Planar Triangles Research Articles

Symplectic embeddings and continued fractions: a survey

As has been known since the time of Gromov’s Non-squeezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. After surveying some of the most important ways of measuring the size of a symplectic set, these notes discuss some recent developments concerning the question of when a 4-dimensional ellipsoid can be symplectically embedded in a ball. This problem turns out to have unexpected relations to the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar triangles.

Density functional study of hydrogen adsorption and dissociation on small Pd n ( n = 1–7) clusters

Density functional theory has been performed to investigate the interaction of H 2 and Pd n clusters ( n = 1–7). The local minima configurations for different H 2 molecule approach modes towards Pd n clusters are presented. Our results show that in some cases H 2 is physically adsorbed around Pd atom, and in other cases H 2 is dissociated to be H atoms. Except for PdH2, Pd n clusters with H atoms dissociatively adsorbed are most stable. For these most stable PdnH2 clusters ( n ⩾ 2), the binding energy of hydrogen atom decreases as the number of Pd atom increases until n = 4, and when n ⩾ 4, the binding energy almost keeps constant with the H atoms bound sites changing from Pd–Pd bonds to Pd triangle planes. Besides, the adsorption of H 2 on other low-lying isomers of Pd n clusters is also discussed.

RESTRICTED MESH SIMPLIFICATION USING EDGE CONTRACTIONS

We consider the problem of simplifying a planar triangle mesh using edge contractions, under the restriction that the resulting vertices must be a subset of the input set. That is, contraction of an edge must be made onto one of its adjacent vertices, which results in removing the other adjacent vertex. We show that if the perimeter of the mesh consists of at most five vertices, then we can always find a vertex not on the perimeter which can be removed in this way. If the perimeter consists of more than five vertices such a vertex may not exist. In order to maintain a higher number of removable vertices under the above restriction, we study edge flips which can be performed in a visually smooth way. A removal of a vertex which is preceded by one such smooth operation is called a 2-step removal. Moreover, we introduce the possibility that the user defines "important" vertices (or edges) which have to remain intact. Given m such important vertices, or edges, we show that a simplification hierarchy of size O(n) and depth O( log (n/m)) can be constructed by 2-step removals in O(n) time, such that the simplified graph contains the m important vertices and edges, and at most O(m) other vertices from the input graph. In some triangulations, many vertices may not even be 2-step removable. In order to provide the option to remove such vertices, we also define and examine k-step removals. This increases the lower bound on the number of removable vertices.

Structural analysis of lithium lanthanum titanate with perovskite structure

AbstractNeutron and high‐energy X‐ray diffraction analysis of polycrystalline La4/3‐xLi3xTi2O6 have been performed to clarify the extent of disorder of the distribution of La and Li ions and to understand the relation of these distributions to ionic conduction. The distributions of the La and Li ions in a 10 × 10 × 20 cubic box (i.e., 10 × 10 × 10 unit cell) super‐structure, in which Ti and O atoms are fixed onto their regular sites, were obtained by the reverse Monte Carlo (RMC) structural modelling of both diffraction data sets. When the occupancy of La ions in the planes perpendicular to the c‐axis is analysed, one can find a La‐rich and La‐poor layers alternating, which is consistent with the results of earlier Rietveld analysis (Stramare et al., Chem. Mater. 15, 3974 (2003) [1]). Of particular interest, the Li ions are found mainly on the interstitial sites between the O‐3 triangle plane of the TiO6 octahedron and a La ion, which is different from the earlier work (Yashima et al., J. Am. Chem. Soc. 127, 3491 (2005) [2]). (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Tangential Distance Fields for Mesh Silhouette Problems

Aabstract We consider a tangent-space representation of surfaces that maps each point on a surface to the tangent plane of the surface at that point. Such representations are known to facilitate the solution of several visibility problems, in particular, those involving silhouette analysis. In this paper, we introduce a novel class of distance fields for a given surface defined by its tangent planes. At each point in space, we assign a scalar value which is a weighted sum of distances to these tangent planes. We call the resulting scalar field a ‘tangential distance field’ (TDF). When applied to triangle mesh models, the tangent planes become supporting planes of the mesh triangles. The weighting scheme used to construct a TDF for a given mesh and the way the TDF is utilized can be closely tailored to a specific application. At the same time, the TDFs are continuous, lending themselves to standard optimization techniques such as greedy local search, thus leading to efficient algorithms. In this paper, we use four applications to illustrate the benefit of using TDFs: multi-origin silhouette extraction in Hough space, silhouette-based view point selection, camera path planning and light source placement.

