Planar Triangles Research Articles

CONSTRUCTING AN APPROACHED SWEEP OF THE SURFACE GELICODE OF THE WIND GENERATOR

It is well-known that experts estimate that traditional energy sources, such as oil, gas and coal, may be depleted in the near future. This raises the question of finding alternative sources. In addition to the use of atomic energy, recent attention has been given to developments in the use of solar and wind energy. In this paper, attention is paid to the design features of the blades of the wind turbine with the vertical axis of rotation. Such wind turbines have a significant advantage over structures with a horizontal axis of rotation, since they do not need to be oriented in the wind direction. The main feature of the design of the considered wind turbines is their blades, which, most often, are compartments of non-expandable helicoidal surfaces, which complicates their manufacture, especially for individual use. In view of all the above, it seems relevant to construct an approximate sweep of the blade, which can be performed using the triangulation method. This method makes it possible, from an engineering point of view, to erroneously construct the contours of the sweep and to determine the approximate area of the blade, and thus to determine the cost of the required material. The paper deals with the method of triangulation for construction of the residential vertical turbine blade sweep, which can be used in a private house. To do this, split the blade height into several equal parts. The second step is to break a certain strip with concentric cylinders. Joining the points of intersection of the edges of the strip with segments, we obtain curvilinear triangles, which after the necessary calculations are transformed into planes. Thus, we create an approximate sweep of one strip across the entire surface. By copying this sweep as many times as we need, we get an approximate sweep for the entire blade. For an approximate calculation of the area of the helical surface, it suffices to find the area of planar triangles with which we have replaced the corresponding curvilinear ones.

Two‐Dimensional Uranyl Borates: From Conventional to Extreme Synthetic Conditions

A systematic investigation of uranyl borates under different synthetic conditions resulted in five new 2D compounds, namely, (H3O)[(UO2)(BO3)], Li[(UO2)(BO3)]·(H2O), α‐K4[(UO2)5(BO3)2O4], β‐K4[(UO2)5(BO3)2O4] and K2.5[(UO2)5(BO3)2O2.5(OH)1.5]·(H2O)2.5. (H3O)[(UO2)(BO3)] and Li[(UO2)(BO3)]·(H2O) were obtained from hydrothermal reactions at 220 °C using the same mineralizer. Both materials possess the uranophane sheet topology with different symmetry of the unit cells. In the structure of (H3O)[(UO2)(BO3)] and Li[(UO2)(BO3)]·(H2O), UO7 pentagonal bipyramids share edges and vertexes with four BO3 planar triangles. α‐K4[(UO2)5(BO3)2O4] and β‐K4[(UO2)5(BO3)2O4] are polytypes. α‐K4[(UO2)5(BO3)2O4] was synthesized from a high‐temperature solid‐state reaction under ambient pressure; whereas β‐K4[(UO2)5(BO3)2O4] was obtained from a high‐temperature/high‐pressure (HT/HP) reaction. Both structures have an identical anion topology, but β‐K4[(UO2)5(BO3)2O4] crystallizes in space group with higher symmetry. In K2.5[(UO2)5(BO3)2O2.5(OH)1.5]·(H2O)2.5, which was obtained from a hydrothermal reaction, UO7 polyhedra share vertexes and edges with two independent BO3 triangles, forming the most complex uranyl borate layers among all five compounds. The different synthetic routes, novel topologies, thermal behavior and Raman spectra are discussed.

A New Coordinate System for Constructing Spherical Grid Systems

In astronomy, physics, climate modeling, geoscience, planetary science, and many other disciplines, the mass of data often comes from spherical sampling. Therefore, establishing an efficient and distortion-free representation of spherical data is essential. This paper introduces a novel spherical (global) coordinate system that is free of singularity. Contrary to classical coordinates, such as Cartesian or spherical polar systems, the proposed coordinate system is naturally defined on the spherical surface. The basic idea of this coordinate system originated from the classical planar barycentric coordinates that describe the positions of points on a plane concerning the vertices of a given planar triangle; analogously, spherical area coordinates (SACs) describe the positions of points on a sphere concerning the vertices of a given spherical triangle. In particular, the global coordinate system is obtained by decomposing the globe into several identical triangular regions, constructing local coordinates for each region, and then combining them. Once the SACs have been established, the coordinate isolines form a new class of global grid systems. This kind of grid system has some useful properties: the grid cells exhaustively cover the globe without overlapping and have the same shape, and the grid system has a congruent hierarchical structure and simple relationship with traditional coordinates. These beneficial characteristics are suitable for organizing, representing, and analyzing spatial data.

