Arbitrary Polyhedra Research Articles

Magnetic Flux Density and Vector Potential of Uniform Polyhedral Sources

The magnetic flux density and the magnetic vector potential produced by a uniform current density or a uniform magnetization inside an arbitrary polyhedron are calculated analytically. The obtained closed-form expressions are remarkably simple and provided in an intrinsic vector form, independently of any reference frame. They are well suited to cope with the data structures, i.e., faces-edges and edges-vertices incidence matrices, provided by unstructured polygonal meshes generators. The expressions obtained contain only elementary functions, and computed results illustrate their effectiveness.

3-D computation of flood processes in sharp river bends

Three different three-dimensional computational fluid dynamics codes—SSIIM, Fluent and RSim-3D—were used to predict water surface and flow field during the 100-year flood of August 2002 in a sharp bend of the Danube River near the Austrian municipality of Grein. All models solved the Reynolds-averaged Navier–Stokes equations using a finite-volume approach. For the SSIIM and Fluent models, a structured grid of hexahedral cells was employed, while the basis for the RSim-3D model was an unstructured grid of arbitrary polyhedrons. The non-hydrostatic pressure field was computed using the Simple method and the standard k−ε model provided turbulence closure. The position of the water surface was derived from the pressure field. Computed water surface elevations were compared with each other and verified using field data observed during the flood event. It was found that the maximum super-elevation of bank water levels was predicted within an error margin of 10–20% of the observed value. Flow velocity magnitudes were found to vary among the different models, which is primarily credited to different pressure distribution schemes. Strong secondary motions with velocity magnitudes of up to 1 m/s were predicted by the computer codes, contributing to the extreme super-elevations observed.

Sphere packing with a geometric based compression algorithm

An efficient algorithm for the random packing of spheres can significantly save the cost of the preparation of an initial configuration often required in discrete element simulations. It is not trivial to generate such random packing at a large scale, particularly when spheres of various sizes and geometric domains of different shapes are present. Motivated by the idea of compression complemented by an efficient physical process to increase packing density, shaking, a new approach, termed compression algorithm, is proposed in this work to randomly fill any arbitrary polyhedral or cylindrical domains with spheres of various sizes. The algorithm features both simplicity and high efficiency. Tests show that it takes 181 s on a 1.4-GHz PC to complete the filling of a cylindrical domain with a total number of 26,787 spheres, achieving a packing density of 52.89%.

Maillage simplicial d'un polyèdre arbitraire

Issues related to the existence of a triangulation of an arbitrary polyhedron are addressed. Given a boundary surface mesh (a set of triangular facets), the problem to decide whether or not a triangulation (with no internal points apart from the Steiner points) exists is reported to be NP-hard. In this paper, an algorithm to triangulate a general polyhedron is used which makes use of a classical Delaunay triangulation algorithm, a phase for recovering the missing boundary facets by means of facet partitioning and a final phase that makes it possible to remove the additional (non-Steiner) points previously defined so as to recover the initial boundary mesh thus resulting in a mesh of the given polyhedron. To cite this article: P.-L. George, H. Borouchaki, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

IBCD: a fast collision detection algorithm based on image space using OBB

AbstractImage‐based collision detection algorithms make efficient use of the graphics rendering hardware and reduce the computational cost of CPU. In this article, a fast collision detection algorithm based on image space is presented, which combines graphics hardware capabilities with a simplified geometric representation (oriented bounding box) in order to rapidly detect collisions between complex objects. The method can deal with arbitrary polyhedra, while preserving the merits of image‐based collision detection algorithms. This is achieved by decomposing the surfaces of an object into a list of convex pieces. High efficiency of the algorithm is obtained by organizing the convex pieces into a binary tree with each node representing a convex piece, and by adopting triangle strip compression. The algorithm has been implemented and compared with related algorithms. Copyright © 2003 John Wiley & Sons, Ltd.

