Trihedral Angle Research Articles

Origin of resonant tunneling through single-point barriers

The physical interpretation of the appearance of resonant transmission through single-point barriers is discussed on the basis of a double-layer heterostructure in the squeezing limit as both the thickness of the layers and the distance between them tend to zero simultaneously. In this limit, the electron transmission through a barrier-well structure is derived to be non-zero at certain discrete values of the system parameters forming the so-called resonance set, while beyond this set, the structure behaves as a perfectly reflecting wall. The origin of this phenomenon is shown to result from the reflection coefficients at the interfaces in the inter-layer space. The transmission amplitude is computed as a set function defined on the trihedral angle surface in a three-dimensional parameter space.

Modeling and Analysis of a 2-DOF Spherical Parallel Manipulator.

The kinematics of a two rotational degrees-of-freedom (DOF) spherical parallel manipulator (SPM) is developed based on the coordinate transformation approach and the cosine rule of a trihedral angle. The angular displacement, angular velocity, and angular acceleration between the actuators and end-effector are thus determined. Moreover, the dynamic model of the 2-DOF SPM is established by using the virtual work principle and the first-order influence coefficient matrix of the manipulator. Eventually, a typical motion plan and simulations are carried out, and the actuating torque needed for these motions are worked out by employing the derived inverse dynamic equations. In addition, an analysis of the mechanical characteristics of the parallel manipulator is made. This study lays a solid base for the control of the 2-DOF SPM, and also provides the possibility of using this kind of spherical manipulator as a 2-DOF orientation, angular velocity, or even torque sensor.

Autonomous Navigation Airborne Forward-Looking SAR High Precision Imaging with Combination of Pseudo-Polar Formatting and Overlapped Sub-Aperture Algorithm

Autonomous navigation airborne forward-looking synthetic aperture radar (SAR) observes the anterior inferior wide area with a short cross-track dimensional linear array as azimuth aperture. This is an application scenario that is drastically different from that of side-looking space-borne or air-borne SAR systems, which acquires azimuth synthetic aperture with along-track dimension platform movement. High precision imaging with a combination of pseudo-polar formatting and overlapped sub-aperture algorithm for autonomous navigation airborne forward-looking SAR imaging is presented. With the suggested imaging method, range dimensional imaging is operated with wide band signal compression. Then, 2D pseudo-polar formatting is operated. In the following, azimuth synthetic aperture is divided into several overlapped sub-apertures. Intra sub-aperture IFFT (Inverse Fast Fourier Transform), wave front curvature phase error compensation, and inter sub-aperture IFFT are operated sequentially to finish azimuth high precision imaging. The main advantage of the proposed algorithm is its extremely high precision and low memory cost. The effectiveness and performance of the proposed algorithm are demonstrated with outdoor GBSAR (Ground Based Synthetic Aperture Radar) experiments, which possesses the same imaging geometry as the airborne forward-looking SAR (short azimuth aperture, wide azimuth swath). The profile response of the trihedral angle reflectors, placed in the imaging scene, reconstructed with the proposed imaging algorithm and back projection algorithm are compared and analyzed.

Lack of relative monotonicity among various measures of trihedral angles

We consider the various measures on trihedral angles that have appeared in the literature and we show that no two of these measures are monotone with respect to each other. In other words, for any measures f, g, there exist trihedral angles α, β, γ, θ such that f(α) > f(β), g(α) f(θ), g(γ) > g(θ). This is done through an elementary and systematic method based on multivariable calculus.

Transition radiation produced in dihedral and trihedral angles

Method of electrical images is applied to the description of transition radiation generated when a charges particle passes through a dihedral angle formed by perfectly conducting planes. It is shown that the electromagnetic field generated when a charged particle passes through a dihedral angle α = π/ m is equivalent to the field generated by 2 m charged particles which instantaneously start to move. Transition radiation in trihedral angle is also considered.

Transient radiation of particles in dihedral and trihedral angles

The image method is used for determining the field of transient radiation emitted by a charged particle intersecting a dihedral angle formed by ideally conducting charged planes. The spectral-angular distributions of radiation intensity in a dihedral angle with different corner angles are calculated. The effect of the direction of motion of the particle and of the position of the point at which the particle intersects the plane on radiation parameters is considered. Transient radiation distributions in a trihedral angle are also obtained.

