Adjacent Triangles Research Articles

Discrete bending forces and their Jacobians

Computation of bending forces on triangle meshes is required for numerous simulation and geometry processing applications. In particular it is a key component in cloth simulation. A common quantity in many bending models is the hinge angle between two adjacent triangles. This angle is straightforward to compute, and its gradient with respect to vertex positions (required for the forces) is easily found in the literature. However, the Hessian of the bend angle, which is required to compute the associated force Jacobians is not documented in the literature. Force Jacobians are required for efficient numerics (e.g., implicit time stepping, Newton-based energy minimization) and are thus highly desirable. Readily available computations of the force Jacobian, such as those produced by symbolic algebra systems, or by autodifferentiation codes, are expensive to compute and therefore less useful. We present compact, easily reproducible, closed form expressions for the Hessian of the bend angle. Compared to automatic differentiation, we measure up to 7× speedup for the evaluation of the bending forces and their Jacobians.

Seamless terrain rendering method based on binary tree and GPU

In this paper a Graphic Processing Unit(GPU)-based and seamless method for terrain rendering was proposed.The method utilized binary tree to construct the terrain Level Of Detail(LOD) which was represented by row-column templates.In this method,the elevation data was transformed to elevation-texture and then the elevation-texture was used to extract elevation for rendering by Vertex Texture Fetch(VTF).And the whole process was handled by GPU so that the data accessing efficiency was greatly improved.The force-split method in Real-time Optimally Adapting Meshes(ROAM) was used in this method which solved terrain cracks by limiting the level of adjacent terrain blocks.The TriangleStrip primitive was used for terrain rendering which avoided transferring the same vertices' coordinates of adjacent triangles to GPU.Finally the paper tested the efficiency of the new method by two terrain datasets,and compared the frame rates between Clipmap and the new method.The results show that the new proposed method solves the terrain crack effectively and fulfills the demand of interactive terrain rendering.

A sufficient condition for a planar graph to be 3-colorable

Steinberg conjectured that every planar graph with neither cycle of length 4 nor cycle of length 5 is 3-colorable. Xu, independently, Borodin et al. proved that planar graphs without adjacent triangles and without 5- and 7-cycles are 3-colorable. As a corollary of this result, planar graphs without 4-, 5- and 7-cycles are 3-colorable. In this paper, we prove a result which is closer to the Steinbergs conjecture than the corollary mentioned above. Let G be a planar graph without 4- and 5-cycles. G is 3-colorable if it has no ( k , 7)-chords for each k ∈ {3, 6, 7}, where a ( k , 7)-chord is a chord of a cycle of length 7 + k -2 whose two ends divide the cycle into two paths, one of which is of length 6 and the other of length k -1.

Novel Design of Electromagnetic Bandgap Using Fractal Geometry

A novel electromagnetic bandgap (EBG) structural design based on Fractal geometry is presented in this paper. These Fractals, which are the Sierpinski triangles, are arranged to repeat each 60° to produce the hexagonal unit cells. By changing the gap between two adjacent Sierpinski triangles inside EBG unit cell, we can produce two EBG structures separately that have broadband and dual bandgap. By using the suspending microtrip method, two arrays 3 × 4 of EBG unit cells are utilized to investigate the bandgap of the EBG structures. The EBG operation bandwidth of the broadband structure is about 87% and of the dual-band structure is about 40% and 35% at the center bandgap frequencies, respectively. Moreover, a comparison between the broadband EBG and the conventional mushroom-like EBG has been done. Experimental results of the proposed design show good agreement in comparison with simulation results.

REPRESENTATION AND RECONCONSTRUCTION OF TRIANGULAR IRREGULAR NETWORKS WITH VERTICAL WALLS

Abstract. Point clouds obtained by aerial laser scanning are a convenient input source for high resolution 2.5d elevation models, such as the Dutch AHN-2. More challenging is the fully automatic reconstruction of 3d city models. An actual demand for a combined 2.5d terrain and 3d city model for an urban hydrology application led to the design of an extension to the well-known Delaunay triangulated irregular networks (TINs) as to accommodate vertical walls. In addition we introduce methods to generate and refine models adhering to our data structure. These are based on combining two approaches: a representation of the TIN using stars of vertices and triangles, together with segmenting the TIN on the basis of coplanarity of adjacent triangles. The approach is supposed to deliver the complete model including walls at the correct locations, without relying on additional map data, as these often lack completeness, actuality and accuracy, and moreover most of the time do not account for parts facades not going down to street level. However, automatic detection of height discontinuities to obtain the exact location of the walls is currently still under implementation.