Polyhedral Objects Metamorphosis Using Convex Decomposition and Morphology

A new technique is presented for computing continuous shape transformations between polyhedral objects. The polyhedron shape transformations can be divided into polyhedron metamorphosis and bi-directional local rigid body rotation transformation. By decomposing two objects into sets of individual convex sub-objects respectively, and establishing the matching between two subsets, the approach can solve the metamorphosis problem of two non-homotopic objects (including concave objects and holey objects). Compared with other methods, this metamorphosis algorithm can be executed automatically for arbitrary polyhedrons and no need user interaction. The user has the ability to choose an automatic matching or to select interactively pairs of corresponding matching convex subsets to obtain special effects. Experiments show that this method can generate natural, high-fidelity, eye-pleasing metamorphosis results with simple computation.

‘Ultimate’ robustness in meshing an arbitrary polyhedron

AbstractGiven a boundary surface mesh (a set of triangular facets) of a polyhedron, the problem of deciding whether or not a triangulation exists is reported to be NP‐hard. In this paper, an algorithm to triangulate a general polyhedron is presented which makes use of a classical Delaunay triangulation algorithm, a phase for recovering the missing boundary facets by means of facet partitioning, and a final phase that makes it possible to remove the additional points defined in the previous step. Following this phase, the resulting mesh conforms to the given boundary surface mesh. The proposed method results in a discussion of theoretical interest about existence and complexity issues. In practice, however, the method should provide what we call ‘ultimate’ robustness in mesh generation methods. Copyright © 2003 John Wiley & Sons, Ltd.

Fast Volume Rendering and Cutting for Finite Element Model

Based on the incremental slicing, a direct volume rendering algorithm for finite element meshes is proposed. To improve the algorithm efficiency, the element compensation method and sliced polygon forming method are presented. The image quality is improved greatly by the hybrid rendering of both the volume data and geometric surfaces. We also integrate a volume cutting method with arbitrary convex polyhedron into the volume rendering pipeline. This offers the user more flexibility to explore the data distribution inside structures.

COMPUTING AN ALMOST MINIMUM SET OF SPANNING LINE SEGMENTS OF A POLYHEDRON

A set of spanning line segments (SLS) is a subset of the edges of a finite polyhedron in E 3 such that an arbitrary plane intersects the polyhedron if and only if the plane intersects at least one of the line segments of the SLS. In this paper an algorithm is presented for computing an almost minimum set of spanning line segments for an arbitrary polyhedron [Formula: see text] . When the number of extreme vertices of [Formula: see text] is odd, the computed SLS is minimum; when the number of extreme vertices of [Formula: see text] is even, the size of the computed SLS is at most the minimum size plus one. The algorithm has linear-time complexity for a convex polyhedron, hence is optimal in this case; its time complexity is Θ(m log m) for an arbitrary polyhedron, where m is the number of vertices of the polyhedron.

Terrain correction computations for a densely sampled DTM in the Bavarian Alps

Various methods are presented for terrain correction computations referred to a digital terrain model (DTM) of a part of the Bavarian Alps to the southwest of Munich. The most commonly used fast Fourier transform (FFT) techniques exhibit some shortcomings for the zone adjacent to the computation point. A combination method is proposed which applies FFT for the whole DTM except for a small zone surrounding the computation point. In the latter an analytical method is employed by decomposing the terrain into elementary bodies. Different modelling techniques are tested, with the arbitrary polyhedron being the most precise one. With the proposed method both the exactness of the analytical expressions and the efficiency and speed of the FFT technique can be exploited. It is shown that this modified approach has two positive effects. First, the convergence of the FFT series is regained, which is not always the case when slopes greater than 45∘ exist in the terrain. Second, it approximates the terrain correction as computed by the classical prism summation method better than its linear approximation as given by FFT.

Continuous field based free-form surface modeling and morphing

In this paper, we present a novel continuous field based free-form surface modeling and morphing algorithm. Polyhedra are used in our algorithm as the skeletons of the free-form surfaces. Unlike the conventional methods, we first construct a continuous field from each skeleton using variational interpolation, all these fields are then fused into a global continuous field using blending, warping and boolean operations. The final surface is approximated by the iso-surface of the global field. Based on the field blending operation, our modeling algorithm can easily perform 3D morphing between two arbitrary polyhedra in an intuitive and controllable manner by introducing certain constraints. Several examples are shown to demonstrate that our method can be used to naturally and intuitively model complex free-form surfaces and morph in a controllable way.