Evaluation of Static Potential Integrals on Triangular Domains

Static potential integrals for constant and linear sources on triangles are derived in a straightforward way. The new representations, as presented, are robust with respect to machine evaluation in important limiting cases. The potential integrals comprise up to six functions, each dependent on the relative position and orientation (with respect to an observation point) of a vertex and edge, respectively, of the source triangle. Gradients of the potentials are derived by differentiation, thus preserving relations between the representations. Each such vertex function reveals any anomalous functional behavior near its associated vertex or edge, which is useful information for devising test integral schemes. Potential plots in the source plane of an equilateral triangle illustrate such behavior, as do similar plots for each vertex function and gradient components near their associated edge and vertex.

Distance Product Cubics

The locus of points that determine a constant product of their distances to the sides of a triangle is a cubic curve in the projectively closed Euclidean triangle plane. In this paper, algebraic and geometric properties of these distance product cubics shall be studied. These cubics span a pencil of cubics that contains only one rational and non-degenerate cubic curve which is known as the Bataille acnodal cubic determined by the product of the actual trilinear coordinates of the centroid of the base triangle. Each triangle center defines a distance product cubic. It turns out that only a small number of triangle centers share their distance product cubic with other centers. All distance product cubics share the real points of inflection which lie on the line at infinity. The cubics' dual curves, their Hessians, and especially those distance product cubics that are defined by particular triangle centers shall be studied.

A family of geometric operators on triangles with two complex variables

Given a plane triangle $\Delta$, one can construct a new triangle $\Delta'$ whose vertices are intersections of two cevian triples of $\Delta$. We extend the family of operators $\Delta\mapsto\Delta'$ by complexifying the defining two cevian parameters and study its rich structure from arithmetic-geometric viewpoints. We also find another useful parametrization of the operator family via finite Fourier analysis and apply it to investigate area-preserving operators on triangles.

Development and Production of Computer Generated Instructional Materials for College Geometry

Printed materials such as worktext and textbooks are considered the best instrument in improving transfer of learning since these materials can simulate a human facilitator to teach efficiently and effectively. This study aimed to develop and produce a computer generated instructional materials (CGIM) in College Geometry. The first phase of the study is assessment. It involved eighty-three (83) Bachelor of Elementary Education students. This research utilized an 80-item researcher-made test. Statistical tools employed were percentage, mean, standard deviation, and t-test for one sample. The second phase is development and production using document analysis methods. The result of the first phase shows that students have significantly low competencies in all the topics of College Geometry compared to the passing mean set by the PRC for teachers. The researchers developed a CGIM, specifically, the worktext paired with multimedia presentation. The parts were objectives, learning concepts, exercises, evaluation, and performance task. The topics included were logic and reasoning, geometry of shapes and size, angles and perpendiculars, triangle congruence, quadrilaterals, similarity, circles, and plane coordinate geometry. Every topic has corresponding subtopics. It is recommended that the CGIM will be used in Mathematics classes to improve students’ achievement, habits of mind, and problem solving skills.

A segmented Adomian algorithm for the boundary value problem of a second-order partial differential equation on a plane triangle area

For the boundary value problem (BVP) of a second-order partial differential equation on a plane triangle area, we propose a new algorithm based on the Adomian decomposition method (ADM) combined with a segmented technique. In addition, we present a new theorem that ensures the convergence of the algorithm. By this algorithm, the model for the effect of regional recharge on the plane triangle groundwater flow region is solved, from which we obtain the segmented exact solution of the problem, which satisfies the governing equation and all of the specified boundary conditions. Then, by the algorithm combined with Taylor’s formula, the heterogeneous aquifer model on the plane triangle groundwater flow region is considered, from which we obtain the segmented high-precision approximate solution of the problem.

