior $$(c \ge 4)$$. The particular case $$c=4$$ solves the conjecture of Basu, Bhattacharya, and Das in [6]. For the case of two colors, it is still unknown whether every sufficiently large bichromatic point set contains an empty monochromatic convex quadrilateral [11]. In this direction we show that if S is sufficiently large, then it always contains a convex monochromatic quadrilateral with at most $$2c-3$$ points in its interior $$(c \ge 2)$$. We also show a general result on the number of empty convex quadrilaterals with disjoint interiors. Let P be a set of n points in general position. It is straightforward to see that if the elements of P are the vertices of a convex polygon, $$n \ge 4$$, P always has $$\left\lceil \frac{n-3}{2} \right\rceil $$ empty convex quadrilaterals with disjoint interiors. In this paper we prove that any point set P in general position has at least $$\left\lfloor \frac{n-3}{2}\right\rfloor $$ interior disjoint empty convex quadrilaterals. We also give direct consequences of this theorem related to simultaneously flippable edges in triangulations and convex decompositions of point sets. We show that for any point set P there is always a triangulation that has $$\left\lfloor \frac{n-3}{2}\right\rfloor $$ simultaneously flippable edges. We also show that there is always a convex decomposition of P consisting of triangles and convex quadrilaterals with at most $${3n-2h\over 2}$$ elements, where h is the number of points in the convex hull of P.

The least symmetric crystallographic derivative of the developable double corrugation surface: Computational design using underlying conic and cubic curves

Flat-foldable origami tessellations are a rich source of inspiration in the design of transformable structures and mechanical metamaterials. Among all such tessellations, the developable double corrugation (DDC) surface, popularly known as the Miura-ori, is perhaps the most ubiquitous origami pattern in science, engineering, and architectural design. Origami artists, designers, and researchers in various fields of science and engineering have proposed a range of symmetric variations for this pattern. While designing many such derivatives is straightforward, some of them present considerable geometric or crystallographic challenges. In general, the problem of finding flat-foldable derivatives for a given origami tessellation is more challenging for less symmetric descendants. This paper studies the existence and design of the least symmetric derivative of the Miura fold pattern with minimal unit cell enlargement in the longitudinal direction. The course of this study raises a fundamental problem in the flat-foldability of quadrilateral-shaped flat sheets on fold lines through their vertices. An analytical solution to this general problem is presented along with solutions for the special cases of convex quadrilaterals.

Area formulas for certain convex quadrilaterals

The purpose of this paper is to collect for the first time area formulas for all convex quadrilaterals that can be expressed in terms of their four sides alone. There are presently 23 such classes that we know of. Many mathematics teachers would probably find it challenging just to name that many different classes of quadrilaterals, and even more difficult to tell how to calculate their different areas. Several of these classes of quadrilaterals are included neither in MathWorld, Wikipedia nor in any other encyclopedia that we know of. The formulas can be sorted into four categories as either well known, a special case of another formula, less known, or previously unknown (at least to us). The last category only includes two formulas and we will prove these.

Full [formula omitted]-approximation of linear elasticity on quadrilateral meshes based on [formula omitted] finite elements

For meshes of nondegenerate, convex quadrilaterals, we present a family of stable mixed finite element spaces for the mixed formulation of planar linear elasticity. The problem is posed in terms of the stress tensor, the displacement vector and the rotation scalar fields, with the symmetry of the stress tensor weakly imposed. The proposed spaces are based on the Arnold–Boffi–Falk (ABFk, k≥0) elements for the stress and piecewise polynomials for the displacement and the rotation. We prove that these finite elements provide full H(div)-approximation of the stress field, in the sense that it is approximated to order hk+1, where h is the mesh diameter, in the H(div)-norm. We show that displacement and rotation are also approximated to order hk+1 in the L2-norm. The convergence is optimal order for k≥1, while the lowest order case, index k=0, requires special treatment. The spaces also apply to both compressible and incompressible isotropic problems, i.e., the Poisson ratio may be one-half. The implementation as a hybrid method is discussed, and numerical results are given to illustrate the effectiveness of these finite elements.

Finit element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for linear convex quadrilaterals of cubic order Serendipity and Lagrange families

This paper presents an explicit integration scheme to compute the stiffness matrix of twelve node and sixteen node linear convex quadrilateral finite elements of Serendipity and Lagrange families using an explicit integration scheme and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique, In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties matrices and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+) in the bivariates over a 2-square (-1 ) with the nodes on the boundary and in the interior of this simple domain. The finite elements up to cubic order have nodes only on the boundary for Serendipity family and the finite elements with boundary as well as some interior nodes belong to the Lagrange family. The first order element is the bilinear convex quadrilateral finite element which is an exception and it belongs to both the families. We have for the present ,the cubic order finite elements which havee 12 boundary nodes at the nodal coordinates {(-1,-1),(1,-1),(1,1),(-1,1),(-1/3,-1), (1/3,-1),(1,-1/3),(1,1/3),(1/3,1),(-1/3,1),(-1,1/3),(-1,-1/3)} and the four interoior nodal coordinates at the points (-1/3,-1/3),(1/3,-1/3),(1/3,1/3),(-1/3,1/3)} in the local parametric space ( In this paper, we have computed the integrals of local derivative products with linear denominator (4+) in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be then applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all 12-node and 16-node linear(straight edge) convex quadrilaterals for a polygonal domain which provides the nodal coordinates and the element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is an unknown physical variable and in with Dirichlet boundary conditions over the convex polygonal domain.