Variational mesh decomposition

The problem of decomposing a 3D mesh into meaningful segments (or parts) is of great practical importance in computer graphics. This article presents a variational mesh decomposition algorithm that can efficiently partition a mesh into a prescribed number of segments. The algorithm extends the Mumford-Shah model to 3D meshes that contains a data term measuring the variation within a segment using eigenvectors of a dual Laplacian matrix whose weights are related to the dihedral angle between adjacent triangles and a regularization term measuring the length of the boundary between segments. Such a formulation simultaneously handles segmentation and boundary smoothing, which are usually two separate processes in most previous work. The efficiency is achieved by solving the Mumford-Shah model through a saddle-point problem that is solved by a fast primal-dual method. A preprocess step is also proposed to determine the number of segments that the mesh should be decomposed into. By incorporating this preprocessing step, the proposed algorithm can automatically segment a mesh into meaningful parts. Furthermore, user interaction is allowed by incorporating the user's inputs into the variational model to reflect the user's special intention. Experimental results show that the proposed algorithm outperforms competitive segmentation methods when evaluated on the Princeton Segmentation Benchmark.

Direct mapping of plasmonic coupling between a triangular gold island pair

Local plasmonic coupling between a triangular gold island pair is observed with high spatial resolution (∼10 nm) by an apertureless scanning near-field optical microscope. The measured local field distribution is fully reproduced by three dimensional numerical simulations. Our results show a strong near-field coupling between two adjacent gold triangles, which gives direct evidence to the local field enhancement in plasmonic nanoantenna effect.

A graph-based technique for semi-supervised segmentation of 3D surfaces

A wide range of cheap and simple to use 3D scanning devices has recently been introduced in the market. These tools are no longer addressed to research labs and highly skilled professionals, but rather, they are mostly designed to allow inexperienced users to acquire surfaces and whole objects easily and independently. In this scenario, the demand for automatic or semi-automatic algorithms for 3D data processing is increasing. In this paper we address the task of segmenting the acquired surfaces into perceptually relevant parts. Such a problem is well known to be ill-defined both for 2D images and 3D objects, as even with a perfect understanding of the scene, many different and incompatible semantic or syntactic segmentations can exist together. For this reason recent years have seen a great research effort into semi-supervised approaches, that can make use of small bits of information provided by the user to attain better accuracy. We propose a semi-supervised procedure that exploits an initial set of seeds selected by the user. In our framework segmentation happens by propagating part labels over a weighted graph representation of the surface directly derived from its triangulated mesh. The assignment of each element is driven by a greedy approach that accounts for the curvature between adjacent triangles. The proposed technique does not require to perform edge detection or to fit parametrized surfaces and its implementation is very straightforward. Still, despite its simplicity, tests made on a standard database of scanned 3D objects show its effectiveness even with moderate user supervision.

Feature tracking on the hierarchical equal area triangular mesh

A novel scheme is presented to enable the identification of weather-like features in meteorological data along with a method to track the features across timesteps. The scheme defines a hierarchical triangular mesh produced by repeatedly subdividing an icosahedron. The surface area of each triangle in the mesh is equalised by an iterative application of vertex repulsion, analogous to Coulomb's law. The scheme regrids meteorological data to the triangular mesh via a point inclusion test and area-weighted averaging. Weather-like feature points are detected in the regridded data via a search for extrema in adjacent triangles. These feature points are transformed into objects, such as a low pressure system, by growing the feature point into an area where the sign of the second derivative of the data remains unchanged. The centre points of these objects are tracked across timesteps by the application of a cost minimising algorithm which guarantees locally optimal tracks. By using this suite of methods, weather-like features are identified consistently at high and low latitudes, all weather-like features remain on the same scale and the tracking of features at high latitudes and across the Poles is enabled. These properties are demonstrated by test cases using synthetic data.