Ground state of an arbitrary triangle with a Calogero-Sutherland-Moser potential

We construct the expression for the ground state eigenfunction of the Schrodinger equation for a particle inside an arbitrary planar triangle under the influence of a potential.

THE NEW ASTRONOMY OF IBN AL-HAYTHAM

In order to get rid of the contradictions he had identified in Ptolemy’s Astronomy, Ibn al-Haytham abandons cosmology and develops a purely kinematic description of the movement of the wandering stars. This description culminates with the proof that such a star, during its daily movement, reaches exactly one time a maximum height above the horizon and that any inferior height is reached exactly twice. The proofs of these facts necessitates new mathematical tools and Ibn al-Haytham is led to establish very sophisticated statements concerning the variation of certain ratios of arcs of circles on the sphere. He also introduces the fruitful idea of assimilating a very small spherical triangle to a plane triangle.

A macro plane triangle element from the individual element test

A simple way to make the closed forms stiffness matrix of macro triangle elements is proposed through the individual element test and the computational algebra system in this paper. The newly developed element consists of three constant strain triangles, which only satisfy the convergence conditions according to the individual element test. Computational algebra system MAPLE and the free formulation framework are used to combine the stiffness matrix with the free parameters in terms of the algebraic symbols. The assembly of three constant strain triangles produces the linear strain modes in the resulted macro elements. The rigid body modes and constant strain modes of elements and the newly produced linear strain modes of the macro element are systematically assembled in the basic stiffness and the higher order stiffness in the free formulation procedure. The parameter of the decomposed higher order stiffness was tuned through the bending test. The results of numerical examples show the advantages of the proposed macro element in the distorted mesh and elements with the high aspect ratio.

Fuzzy Set Theoretical Analysis of Human Membership Values on the Color Triangle

The present study considers a fuzzy color system in which three membership functions are constructed on the RGB color triangle. This system can process a fuzzy input (as the membership values of subjects) to an RGB system and output the center of gravity of three weights associated with respective grades. Three membership functions are applied to the RGB color triangle relationship. By treating three membership functions of redness, greenness, and blueness on the RGB color triangle, an average color value can be easily obtained as the center of gravity of the fuzzy output. The differences between fuzzy input and inference output are described, and the relationship between the centers of gravity of fuzzy inputs and inference outputs for fuzzy inputs are shown in the present paper. A technique for obtaining expressions of the RGB color triangle using the fuzzy set theoretical method has been reported [5] and improved [6]. In the previous study, the relationship between input fuzzy sets with a plateau on the RGB triangle and fuzzy inputs of conical membership functions was examined. The RGB color triangle (plane) represents the hue and saturation of a color [8]. The six fundamental colors and white can be represented on the same color triangle (See Fig. 1). Vague colors on the RGB color triangle were clarified. In the present study, the membership value on the RGB triangular system are examined to determine the average color value as the center of gravity of the attribute information of vague colors. This fuzzy set theoretical approach is useful for vague color information processing, color-naming systems, and similar applications.

Ab initio investigation of the non-collinear magnetic structure and the lowest magnetic excitations in dysprosium triangles

The unusual magnetism exhibited by dysprosium triangles [Dy3(μ3-OH)2L3Cl2(H2O)4][Dy3(μ3-OH)2L3Cl(H2O)5]Cl5·19H2O is explained using the recently developed ab initio methodology for the simulation of magnetic properties of complexes. The local anisotropy axes on the dysprosium sites are found to lie in the plane of the Dy3 triangle and to make angles of ca. 120° with each other. The small antiferromagnetic exchange interaction between sites leads to a non-magnetic Kramers doublet in the ground state of the complex. The arrangement of the local magnetization vectors in this state is close to toroidal. By contrast, the lowest excited states are characterized by a huge magnetic moments of ca. 20 μB and show very different behavior of magnetization for fields applied along and perpendicular to the plane of the Dy3 triangle.

