Convex Quadrilaterals Research Articles

Numbers of Weights of Convex Quadrilaterals in Weighted Point Sets

<p style="text-align: justify;">Let ℘<sub>n</sub> denote the family of sets of points in general position in the plane each of which is assigned a different number, called a weight, in {1,2,...,<em>n</em>}. For <em>P</em>∈℘<sub>n</sub> and a polygon Q with vertices in <em>P</em>, we define the weight of <em>Q</em> as the sum of the weights of its vertices and denote by <em>W</em><sub>k</sub>(<em>P</em>) the set of weights of convex <em>k</em>-gons with vertices in <em>P</em>∈℘<sub>n</sub>. Let <em>f</em><sub>k</sub>(<em>n</em>) = min<sub><em>P</em>∈℘<sub>n</sub></sub> |<em>W</em><sub>k</sub>(<em>P</em>)|. It is known that <em>n</em>-5 ≤ <em>f</em><sub>4</sub>(<em>n</em>) ≤ 2<em>n</em>-9 for <em>n</em>≥7. In this paper, we show that <em>f</em><sub>4</sub>(<em>n</em>)≥ 4<em>n</em>/3-7.</p>

An Application of Einsteinian-Phythagorean Theorem in Einstein Gyrovector Spaces

In [Ungar 2008; Ungar 2015] A.A. Ungar, employs the Einstein gyrovector spaces for the introduction of the gyrotrigonometry, Ungar’s and other researcher’s works play a major role in translating some theorems from Euclidean geometry to corresponding theorems in Einstein gyrovector spaces. In Euclidean geometry, the sum of the squares of the lengths of opposite sides of convex or concave quadrilaterals whose diagonals intersect perpendicularly is equal to each other. In this paper, we present this theorem in Einstein gyrovector spaces in terms of gamma factors.

ON SELF-SIMILARITY OF QUADRANGLE

Wen [Some problems in fractal geometry, in Report in Chinese Conference on Fractal Geometry and Dynamical Systems, 2023] posed the following questions: when a quadrangle is a self-similar set with the open set condition (OSC). It is well known that any convex quadrangle is a self-similar set. For the concave quadrangles, this case is more complicated. Loosely speaking, some concave polygons may be not self-similar. In this paper, we prove two results. First, we give a necessary and sufficient condition such that a concave quadrangle is a self-similar set. Moreover, we also give a necessary condition for a concave quadrangle to be a self-similar set with the OSC. For the convex quadrangle, we investigate the trapezoid. We show under some condition that the trapezoid is a self-similar set with the OSC. In particular, for any trapezoid, if the ratio of the lengths of two bases is rational, then the trapezoid is a self-similar set with the OSC.

볼록 사각형의 Macbeath inconic과 볼록 사각형의 Orthic inconic에 대한 연구

This study was based on the research results conducted as an R&E project for gifted students with financial support from the Korea Foundation for the Advancement of Science and Creativity. In this study, the Macbeath inconic and Orthic inconic defined in triangles are expanded to convex quadrilateral to study the Macbeath inconic of convex quadrilaterals and the Orthic inconic of convex quadrilaterals. Through this study, the following research results are obtained. First, it is discovered that the condition for a quadrilateral in which the Macbeath inconic of a convex quadrilateral exists. Second, the locations of the two foci of the Macbeath inconic of convex quadrilateral were found. Third, this study also reveals the formulas for the length of the major axis, the length of the minor axis, and the area of the Macbeath inconic of convex quadrilaterals. Fourth, the study unveils the division ratio of the Macbeath inconic of convex quadrilateral. Fifth, it is also discovered that the condition for a quadrilateral in which an orthic inconic of a convex quadrilateral exists. Sixth, the uniqueness of the orthic inconic of the convex quadrilateral is proven. Seventh, the area formula for the orthic inconic of a convex quadrilateral is derived. Lastly, this study shows the division ratio of the orthic inconic of the convex quadrilateral. It is expected that expanding and studying mathematical concepts into a wider area, as in this study, will contribute to the development of mathematics.