New SiC Epitaxial Growth Process with up to 100% BPD to TED Defect Conversion on 150mm Hot-Wall CVD Reactor

The future challenges for SiC device technology are cost reduction and increased reliability. A key point to achieve that is the increase of yield during epitaxial layer growth through the reduction of structural defects (such as basal plane dislocations and triangle defects), an increased thickness and doping uniformity, and a high growth rate. Despite significant advancements in SiC epitaxial growth technology, it still constitutes a big challenge to find the optimum working point at which all those requirements are fulfilled. By implementing a new epitaxial layer growth process, we are able to grow basal plane dislocation free epitaxial layers, while the density of other structural defects remains low. Additionally, intra-wafer thickness and doping uniformities of the epitaxial layers are further improved.

Anatomic Variation of the Cystic Artery: New Findings and Potential Implications

Purpose: To determine the anatomy of the cystic artery by dual-source CT, and correlate imaging findings with those patients who had laparoscopic cholecystectomy (LC). Materials and Methods: Following institutional review board approval, a total of 289 consecutive patients (204 men and 85 women) were evaluated with CT for abdominal pain, including 55 patients subsequently underwent LC. Location of the cystic artery termination, distance between the cystic artery origin and the gallbladder, and angle between the cystic artery and its parent artery were evaluated by two radiologists. The laparoscopic surgical video record (gold standard) was similarly evaluated by a surgeon. Results: A total of 256 cystic arteries in the 247 patients were included. Cystic artery terminations are predominately found in ventral Calot triangle plane (50.8%, type II). Cystic artery origin immediately adjacent to the gallbladder surface was seen in 11/256 (4.3%). Zero angle between the cystic artery and its parent artery was found in 17 of 256 cystic arteries (6.6%). The cystic arteries and the Calot triangle were depicted in 49 patients (95% confidence interval: 85%, 97%). For all 49 patients, CT imaging findings were consistent with surgical video records. No case involved vascular and biliary injury occurred. Conclusions: Given the large number of LC performed each year, better knowledge of anatomic variation of the cystic artery could potentially prevent arterial injury and bile duct injury, particularly for patients with unusual anatomy.

Asymptotic Boundary Observability for the Wave Equation on One Side of a Planar Triangle

We consider the wave equation $(\partial_t^2-\Delta)u=0$ on a planar triangular domain $\Omega\subset\mathbb{R}^2$ with Dirichlet boundary conditions. We use a commutator and integration by parts argument similar to that in \cite{Chr2DTriangles} by the first author to obtain an observability asymptotic for any one side of the triangle. Our result is particular to triangular domains and does not hold for polygons in general.

The icosahedra of edge length 1

Retaining the combinatorial Euclidean structure of a regular icosahedron, namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we investigate variations of the regular icosahedron admitting self-intersections of faces. We determine all rigid equivalence classes of these icosahedra with non-trivial automorphism group and find one curve of flexible icosahedra. Visualisations and explicit data for this paper are available under http://algebra.data.rwth-aachen.de/Icosahedra/visualplusdata.html.

Coordination Compounds of Lanthanides with Hydrazones of 3-Methyl-1-Phenyl-4-Formylpyrazole-5-One and 1,3,5-Benzenetricarboxylic acid

We describe the results of the X-ray diffraction study of a dysprosium(III) complex with trihydrazone of 3-methyl-1-phenyl-4-formylpyrazole-5-one and 1,3,5-benzenetricarboxylic acid, as well as a heteroligand gadolinium(III) complex with tri- and dihydrazone of 3-methyl-1-phenyl-4-formylpyrazole-5-one and 1,3,5-benzenetricarboxylic acid (H3L and H3L1 respectively). It is determined that the homoligand complex of dysprosium has a M4L4 composition and a pseudotetrahedral structure in which metal ions occupy vertices of a trigonal pyramid and trinucleating ligands are located on its sides. The heteroligand gadolinium complex Gd3(L)(HL1)3 has a trigonal structure. Metal ions at the triangle vertices are linked by a trinucleating ligand located above the triangle plane, and three binucleating ligands link the central atoms in pairs. The geometry of dysprosium and gadolinium coordination polyhedra correspond to a three-capped trigonal prism.