Change in the emitter surface during high-current-density field emission from the 〈111〉 trihedral angle of a reconstructed tungsten tip

Changes in the shape and emission characteristics of the atomically sharp trihedral 〈111〉 angle of a tungsten tip reconstructed in an electric field are studied by continuous-mode field desorption microscopy during high-current-density field emission. The main changes in the tip shape and the slope of the Fowler-Nordheim characteristic occur at an emission current of 1–5 μA. At a current of 50–100 μA taken from the angle, the tip shape and emission characteristics stabilize and remain unchanged in the range 0–150 μA. The new tip shape is characterized by the widening of the angle edges; the appearance of {112} and {001} plane steps in them; a decrease in the sizes of the {011} planes forming the angle faces; and the appearance of steplike transition regions between {011}, {001}, and {112} faces. These changes in the tip shape are related to the fact that the emission field is weaker than the electric field used for preliminary tip reconstruction, the weakening of the field by the space charge of emitted electrons, and a nonuniform temperature distribution in the tip.

Atomically sharp 〈111〉 trihedral angle of a tungsten tip

The processes occuring in the course of heating of a tungsten tip in an electric field and resulting in the formation of the 〈111〉 trihedral angle at the intersection of three {011} closest packed planes in the crystal lattice of tungsten are investigated using field-emission microscopy, continuous-mode field-desorption microscopy, and high-temperature field evaporation microscopy. It is demonstrated that atomically sharp angles can be formed at temperatures above 2200 K in the absence of field evaporation. An atom forming the apex of the trihedral angle lies in the triangle of atoms arranged in the (111) plane. In the triangle, each atom is located at the intersection of the 〈111〉 close-packed atomic rows, which are the boundaries of the {011} planes forming the trihedral angle and the {112} planes forming the angle edges two rows in width.

Field desorption microscopy of the 〈111〉 trihedral angle of a reconstructed tungsten tip

The surface structure near the 〈111〉 trihedral angle, which forms in an electric-field-heated tungsten tip, is studied by field electron microscopy, continuous-mode field desorption microscopy, and high-temperature field evaporation microscopy. The shape and structure of the surface depend on the temperature, field, and time. The angle is formed by three {011} planes, with the (111) plane at its vertex being retained in the form of a triangle or a hexagon with randomly arranged atomic clusters. The edges between {011} faces represent long and narrow {112} planes having longitudinal or transverse steps. In the absence of field evaporation, the edges and angle sharpen, becoming monoatomic. Field evaporation from the angle or microprotrusions on the edges extends these edges and causes transverse steps to appear on them. The explanation of the changes in the shape and structure of the surface is based on considering the competition of surface diffusion, crystal growth in an electric field, and field evaporation.

Preequilibrium thermofield microprotrusions as effective field point sources of electrons and ions

Field emission methods are employed for studying the conditions of formation, crystallographic localization, and emissive properties of preequilibrium thermofield microprotrusions for a number of refractory metals. Individual preequilibrium microprotrusions can be easily obtained using a W emitter of the ordinary 〈110〉 orientation; however, the number of such protrusions on the surface changes with time in the course of ionic emission, as well as their emission parameters (the parameters and the number of microprotrusions do not change in the case of electron emission). Trihedral angles of the rearranged tip, which are formed in the {111} regions, exhibit higher stability to ionic emission. A single trihedral angle stably emitting ions and located on the geometrical axis of the emitter can easily be obtained with the help of a W emitter with the 〈111〉 orientation. Two stable preequilibrium microprotrusions arranged symmetrically about the axis of the emitter in the {111} regions can be obtained using a Ta emitter of the conventional 〈110〉 orientation. Such microprotrusions virtually do not change the emission parameters during long-term extraction of ionic current.

Morphometry of otoconia in the utricle and saccule of developing Japanese quail

The development of otoconia in the utricular and saccular maculae from initial embryonic formation to adult stages was examined in Japanese quails. Both the morphology and size of the otoconia were quantified at different developmental stages. It was observed that the otoconia were initially formed on embryologic stage E5 in the saccule and E6 in the utricle. Otolith mass areas increased in a sigmoidal growth pattern, with saccular otolith areas being smaller than the utricular mass areas. Saccular otolith masses reached adult values at embryonic stage E12 and utricular areas reached adult values at post-hatch day 7. Mature individual otoconia were characterized by a barrel shape with two trihedral faceted ends. However, initial formation of otoconia at E5 (saccular) and E6 (utricular) maculae was characterized by a double fluted morphology that consisted of an hourglass shape with extended fins forming trihedral angles of 120°. Double fluted otoconia rapidly filled, so that by embryonic day 8 mature otoconia dominated the maculae for the remainder of development through adulthood. Thus, a progression from double fluted to mature forms was noted. Mature utricular otoconia in adult quails averaged 11 μm in length and 5 μm in width, with length/width ratios of approximately 2.5:1, for all size ranges. Saccular otoconia were smaller, having about 70% the size of utricular otoconia in both length and width. During development, the average size and range of individual otoconia increased nearly linearly for both otolith organs. In the utricular macula, large otoconia were concentrated in the lateral regions of the epithelium. In contrast, otoconia of various sizes were distributed uniformly across the surface of the saccular macula.