볼록 사각형의 Siebeck-Marden의 정리에 대한 연구

This study was based on the research results conducted as an R&E project for gifted students with financial support from the Korea Foundation for the Advancement of Science and Creativity. In this study, we extended Siebeck-Marden’s Theorem, which holds for triangles, into arbitrary convex quadrilaterals. Through this study, the following results were obtained. First, we found and proved the ratio in which the inellipse of a parallelogram divides each side. Second, we discovered Siebeck-Marden’s Theorem for exellipses. We discovered Siebeck-Marden’s Theorem regarding the location of the foci of an exellipse of a triangle. Third, we defined and proved Siebeck-Marden’s Theorem of arbitrary convex quadrilaterals. We defined Siebeck-Marden’s Theorem regarding the location of the foci of an inellipse of an arbitrary convex quadrilateral. In this study, we extended Siebeck-Marden’s Theorem from triangles to convex quadrilaterals. It is expected to contribute to the development of mathematics by enhancing mathematical concepts and properties, just as we extended Siebeck-Marden’s Theorem to arbitrary convex quadrilaterals. Furthermore, this study is expected to encourage further research on the inellipse of convex quadrilaterals.

볼록 연꼴 및 정다각형의 내접 타원에 관한 연구

This study was based on the research results conducted as a R&E project for the gifted students with a financial support from the Korea Foundation for the Advancement of Science and Creativity. Any triangle or convex quadrilateral has an infinite number of inscribed ellipses (Agarwal, Clifford, & Lachance, 2015). In the case of triangles, there is an inscribed ellipse with the focus on any point inside (Park, Park, & Cho, 2020), while the set of points that can be the focus of the inscribed ellipse of a parallelogram forms a specific trace (Park, Park, & Cho, 2021). Therefore, in the case of convex quadrilaterals other than parallelogram, there was a question about how the traces of the focus of the inscribed ellipse were drawn, and among those quadrilateral, the nature of the focus was investigated for the inscribed ellipse of the kite. As a result, in this study, it was proved that these three propositions are equivalent to each other: the point is the focus of the inscribed ellipse; four points, that each is symmetrical to the point for one of the sides, lie on the same circle; the point is a square focus. Based on this, the traces of the focal point of the inscribed ellipse of the kite were shown to be the diagonal which is the axis of symmetry, and the arc that passes through incenter and two vertices that not on the axis of symmetry. Furthermore, using the traces, it could be proved that the only inner ellipse of the regular polygon with 5 or more sides was the incircle, which is meaningful in that it showed the uniqueness of the inscribed ellipse in a different way from the existing projective geometric approach (Agarwal, et al., 2015).

Optimal [formula omitted](div) flux approximations from the Primal Hybrid Finite Element Method on quadrilateral meshes

In this work, we propose and analyze an element-wise H(div,Ω)-conforming computational strategy to post-process the flux vector from the approximated solution of a second-order elliptic equation, obtained by the Primal Hybrid Finite Element Method on regular meshes of convex quadrilaterals generated by bilinear isomorphisms. The recovery strategy makes use of the Arnold–Boffi–Falk (ABFt, t≥0) spaces, leading to locally conservative approximated fluxes with continuous normal components. The proposed method is proven to provide optimal order approximations in H(div,Ω), in the sense that both the flux and its divergence achieve optimal order ht+1 in the L2(Ω)-norm, on quadrilateral meshes. We also show that, within the recovery strategy, it is possible to achieve higher-order approximations for the divergence of the flux by increasing the index t of the local spaces, with almost no extra overall computational cost. Numerical results on affine and bilinear quadrilateral meshes are presented, confirming the theoretical predictions.

On the convergence of the primal hybrid finite element method on quadrilateral meshes

We consider the approximation of a second order elliptic equation (Darcy problem) by using the Primal Hybrid Finite Element Method on quadrilateral meshes. We present new results in terms of sufficient, and in some cases also necessary, conditions to obtain the optimal convergence rates on convex quadrilaterals obtained from bilinear mappings. Numerical experiments are performed to illustrate the theoretical results.