Structure refinement and thermal properties of novel cubic borate Lu2Ba3B6O15

Abstract Novel borate Lu2Ba3B6O15 was prepared using a multi-step high-temperature solid-state synthesis. It crystallizes in the cubic system, Ia 3 ¯ space group, a = 14.18197(3) A, V = 2852.396(18) A3, Z = 8, and it is isostructural with Y2Ba3B6O15. The crystal structure is described as a framework composed of the edge-sharing large irregular BaO9 polyhedra. The intersecting tunnels in this framework are filled with the lutetium atoms. Two planar BO3 triangles are connected to each other by the vertices, forming the isolated B2O5 pyroborate groups. The Lu2Ba3B6O15 thermal behavior was investigated by thermal analysis and high-temperature X-ray powder diffraction (HTXRD). The Lu2Ba3B6O15 sample starts to melt at 1058 °C. As the temperature increases, the coefficient of linear expansion increases (αa = 5.7 at 25 °C and 8.9 × 10−6 C−1 at 800 °C). The Rietveld refinement of the HTXRD data showed that thermal deformations of the non-rigid BaO9 polyhedra have the greatest influence on the Lu2Ba3B6O15 expansion among those of the other constituent polyhedra.

High-order triangular finite elements applied to visco-hyperelastic materials under plane stress

A finite element formulation for analysis of highly deformable viscoelastic materials under plane stresses is presented. The element is the isoparametric plane triangle of any-order. The constitutive modeling is based on a Lagrangian visco-hyperelastic framework, in which the multiplicative split of the deformation gradient is used. The internal variable employed is the viscous right Cauchy–Green stretch tensor. The work is restricted to the compressible neo-Hookean hyperelastic law, the Zener rheological model and an isochoric nonlinear evolution equation, although other models can be further implemented. The plane stress condensation and the compact 2D constitutive relation are described. Three large deformation problems are numerically analyzed to validate the methodology: a membrane with high compressive strain levels and mesh distortion; a perforated plate with stress concentration; and a circular ring under nonlinear bending. Meshes with different element orders (from linear to sixth) are employed. Results confirm that, except for the linear degree, mesh refinement completely remove locking problems, providing an accurate and, thus, a reliable prediction of the mechanical behavior. It is also demonstrated that the creep phenomenon can be reproduced in finite strain regime. The novelty of the paper is the convergence analysis regarding displacements, strains and stresses in the context of plane stress visco-hyperelasticity.

Mechanism of Crystal Structure Transformation and Abnormal Reduction in Ca5- y(BO3)3- x(PO4) xF (CBP xF): yBi3.

Tricoordinated planar triangle (PO4)3- may be formed due to the structural differences between planar triangular (BO3)3- and tetrahedral (PO4)3- when (BO3)3- is gradually substituted by (PO4)3-. This transformation of structure may affect the luminescence properties of phosphor. Therefore, a series of Ca5- y(BO3)3- x(PO4) xF (CBP xF): yBi3+ ( y = 0.05, 0.15; x = 0-3), Ca5- y(PO4)3- X(BO3) XF (CPB XF): yBi3+ ( y = 0.05, 0.15; X = 0-1), Ca4.9(PO4)3F (CPF):0.1Eu3+, Ca4.95(PO4)3F (CPF):0.05Bi3+, and nCaF2/CaCl2 ( n = 0-0.1) are synthesized to explore transformation of the crystal structure on luminescence properties. In CBP xF:0.15Bi3+ ( x = 0-3), (PO4)3- is doped to substitute for (BO3)3-, the position of emission spectra remains unchanged and the emission intensity decreases rapidly with increasing x. The underlying main reason for that is formation of the triangular plane (PO4)3-, which has been verified by performing a series of verification experiments of CPB XF: yBi3+ ( y = 0.5, 0.15; X = 0-1). In CPB XF: yBi3+ ( y = 0.5, 0.15; X = 0-1), (BO3)3- is doped to substitute for (PO4)3-, P-O2 bond breaks and the coordination of (PO4)3- varies from four to three when 0.5 < X < 1; meanwhile, the crystal structure transforms from Ca5(PO4)3F (ICSD-9444) to Ca5(PO4)3F (ISCD-30261), which impedes abnormal reduction from Bi3+ to Bi2+. Furthermore, Bi3+ should non-luminance in the plane triangular (PO4)3-, but luminescence in (BO3)3-. Therefore, the emission intensity starts to increase and the emission position suddenly changes from 553 to 474 nm in CPB XF: yBi3+ ( y = 0.05, 0.15; 0.5 < X < 1). From this, the crystal structures of CBP xF: yBi3+ ( y = 0.05, 0.15; x = 0-3) has been inferred to transform from Ca5(BO3)3F (ISCD-65763) to Ca5(PO4)3F (ISCD-30261), and then to Ca5(PO4)3F (ISCD-9444) with x increasing. Emission position remains unchanged and the emission intensity decreases rapidly in CBP xF: yBi3+ ( y = 0.05, 0.15; x = 0-3) do to formation of the triangular plane (PO4)3-. In addition, the rate of abnormal reduction from Bi3+ to Bi2+ can be improved by reducing the electronegativity of the environment around the activator or increasing the ionization energy of the activator, which has been confirmed by verification experiments of CPF:0.05Bi3+, nCaF2/CaCl2 ( n = 0-0.1), and CPF:0.1Eu3+.