Does any convex quadrilateral have circumscribed ellipses?

Abstract The past decades have witnessed several well-known beautiful conclusions on four con-cyclic points. With highly promising research value, we profoundly studied circumscribed ellipses of convex quadrilaterals in this paper. Using tools of parallel projective transformation and analytic geometry, we derived several theorems including the proof of the existence of circumscribed ellipses of convex quadrilaterals, the properties of its minimal coverage area, and locus center, respectively. This simple approach lays a solid foundation for its application to three-dimensional situations, which is namely the circumscribed quadric surface of a solid figure and its wide-range utility in construction engineering. Meanwhile, we have a new insight into innate connection of conic sections as well as a taste of beauty and harmony of geometry.

Finite element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for nine node linear convex quadrilateral of Lagrange family

This paper presents an explicit integration scheme to compute the stiffness matrix of a nine node linear convex quadrilateral element of Lagrange family using symbolic mathematics and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique , In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+ ) in the bivariates over a nine node 2-square (-1 ) with nodes at the points {(-1,-1),(1,-1),(1,1),(-1,1),(0,-1),(1,0).(0,1),(- 1,0),(0,0)} in local parametric space. In this paper, we have computed these integrals in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all nine node linear convex quadrilaterals for a polygonal domain which provides the nodal coordinates and element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is unknown physical variable and in with given Dirichlet boundary conditions over the given convex polygonal domain.

Table cartogram

A table cartogram of a two dimensional m×n table A of non-negative weights in a rectangle R, whose area equals the sum of the weights, is a partition of R into convex quadrilateral faces corresponding to the cells of A such that each face has the same adjacency as its corresponding cell and has area equal to the cell's weight. Such a partition acts as a natural way to visualize table data arising in various fields of research. In this paper, we give a O(mn)-time algorithm to find a table cartogram in a rectangle. We then generalize our algorithm to obtain table cartograms inside arbitrary convex quadrangles, circles, and finally, on the surface of cylinders and spheres.

An All Quadrilateral Automesh Generation Technique and Explicit Integration Scheme for Finite Elements to Solve Some Elliptic Boundary Value Problems in Two Space

This paper proposes the explicit integration scheme for a unique linear convex quadrilateral which can be obtained from an arbitrary linear triangle by joining the centroid to the midpoints of sides of the triangle. The explicit integration scheme proposed for these unique linear convex quadrilaterals is derived by using the standard transformations in two steps. We first map an arbitrary triangle into a standard right isosceles triangle by using an affine linear transformation from global (x, y) space into a local space (u, v). We discretise the standard right isosceles triangle in (u, v) space into three unique linear convex quadrilaterals. We have shown that any unique linear convex quadrilateral in (x, y) space can be mapped into one of the unique quadrilaterals in (u, v) space. We can always map these linear convex quadrilaterals into a 2-aquare in a natural space by an appropriate bilinear transformation. Using these two mappings, we have established an integral derivative product relation between the linear convex quadrilaterals in the (x, y) space interior to the arbitrary triangle and the linear convex quadrilaterals of the local space (u, v) interior to the standard right isosceles triangle. Further , we have shown that the product of global derivative integrals in (x, y) space can be expressed as a matrix triple product P ( X (2 * area of the arbitrary triangle in (x, y) space ) in which P is a geometric properties matrix and is the product of global derivative integrals in (u, v) space.We have shown that the explicit integration of the product of local derivative integrals in (u, v) space over the unique quadrilateral spanning vertices (1/3 ,1/3), (0, ½), (0, 0) , (1/2, 0) is now possible by use of symbolic processing capabilities in MATLAB which are based on Maple – V software package. In a similar manner we have found explicit integrals of ihe shape function and global derivative products as well as the product of shape functions.We use these integral values in computing stiffness matrix for some elliptic equations with constant coefficients.The proposed explicit integration scheme is a useful technique for boundary value problems governed by either a single equation or a system of second order partial differential equations. The physical applications of such problems are numerous in science, and engineering, the examples of single equations are the well known Laplace and Poisson equations with suitable boundary conditions. These problems are already studied in authors recent paper. We have demonstrated the proposed explicit integration scheme to solve the elliptic equations with constant coefficients subject to Dirichlet boundary conditions over simple polygonal domains like 1-square and 2-square.We have demonstrated the propsed scheme and technique to study five typical elliptic boundary value problems.Three FEM models with 2400,5400 and 7776 elements and 2481,5521 and 7921 nodes respectively are used.Tables of fem computed values,theoretical values at selected nodes and the finite element meshes are appended.The findings confirm with surface plots of exact solutions for the considered elliptic boundary value problems. Key words: Explicit Integration, Finite Element Method, Matlab Symbolic Mathematics, All Quadrilateral Mesh Generation Technique,Elliptic equations with constant coefficients ,Dirichlet Boundary Conditions ,Polygonal Domain,2-square,1-square