Non-oscillatory central schemes on unstructured grids for two-dimensional hyperbolic conservation laws

We propose a new central finite volume scheme on unstructured triangular grids to approximate the solution of general two-dimensional hyperbolic systems of conservation laws. The proposed method is an unstructured two-dimensional extension of the original Nessyahu and Tadmor scheme, and a generalization of the barycentric central methods of Arminjon et al. Starting with a conformal finite element triangulation, the proposed method evolves a piecewise linear numerical solution on two staggered grids, thus avoiding the resolution of the Riemann problems arising at the cell interfaces. The control cells of the original grid are the triangles of a finite element mesh, while the dual cells are the staggered quadrilaterals constructed on adjacent triangles. The resulting central scheme is second-order accurate both in space and time and is oscillations-free thanks to numerical gradients limiting. The extension of the staggered Lax–Friedrichs scheme on unstructured grids is easily obtained from the Nessyahu and Tadmor extension by simply evolving a piecewise constant solution instead of a linear one. We validate the developed schemes and solve classical two-dimensional problems arising in gas dynamics. The quality of the obtained numerical results confirms the efficiency and robustness of the proposed schemes.

Geometrical phase and surface plasmon focusing with azimuthal polarization

Owing to a geometric phase effect, an isosceles triangular aperture etched into thin metal film leads to constructive or destructive interference of surface plasmons excited at the two equal sides under linearly polarized illumination. Through appropriate spatial arrangement of an array of triangles, a highly confined focal spot beyond the diffraction limit can be achieved at the geometric center under azimuthally polarized excitation with field enhancement comparable to a bull's eye plasmonic lens under radially polarized illumination. Through simply rotating the orientation of each triangle aperture by 90°, the plasmonic structure defocuses the same azimuthal polarization illumination due to destructive interference caused by a geometric π-phase difference between the two sides of the triangle and between the adjacent triangles.

Decay of correlations in a topological glass

In this paper we continue the study of a topological glassy system. The state space of the model is given by all triangulations of a sphere with N nodes, half of which are red and half are blue. Red nodes want to have five neighbors while blue ones want seven. Energies of nodes with other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal between two adjacent triangles, with a temperature-dependent probability. We consider the system at very low temperatures. We concentrate on several new aspects of this model: Starting from a detailed description of the stationary state, we conclude that pairs of defects (nodes with the “wrong” degree) move with very high mobility along one-dimensional paths. As they wander around, they encounter single defects, which they then move “sideways” with a geometrically defined probability. This induces a diffusive motion of the single defects. If they meet, they annihilate, lowering the energy of the system. We both estimate the decay of energy to equilibrium, as well as the correlations. In particular, we find a decay like t −0.4.

List total colorings of planar graphs without triangles at small distance

Suppose that G is a planar graph with maximum degree Δ. In this paper it is proved that G is total-(Δ + 2)-choosable if (1) Δ ≥ 7 and G has no adjacent triangles (i.e., no two triangles are incident with a common edge); or (2) Δ ≥ 6 and G has no intersecting triangles (i.e., no two triangles are incident with a common vertex); or (3) Δ ≥ 5, G has no adjacent triangles and G has no k-cycles for some integer k ∈ {5, 6}.

On edge colorings of 1-planar graphs without adjacent triangles

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that every 1-planar graph without adjacent triangles and with maximum degree Δ ⩾ 8 can be edge-colored with Δ colors.

Hexagonal resonance of (3,6)-fullerenes

A (3,6)-fullerene G is a plane cubic graph whose faces are only triangles and hexagons. It follows from Euler’s formula that the number of triangles is four. A face of G is called resonant if its boundary is an alternating cycle with respect to some perfect matching of G. In this paper, we show that every hexagon of a (3,6)-fullerene G with connectivity 3 is resonant except for one graph, and there exist a pair of disjoint hexagons in G that are not mutually resonant except for two trivial graphs without disjoint hexagons. For any (3,6)-fullerene with connectivity 2, we show that it is composed of n(n ≥ 1) concentric layers of hexagons, capped on each end by a cap formed by two adjacent triangles, and none of its hexagons is resonant.