First Demonstration of Magnetoelectric Coupling in a Polynuclear Molecular Nanomagnet: Single-Crystal EPR Studies of [Fe3 O(O2 CPh)6 (py)3 ]ClO4 ⋅py under Static Electric Fields.

Single-crystal EPR experiments show that the highly symmetric antiferromagnetic half-integer spin triangle [Fe3 O(O2 CPh)6 (py)3 ]ClO4 ⋅py (1, py=pyridine) possesses a ST =1/2 ground state exhibiting high g-anisotropy due to antisymmetric exchange (Dzyaloshinskii-Moriya) interactions. EPR experiments under static electric fields parallel to the triangle's plane (i.e., perpendicular to the magnetic z-axis) reveal that this ground state couples to externally applied electric fields. This magnetoelectric coupling causes an increase in the intensity of the intradoublet EPR transition and does not affect its resonance position when B0 ∥z. The results are discussed on the basis of theoretical models correlating the spin chirality of the ground state with the magnetoelectric effect.

Numerical limitations of the attainment of the orientation of geological planes

Abstract The paper discusses limitations of analytical attainment of the attitude of a geological plane by using three non-collinear points. We present problems that arise during computing the orientation of a plane generated by almost collinear points. We referred these errors to floating-point arithmetic inaccuracies. To demonstrate the problem, we examined a surface of constant orientation. We used Delaunay triangulation to calculate its local orientation parameters. We introduced a new measure of collinearity applicable for collecting attitude of planar triangles. Using this measure we showed that certain planes generated by the triangulation cannot be treated as a reliable source of measurement. To examine the relationship between collinearity and orientation, we used a combinatorial algorithm to obtain all possible planes from the given set of points. A statistical criterion of rejecting almost collinear planes was suggested.

The statistical shape of geometric reasoning

Geometric reasoning has an inherent dissonance: its abstract axioms and propositions refer to perfect, idealized entities, whereas its use in the physical world relies on dynamic perception of objects. How do abstract Euclidean concepts, dynamics, and statistics come together to support our intuitive geometric reasoning? Here, we address this question using a simple geometric task – planar triangle completion. An analysis of the distribution of participants’ errors in localizing a fragmented triangle’s missing corner reveals scale-dependent deviations from a deterministic Euclidean representation of planar triangles. By considering the statistical physics of the process characterized via a correlated random walk with a natural length scale, we explain these results and further predict participants’ estimates of the missing angle, measured in a second task. Our model also predicts the results of a categorical reasoning task about changes in the triangle size and shape even when such completion strategies need not be invoked. Taken together, our findings suggest a critical role for noisy physical processes in our reasoning about elementary Euclidean geometry.

From map reading to geometric intuitions.

The origins and development of our geometric intuitions have been debated for millennia. The present study links children's developing intuitions about the properties of planar triangles to their developing abilities to read purely geometric maps. Six-year-old children are limited when navigating by maps that depict only the sides of a triangle in an environment composed of only the triangle's corners and vice versa. Six-year-old children also incorrectly judge how the angle size of the third corner of a triangle varies with changes to the other two corners. These limitations in map reading and in judgments about triangles are attenuated, respectively, by 10 and 12 years of age. Moreover, as children get older, their map reading predicts their geometric judgments on the triangle task. Map reading thus undergoes developmental changes that parallel an emerging capacity to reason explicitly about the distance and angle relations essential to euclidean geometry. (PsycINFO Database Record