New method for fast computation of gravity and magnetic anomalies from arbitrary polyhedra

We show that at any point the gravity field from a solid body bounded by plane surfaces and having uniform density can be computed as a field from a fictitious distribution of surface mass‐density on the same body. The surface mass density at every surface element is equal to the product of the volume density of the body and the scalar product of (1) the unit outward vector normal to that surface element and (2) the position vector of the surface element with respect to the point of observation. Accordingly, the contribution to the gravity field from any plane surface of the body vanishes if the observation point lies in the plane of that surface. As a result, we can compute the gravity field everywhere, including points inside, on the surface, on an edge, or at a corner of the body where more than two surfaces meet. This new result lets us compute the gravity field using exactly the same simple procedure as for the magnetic field of a uniformly magnetized object, computed from an equivalent surface distribution of magnetic pole density. To get the gravity field while computing the magnetic field, one simply uses the product of this surface mass density and the universal gravitational constant instead of the surface magnetic pole density. Therefore, the same computer program can be used to compute the gravity, the magnetic field, or both simultaneously. This simple and novel approach makes the numerical computations much faster than all other previously published schemes.

Systems and Experiment Paper: Construction of Panoramic Image Mosaics with Global and Local Alignment

This paper presents a complete system for constructing panoramic image mosaics from sequences of images. Our mosaic representation associates a transformation matrix with each input image, rather than explicitly projecting all of the images onto a common surface (e.g., a cylinder). In particular, to construct a full view panorama, we introduce a rotational mosaic representation that associates a rotation matrix (and optionally a focal length) with each input image. A patch-based alignment algorithm is developed to quickly align two images given motion models. Techniques for estimating and refining camera focal lengths are also presented. In order to reduce accumulated registration errors, we apply global alignment (block adjustment) to the whole sequence of images, which results in an optimally registered image mosaic. To compensate for small amounts of motion parallax introduced by translations of the camera and other unmodeled distortions, we use a local alignment (deghosting) technique which warps each image based on the results of pairwise local image registrations. By combining both global and local alignment, we significantly improve the quality of our image mosaics, thereby enabling the creation of full view panoramic mosaics with hand-held cameras. We also present an inverse texture mapping algorithm for efficiently extracting environment maps from our panoramic image mosaics. By mapping the mosaic onto an arbitrary texture-mapped polyhedron surrounding the origin, we can explore the virtual environment using standard 3D graphics viewers and hardware without requiring special-purpose players.

A FINITE-VOLUME SCHEME FOR RADIATIVE HEAT TRANSFER IN SEMITRANSPARENT MEDIA

A procedure for computing radiation in semitransparent media is presented. A conservative cell-based finite-volume method is developed for unstructured meshes composed of arbitrary polyhedra. The angular domain is discretized into a finite number of control angles over which radiant energy is conserved. At Fresnel interfaces, numerical procedures are developed to conservatively transfer radiant energy from one angular direction to another as a result of reflection and refraction, while accounting for control angle overhang. Similar procedures are also employed at specular surfaces and symmetry boundaries. The method is tested against analytical solutions and shown to perform satisfactorily.

New scheme for computing the magnetic field resulting from a uniformly magnetized arbitrary polyhedron

The magnetic field at any point outside a uniformly magnetized polyhedron of arbitrary shape is obtained by adding the fields resulting from the effective free magnetic poles on each of the polygonal surfaces of the polyhedron. For each polygonal surface, the components of the field at the point of observation are expressed in terms of new line integrals around the edges of the polygon and the solid angle subtended by the polygon at the point of observation. The line integrals are standard elementary forms. This new approach makes the numerical evaluation of the magnetic fields for such models much simpler and faster than previously published methods.

Improved formulation for geometric properties of arbitrary polyhedra

Improved formulation for geometric properties of arbitrary polyhedra

A three-dimensional magnetostatics computer code for insertion devices.