Mechanism of molecular hydrogen dissociation on gold chains and clusters as model prototypes of nanostructures

The reactivity of H(2) on several gold clusters is studied using density functional theory with generalized gradient approximation methods, as model systems designed to study the main effects determining their catalytic properties under controlled conditions. Border effects are studied in finite linear gold chains of increasing size and compared with the corresponding periodic systems. In these linear chains, the reaction can proceed with no barrier along the minimum energy path, presenting a deep chemisorption well of approximately 1.4 eV. The mechanism presents an important dependence on the initial attacking site of the chain. Linear Au(4) chains joined to model-nanocontacts, formed by 2 or 3 gold atoms, in a planar triangle or in a pyramid, respectively, are also studied. The reaction barriers found in these two cases are approximately 0.24 and 0.16 eV, respectively, corresponding to H(2) attacking the more coordinated edge atom of the linear chain. The study is extended to planar clusters with coordinations IV and VI, for which higher H(2) dissociation barriers are found. However, when the planar gold clusters are folded, and the Au-Au distances elongated, the reactivity increases considerably. This is not due to a change of coordination, but to a larger flexibility of the gold orbitals to form bonds with hydrogen atoms, when the planar sd-hybridization is broken. Finally, it is concluded that the major factor determining the reactivity of gold clusters is not strictly the coordination of gold atoms but their binding structure and some border effects.

A new method to generate an almost-diagonal matrix in the boundary integral equation formulation

In this work, a new numerical procedure is developed to generate an almost-diagonal matrix for the solution of boundary integral equation formulation dealing with acoustic scattering problems. The major drawback of the traditional boundary integral equation procedure resulting in a dense system matrix is eliminated in this new procedure by grouping the basis functions into a cluster. The geometry of the structure is modeled by planar triangles and the basis functions are defined on the nodes. By doing so, one can get a benefit of a smaller size matrix to begin with. Furthermore, by grouping these node-based basis functions into a cluster, an almost-diagonal matrix is generated. Thus, the solution procedure developed in this work may be utilized for very large scattering problems since the required computer resources are very low. The solution procedure developed in this work is validated for the scattering cross section of the simple shapes with the closed form solutions wherever available and with the other numerical solution procedures.

Quasiconformal extension from curvilinear triangles

We consider the problem of extension of a quasi-Mobius (quasisymmetric) mapping from a planar curvilinear triangle to a quasiconformal automorphism of the extended plane. Existence of an extension of the sort from the polygons whose sides are quasiarcs was proven by A. K. Varisov but without any upper estimates for the quasiconformality coefficient of the extended mapping. The application of another method makes it possible to obtain, for triangular domains, an upper estimate of dilation of the quasiconformal extension depending only on the distortion function of the initial quasi-Mobius mapping, on the upper estimate of the turning of the sides and the Mobius-invariant characteristic of nondegeneracy of a curvilinear triangle which is introduced in this article and is analogous to the relative radius of the inscribed circle.

Nucleation and Growth of Palladium Clusters on Anatase TiO2(101) Surface: A First Principle Study

The nucleation and growth of palladium clusters supported on anatase TiO2(101) surface has been studied using periodic supercell models and density functional theory. The most active site for single Pd adatom on the perfect TiO2(101) surface is the bridge site formed by two two-coordinated oxygen (2cO) atoms at the step edge with the highest adsorption energy of 2.18 eV. On the defective surface, the effect of oxygen vacancy on Pd nucleation is not as strong as that in the Pt or Au deposition case. The shift of active sites for the Pd dimer growing to trimer on the perfect anatase TiO2(101) surface is observed. On both perfect and defective anatase surfaces, the adsorbed Pd3 clusters prefer to form planar triangles and the adsorbed Pd4 and Pd5 clusters tend to three-dimensional structures. The nucleation and growth of Pd clusters at the anatase TiO2(101) surface is mainly driven by the interaction between Pd and surface atoms when the cluster size is less than four. The strength of Pd−Pd interaction turns out to control dominantly the Pd deposition process as the Pd cluster gets larger.

A finite volume method on general surfaces and its error estimates

In this paper, we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface. The discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons. The optimal error estimates of the approximate solution are proved in both the H 1 and L 2 norms which are of first order and second order respectively under mesh regularity assumptions. Some numerical tests are also carried out to experimentally verify our theoretical analysis.