Iterative TIN-based automatic filtering of sparse LiDAR data

In this letter, a novel method for automatic separation of terrain points and object points using sparse Light Detection And Ranging (LiDAR) data is developed. The proposed method is based on iterative elimination of step edges connecting terrain points to object points. The first stage is to detect these edges. Using a triangulated irregular network (TIN) interpolation of the LiDAR raw points, each triangle is assigned to one of two classes: edge triangle or non-edge triangle, using the slope as the discriminative function. Edge triangles are located at the boundary between terrain and non-terrain points; therefore, the vertices of each triangle consist of terrain points and object points. Initially the lower points are considered as terrain points and the higher points as object points. The elevation of object points is adjusted using an interpolation method based on the estimated local slope. The local slope is calculated using non-edge adjacent triangles to the step triangle. The slopes of modified triangles are recalculated using the new elevation. This process is repeated until no triangle is assigned to the edge triangle class. At the end of this process, all the adjusted points are classified as object points and the other points as terrain points. Validation is done by computing the type I (terrain points misclassified as object points) and type II (object points misclassified as terrain points) errors. We used two large datasets containing many complex objects. The proposed approach achieved an overall accuracy higher than 90% and an average error of less than 10%.

Backbone coloring of planar graphs without special circles

In this paper, we prove that if G is a connected planar graph that is C6-free or C7-free and without adjacent triangles, then there exists a spanning tree T of G such that χb(G,T)≤4.

Acyclic chromatic index of planar graphs with triangles

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ a ′ ( G ) , is the least number of colors in an acyclic edge coloring of G. Let G be a planar graph with maximum degree Δ ( G ) . In this paper, we show that χ a ′ ( G ) ⩽ Δ ( G ) + 4 , if G contains no 4-cycle; χ a ′ ( G ) ⩽ Δ ( G ) + 5 , if G contains no intersecting triangles; and χ a ′ ( G ) ⩽ Δ ( G ) + 6 if G contains no adjacent triangles.

Storminess in northern Italy and the Adriatic Sea reaching back to 1760

This study investigates storminess in northern Italy and the northern Adriatic Sea through the examination of several storm proxies. These proxies are based on homogenized daily mean pressure series given at a set of stations (Genoa, Milan, Padua, Turin, and Hvar). The application of widely accepted and well-known methods on pressure series allows for a long-term year-to-year analysis of the intra-seasonal storm variability. As storminess is usually more intense throughout the cold season, our analysis is limited to the October–March period of each year. The following proxies are considered in this study: First, we assess the statistics of geostrophic wind speed. These statistics are derived from two adjacent triangles that are located across the Adriatic Sea (Padua–Hvar–Genoa) and in northern Italy (Genoa–Padua–Turin). Second, we evaluate annual statistics of time series of pressure tendency. Last, intra-seasonal low percentiles of pressure are also made use of. These proxies are used to describe the evolution of the storm climate far back in time, covering in some cases a 260-year long period. The proxies show pronounced interannual and interdecadal variability, but no sustained long-term trend.

Local Condition for Planar Graphs of Maximum Degree 6 to be Total 8-Colorable

Recently Sun et al [X.-Y. Sun, J.-L. Wu, Y.-W. Wu, J.-F. Hou, Total colorings of planar graphs without adjacent triangles, Discrete Math 309:202-206 (2009)] proved that planar graphs with maximum degree six and with no adjacent triangles are total $8$-colorable. This results implies that if every vertex of a planar graph of maximum degree six is missing either a $3$-cycle or a $4$-cycle, then the graph is total $8$-colorable. In this paper we strengthen that condition by showing that if every vertex of a planar graph of maximum degree six is missing some $k_v$-cycle for $k_v \in \{3,4,5,6,7,8\}$, then the graph is total $8$-colorable.