RADIA is a three-dimensional magnetostatics computer code optimized for the design of undulators and wigglers. It solves boundary magnetostatics problems with magnetized and current-carrying volumes using the boundary integral approach. The magnetized volumes can be arbitrary polyhedrons with non-linear (iron) or linear anisotropic (permanent magnet) characteristics. The current-carrying elements can be straight or curved blocks with rectangular cross sections. Boundary conditions are simulated by the technique of mirroring. Analytical formulae used for the computation of the field produced by a magnetized volume of a polyhedron shape are detailed. The RADIA code is written in object-oriented C++ and interfaced to Mathematica [Mathematica is a registered trademark of Wolfram Research, Inc.]. The code outperforms currently available finite-element packages with respect to the CPU time of the solver and accuracy of the field integral estimations. An application of the code to the case of a wedge-pole undulator is presented.

The geometric cone relations for simplicial and cubical complexes

This note gives several geometric cone relations for arbitrary polyhedron with a regular decomposition. When the decomposition is simplicial or cubical, we obtain the geometric analogs of Dehn-Sommerville and angle-sum relations for simplicial and cubical complexes.

Construction of convex continuations for functions defined on a hypersphere

Construction of convex continuations for functions defined on the vertices of some combinatorial polyhedra, in particular the permutation polyhedron and the arrangement polyhedron, has been studied in [1, 2]. Subsequently this result has been generalized to functions defined at the extreme points of an arbitrary polyhedron [3]. For purposes of combinatorial optimization [4-6] it is relevant to consider the existence and construction of convex continuations from continua, in particular, when the function is defined on a hypersphere in thek-dimensional space. Unfortunately, passage to the limit from discrete sets to continua does not produce positive results in this case. We are thus forced to develop special approaches to investigating the existence of convex continuations of functions defined on continua.

Dynamics of Orbits Close to Asteroid 4179 Toutatis

We use a radar-derived physical model of 4179 Toutatis (975870) to investigate close-orbit dynamics around that irregularly shaped, non-principal-axis rotator. The orbital dynamics about this body are markedly different than the dynamics about uniformly rotating asteroids. The results of this paper are generally applicable to orbit dynamics about bodies in a non-principal-axis rotation state. The radar results support the hypothesis that Toutatis has a homogeneous density distribution, and we assume a density of 2.5 g/cc. The asteroid's gravity field is computed using a truncated harmonic expansion when outside of its circumscribing sphere and a closed-form expression for the potential field of an arbitrary polyhedron when inside that sphere. The complete equations of motion are time-periodic in the Toutatis-fixed frame due to the complex rotation of the asteroid. The system is Hamiltonian and has all the characteristics of such a system, including conservation of phase volume, but there is no Jacobi constant of the motion and zero velocity surfaces cannot be used to analyze the system's behavior. We also examine some of the close-orbit dynamics with the Lagrange planetary form of the equations of motion. Families of quasi-periodic “frozen orbits” that show minimal variations in orbital elements are found to exist very close to the asteroid; some of them are stable and hence can hold natural or artificial satellites. A retrograde family of frozen orbits is especially robust and persists down to semi-major axes of about 2.5 km, comparable to half of Toutatis' longest dimension. We identify families of periodic orbits, which repeat in the Toutatis-fixed frame. Due to the time-periodic nature of the equations of motion, all periodic orbits about Toutatis in its body-fixed frame must be commensurate with the 5.42-day period associated with those equations. Exact calculations of both stable and unstable periodic orbits are made. The sum of surface forces acting on a particle on Toutatis is time-varying, so particles on and in the asteroid are being continually shaken with a period of 5.42 days, perhaps enhancing the uniformity of the regolith distribution. A global map of the gravitational slope reveals that it is surprisingly shallow for such an elongated, irregularly shaped object, averaging 16° globally and less than 35° over 96% of the surface. A global map of tangential accelerations shows no values larger than 0.5 mm/s2, an average value of 0.2 mm/s2, and less than 0.25 mm/s2over 70% of the surface. A global map of the escape speed for launch normal to the surface shows that quantity to be between 1.2 and 1.8 m/s over most of the surface. Each of these mapped quantities has small periodic variations. We have found trajectories that leave the surface, persist in the region of phase space around a frozen orbit, and then impact the surface after a flight time of more than 100 days. Return orbit durations of years seem possible. Whereas a uniformly rotating asteroid preferentially accumulates non-escaping ejecta on its leading sides, Toutatis accumulates ejecta uniformly over its surface. We render a variety of close orbits in inertial and body-fixed frames.