New dualities in convex quadrilaterals

In [1] de Villiers points out the duality between sides and angles of quadrilaterals. The first objective of this article is make explicit two new dualities in the quadrilaterals. For this we will consider the diagonal segments: if diagonals AC and BD of a quadrilateral ABCD intersect at O, we call the diagonal segments OA, OB, OC, OD. According to [2, pp. 179-188], hierarchical classifications of the convex quadrilaterals are made taking as classification criteria the quantity and position of sides, angles and diagonal segments. In hierarchical classifications the more particular concepts form subsets of the more general concepts. Through classification according to the number and position of equal sides it is possible to define six families: four sides equal, at least three equal, two opposite pairs equal, two consecutive pairs equal, at least one opposite pair equal, at least one consecutive pair equal. In the families two opposite pairs equal and two consecutive pairs equal, the pairs may be equal to each other and then the four sides are equal. So in these two families the possibility DA = AB and AB = BC is excluded. Families analogous to the previous ones can be defined taking as a criterion of hierarchical classification the quantity and position of equal angles or the quantity and position of equal diagonal segments.

Boundary-Rigidity of Projective Metrics and the Geodesic X-Ray Transform

We prove that given a compact convex non-empty domain {mathcal {M}} in the plane, a function delta :partial mathcal Mtimes partial {mathcal {M}}rightarrow {mathbb {R}}_+ can be extended to a projective metric d on {mathcal {M}} if and only if delta (P,R)+delta (Q,S)-delta (P,S)-delta (Q,R)>0 for any convex quadrangle Box (PQRS) inscribed in partial {mathcal {M}}. Moreover, this extension is unique.

Critical points of Laplace eigenfunctions on polygons

We study the critical points of Laplace eigenfunctions on polygonal domains with a focus on the second Neumann eigenfunction. We show that if each convex quadrilaterals has no second Neumann eigenfunction with an interior critical point, then there exists a convex quadrilateral with an unstable critical point. We also show that each critical point of a second-Neumann eigenfunction on a Lip-1 polygon with no orthogonal sides is an acute vertex.

Direct serendipity and mixed finite elements on convex quadrilaterals

The classical serendipity and mixed finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop families of direct serendipity and direct mixed finite element spaces, which achieve optimal approximation properties and have minimal local dimension. The set of local shape functions for either the serendipity or mixed elements contains the full set of scalar or vector polynomials of degree r, respectively, defined directly on each element (i.e., not mapped from a reference element). Because there are not enough degrees of freedom for global H^1 or H(text {div}) conformity, exactly two supplemental shape functions must be added to each element when rge 2, and only one when r=1. The specific choice of supplemental functions gives rise to different families of direct elements. These new spaces are related through a de Rham complex. For index rge 1, the new families of serendipity spaces {mathscr {DS}}_{r+1} are the precursors under the curl operator of our direct mixed finite element spaces, which can be constructed to have reduced or full H(text {div}) approximation properties. One choice of direct serendipity supplements gives the precursor of the recently introduced Arbogast–Correa spaces (SIAM J Numer Anal 54:3332–3356, 2016. https://doi.org/10.1137/15M1013705). Other fully direct serendipity supplements can be defined without the use of mappings from reference elements, and these give rise in turn to fully direct mixed spaces. Our development is constructive, so we are able to give global bases for our spaces. Numerical results are presented to illustrate their properties.

Universal bounds for fixed point iterations via optimal transport metrics

We present a self-contained analysis of a particular family of metrics over the set of non-negative integers. We show that these metrics, which are defined through a nested sequence of optimal transport problems, provide tight estimates for general Krasnosel'skii-Mann fixed point iterations for non-expansive maps. We also describe some of their very special properties, including their monotonicity and the so-called convex quadrangle inequality that yields a greedy algorithm to compute them efficiently.

Construction of C1 polygonal splines over quadrilateral partitions

In this paper we construct smooth bivariate spline functions over a polygonal partition, e.g. a convex quadrilateral partition by using vertex spline techniques. Vertex splines, introduced in Chui and Lai (1985) are smooth piecewise polynomial functions supported over a collection of triangles sharing a vertex. In this paper, we extend the concept of vertex splines to the partition of polygons and describe how to construct C1 vertex polygonal splines over a collection of quadrilaterals. We begin with our construction of C1 vertex splines over a collection of parallelograms, although they may not be axis-orientated. Then the construction is generalized to the setting of general convex quadrilaterals. We will use various monomials of Wachspress GBC functions of degrees 5 and 7 to explain how to construct C1 vertex splines together with additional special splines called edge and face splines. With these splines at hand, we construct quasi-interpolatory formulas, whose approximation properties will be shown. Numerical interpolation and approximation results will be presented. Finally, three applications of these splines are explained: the first one is to form smooth locally supported GBC functions, the second one is to construct smooth suitcase corners, and the third one to construct C1 surfaces over quadrilateral partitions with extraordinary points (EP). Several examples will be demonstrated to show the convenience of using these splines.