CI2 for creating and comparing confidence-intervals for time-series bivariate plots

Currently no method exists for calculating and comparing the confidence-intervals (CI) for the time-series of a bivariate plot. The study’s aim was to develop ‘CI2′ as a method to calculate the CI on time-series bivariate plots, and to identify if the CI between two bivariate time-series overlap. The test data were the knee and ankle angles from 10 healthy participants running on a motorised standard-treadmill and non-motorised curved-treadmill. For a recommended 10+ trials, CI2 involved calculating 95% confidence-ellipses at each time-point, then taking as the CI the points on the ellipses that were perpendicular to the direction vector between the means of two adjacent time-points. Consecutive pairs of CI created convex quadrilaterals, and any overlap of these quadrilaterals at the same time or ±1 frame as a time-lag calculated using cross-correlations, indicated where the two time-series differed. CI2 showed no group differences between left and right legs on both treadmills, but the same legs between treadmills for all participants showed differences of less knee extension on the curved-treadmill before heel-strike. To improve and standardise the use of CI2 it is recommended to remove outlier time-series, use 95% confidence-ellipses, and scale the ellipse by the fixed Chi-square value as opposed to the sample-size dependent F-value. For practical use, and to aid in standardisation or future development of CI2, Matlab code is provided. CI2 provides an effective method to quantify the CI of bivariate plots, and to explore the differences in CI between two bivariate time-series.

Finite Element Analysis Of Linear Elastic Torsion For Regular Polygons

This paper presents an explicit finite element integration scheme to compute the stiffness matrices for linear convex quadrilaterals. Finite element formulationals express stiffness matrices as double integrals of the products of global derivatives. These integrals can be shown to depend on triple products of the geometric properties matrix and the matrix of integrals containing the rational functions with polynomial numerators and linear denominator in bivariates as integrands over a 2-square. These integrals are computed explicitely by using symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be applied to solve boundary value problems in continuum mechanics over convex polygonal domains.We have also developed an automatic all quadrilateral mesh generation technique for convex polygonal domain which provides the nodal coordinates and element connectivity.We have demonstrated the proposed explicit integration scheme to solve the Poisson Boundary Value Problem for a linear elastic torsion of a non-circular bar with cross sections having profiles of equilateral triangle, a square and regular polygons (pentagon(5-gon)to icosagon(20gon)) which are inscribed in a circle of unit radius. Monotonic convergence from below is observed with known analytical solutions for the Prandtl stress function and torsional constant .We have shown the solutions in Tables which list both the FEM and exact solutions. The graphical solutions of contour level curves and the corresponding finite element meshes are also displayed.

COMPACTNESS OF SPACES OF CONVEX AND SIMPLE QUADRILATERALS

The space of shapes of quadrilaterals can be identified with $\mathbb{CP}^{2}$. We deal with the subset of $\mathbb{CP}^{2}$ corresponding to convex quadrilaterals and the subset which corresponds to simple (that is, without self-intersections) quadrilaterals. We provide a complete description of the topological closures in $\mathbb{CP}^{2}$ of both spaces. Although the interior of each space is homeomorphic to a disjoint union $\mathbb{R}^{4}\sqcup \mathbb{R}^{4}$, their closures are topologically different. In particular, the boundary of the space corresponding to convex quadrilaterals is homeomorphic to a pair of three-dimensional spheres glued along a Möbius strip while the boundary of the space corresponding to simple quadrilaterals is more complicated.

Optimal Quadrilateral Finite Elements on Polygonal Domains

We propose three quadrilateral mesh refinement algorithms to improve the convergence of the finite element method approximating the singular solutions of elliptic equations, which are due to the non-smoothness of the domain. These algorithms result in graded meshes consisting of convex and shape-regular quadrilaterals. With analysis in weighted spaces, we provide the selection criteria for the grading parameter, such that the optimal convergence rate can be recovered for the associated finite element approximation. Various numerical tests verify the theory. In addition to the bi-k elements, we also investigate the serendipity elements on the graded quadrilateral meshes in the numerical experiments.