The crystal structure of Li2B4O7 compound in the temperature range 10–290K

Structural investigation of Li2B4O7 in the 10–290 K temperature range has been carried out by means of high-resolution powder diffraction technique applying synchrotron radiation. The obtained results show high thermal stability of tetraborate groups in the 10–290 K temperature range. The planar BO3 triangles somewhat enlarge at heating, while the BO4 tetrahedra compress excessively at heating from 10 to 290 K. In the 10–290 K temperature range maximal changes were observed for structural parameters of Li atoms, while the parameters of O atoms show minimal temperature variations.

X-ray and conformation analysis of the new trinuclear cluster of osmium Os3(μ,η2-OCC6H5)(η3-C3H5)(CO)9

The crystal structure of the Os3(μ,η2-O=CC6H5)(η3-C3H5)(CO)9 cluster synthesized by the reaction of the (μ-H)Os3(μ-O=CC6H5)(CO)10 complex with allylamine in chloroform was determined by X-ray analysis. Prolonged storage of the reaction mixture led to N-C bond cleavage in allylamine and η3-addition of the allyl fragment at one of the Os atoms (Os-C 2.246 A, 2.248 A, and 2.273 A). The unit cell parameters of the complex are a = 9.494(1) A, b = 10.479(1) A, c = 12.474(2) A, α = 84.55(1)°, β = 70.08(1)°, γ = 70.72(1)°, V = 1255.8(4), A3, space group P\( \bar 1 \), Z = 2; C19H10O10Os3; dcalc = 2.922 g/cm3, 3085 Ihkl > 2σI of 3611 collected reflections; R = 0.0252. The structure of Os3(μ,η2-O=CC6H5)(η3-C3H5)(CO)9 is molecular. The plane of the Os3 triangle and the OsCOOs plane are connected according to the “butterfly” principle with an angle of 103.4° between them. The Os-Os distances in the cluster core vary from 2.836(1) A to 2.844(1) A; the Os-Ccarb distances are 1.88(1)–1.97(1) A; the distances to the atoms of the bridging ligands are Os-C 2.11(1) A, Os-O 2.14(1) A; the O-C bridging bond is 1.24(1) A. of the Os3(μ,η2-O=CC6H5)(η3-C3H5)(CO)9 triosmium cluster were studied theoretically. The potential curve of the internal rotation of the allyl ligand relative to the Os(1)-C(9) bond was determined. The rotation barrier of the allyl ligand in crystal relative to the Os(1)-C(9) bond is 8.38 kJ/mol, and the rotation of the ligand is not hindered. The effects of the intra-and intermolecular interactions on the conformation state of the cluster complex are considered.

A new smoothing strategy for computing nearly singular integrals in 3D Galerkin BEM

In this paper we propose an efficient strategy to compute nearly singular integrals over planar triangles in R 3 , arising in 3D Galerkin BEM. The strategy is based on a proper use of various nonlinear transformations, which smooth, or move away or quite eliminate all the singularities close to the domain of integration. We will deal with near singularities of the form 1 / r and 1 / r 3 , r = ‖ x − y ‖ being the distance between two generic points x and y lying on triangles which are disjoint, but very close to each other. The approach proposed here is demonstrated by numerical experiments.

Ray tracing algorithm based on octree space partition method

Ray tracing algorithm is one of the main algorithms of creating the real graphic.To improve the speed,accelerating algorithm for ray tracing was studied and compared and a ray tracing algorithm based on octree data structure was presented.The intersecting algorithm based on barycentric coordinate can advance the intersection efficiency.Using barycentric coordinate to express triangular parameter surfaces does not need long-term storage as the planar triangle does,and can quickly tell whether the triangular intersects with the light and calculate the intersection.Experiments show that the algorithm can guarantee image quality and improving the rendering speed.

Conformal self-organizing map on curved seamless surface

This paper presents a new mapping to construct the self-organizing map on the curved seamless surface. This mapping is developed for the planar triangle surface derived from the conformal self-organizing map [C.-Y. Liou, Y.-T. Kuo, Conformal self-organizing map for a genus-zero manifold, Visual Comput. 21(5) (2005) 340–353]. It shows how to construct a seamless surface for the genus-zero manifold. The constructed surface is both seamless and continuous. The mapping between the model surface and the sphere surface is one-to-one and onto. This kind of surface can facilitate many applications of the self-organizing map. We show experiments in surface reconstructions and texture mappings.