Pre-service Middle School Mathematics Teachers’ (Mis)conceptions of Definitions, Classifications, and Inclusion Relations of Quadrilaterals

The purpose of this study is to examine pre-service middle school mathematics teachers’ (mis)conceptions related to definitions, classifications, and inclusion relations of convex quadrilaterals through a case study research design. The participants of the study were 20 pre-service middle school mathematics teachers who attended a must course, “The Methods of Teaching Mathematics” in a public university in Ankara, Turkey. Purposive sampling strategy was used to select the participants. The data were collected through an achievement test before and after taking the methods of teaching mathematics course. The findings of the study revealed that pre-service teachers had misconceptions in hierarchical classification of quadrilaterals, dual inclusion relations (e.g., rhombus-trapezoid), and difficulty in formulating minimal definitions of quadrilaterals before taking the course. However, the findings also indicated that the course had a positive impact for eliminating those misconceptions and supporting their conceptions. Several suggestions were made based on these findings.

Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams

A continuous measure of symmetry and the Voronoi entropy of 2D patterns representing Voronoi diagrams emerging from the Penrose tiling were calculated. A given Penrose tiling gives rise to a diversity of the Voronoi diagrams when the centers, vertices, and the centers of the edges of the Penrose rhombs are taken as the seed points (or nuclei). Voronoi diagrams keep the initial symmetry group of the Penrose tiling. We demonstrate that the continuous symmetry measure and the Voronoi entropy of the studied sets of points, generated by the Penrose tiling, do not necessarily correlate. Voronoi diagrams emerging from the centers of the edges of the Penrose rhombs, considered nuclei, deny the hypothesis that the continuous measure of symmetry and the Voronoi entropy are always correlated. The Voronoi entropy of this kind of tiling built of asymmetric convex quadrangles equals zero, whereas the continuous measure of symmetry of this pattern is high. Voronoi diagrams generate new types of Penrose tiling, which are different from the classical Penrose tessellation.

METHODOLOGICAL APPROACH TO CONGRUENCE OF QUADRILATERALS IN HYPERBOLIC GEOMETRY

In this paper we will prove new criteria for the congruence of convex quadrilaterals in Hyperbolic geometry and consequently, display the appropriate methodological approach in teaching the same. There are seven criteria for the congruence of hyperbolic quadrilaterals, while there are five for the congruence of Euclidean quadrilaterals. Using a comparative geometric analysis of quadrilateral congruence criteria in Euclidean and Hyperbolic geometry we described all possible cases and made a methodological approach to the problem. The obtained results can influence the approaches to the study of these contents with students in the hyperbolic geometry teaching.

H(curl2)-conforming quadrilateral spectral element method for quad-curl problems

In this paper, we propose an [Formula: see text]-conforming quadrilateral spectral element method to solve quad-curl problems. Starting with generalized Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for vector fields over rectangles. [Formula: see text]-conforming elements over arbitrary convex quadrilaterals are then constructed explicitly in a hierarchical pattern using the contravariant transform together with the bilinear mapping from the reference square onto each quadrilateral. It is worth noting that both the simplest rectangular and quadrilateral spectral elements possess only 8 degrees of freedom on each physical element. In the sequel, we propose our [Formula: see text]-conforming quadrilateral spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Numerical results show the effectiveness and efficiency of our method.

Incongruent equipartitions of the plane into quadrangles of equal perimeters

Motivated by a question of R. Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex quadrangles of the same area and the same perimeter. As a byproduct we obtain vertex-to-vertex dissections of the plane by mutually incongruent triangles of unit area that are arbitrarily close to the periodic vertex-to-vertex tiling by equilateral triangles.

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.