A Vector Proof of a Theorem of Bang

(2001). A Vector Proof of a Theorem of Bang. The American Mathematical Monthly: Vol. 108, No. 6, pp. 562-564.

The Remarkable Tetron

(2001). The Remarkable Tetron. The American Mathematical Monthly: Vol. 108, No. 4, pp. 368-370.

The trihedral angle

Just as in the common triangle, solving the trihedral angle means getting some of the data and inferring the rest mathematically. Most such problems are solved by solid geometry and by spherical trigonometry. Here we present an approach which facilitates the handling of such problems. Moreover, the two basic theorems (namely, the law of sines and the law of cosines) of spherical trigonometry are derived as relatively simple by‐products of the general approach. Spherical trigonometry is a simple and vivid example of non‐Euclidian geometry and also a vital tool in navigation, astronomy and geodesy, etc. So it is worthwhile to consider it in the early stages of mathematical education.

Determining the aspect ratio of a CCD camera

The aspect ratio is an essential parameter of a camera. It is the ratio of the distances between two adjacent pixels in the frame buffer along the horizontal and vertical directions, respectively. A simple and fast method is proposed for calculating the aspect ratio using a trihedral normal angle. The focal length can also be derived. Neither nonlinear equations nor iterative procedures are used in this approach, and therefore the calculating procedure is analytic and easy to implement.

A new generalized computational framework for finding object orientation using perspective trihedral angle constraint

This paper investigates a fundamental problem of determining the position and orientation of a three-dimensional (3-D) object using a single perspective image view. The technique is focused on the interpretation of trihedral angle constraint information. A new closed form solution based on Kanatani's formulation is proposed. The main distinguishing feature of the authors' method over the original Kanatani formulation is that their approach gives an effective closed form solution for a general trihedral angle constraint. The method also provides a general analytic technique for dealing with a class of problem of shape from inverse perspective projection by using "angle to angle correspondence information." A detailed implementation of the authors' technique is presented. Different trihedral angle configurations were generated using synthetic data for testing the authors' approach of finding object orientation by angle to angle constraint. The authors performed simulation experiments by adding some noise to the synthetic data for evaluating the effectiveness of their method in a real situation. It has been found that the authors' method worked effectively in a noisy environment which confirms that the method is robust in practical application.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

π-Orbital conjugation and rehybridization in bridged annulenes and deformed molecules in general: π-orbital axis vector analysis

Abstract

An Infinite Class of Deltahedra

The appropriate designation deltahedra (A-hedra) has been proposed by Cundy [1] for polyhedra all faces of which are equilateral triangles. Three of the regular (Platonic) solids the tetrahedron, octahedron, and icosahedron are convex deltahedra with 4, 8, and 20 equilateral triangular faces, respectively. Models of deltahedra can be made conveniently by using No. 10 rubber bands to join the 3-inch flanged equilateral triangle cardboard invented by Seattle architect, Fred Bassetti [2]. Kits of these Poly-O panels can be purchased from Book-Lab, Inc. [3]. Or, home-made can be cut from railroad board in the manner clearly described by Stewart [4], Pritchett [5], and Woolaver [6]. Three convex deltahedra can be formed by placing together the bases of two congruent pyramids: the 6-faced triangular dipyramid or ditetrahedron, the 8-faced square dipyramid or octahedron, and the 10-faced pentagonal dipyramid. Six equilateral triangles can be assembled into a regular hexagon. Two such hexagons joined around their perimeters form a degenerate dipyramid. If this is separated into two parts along two coincident long diagonals, the two parts can be opened up by pressing inward on the ends of the diagonals. After one of the opened parts is rotated through 900, the two parts can be rejoined to form a 12-faced Siamese dodecahedron which is convex. There are only eight convex deltahedra [7]. Completing the set are the 14-faced triaugmented triangular prism, formed by attaching square pyramids to the square lateral faces of a triangular prism; and the 16-faced dicapped square antiprism, formed by attaching square pyramids to the two bases of a square antiprism. Thus the eight convex deltahedra are formed from 4, 6, 8, 10, 12, 14, 16 and 20 equilateral triangles. There are no convex deltahedra with 18 faces, nor any with more than 20 faces. Once convexity is abandoned, the possibilities of forming deltahedra are endless. Members of one particularly interesting and attractive type are made by attaching regular tetrahedra to some or all of the faces of other deltahedra. The deltahedra thus made by augmentation are pseudo-stellated, since they are not the same as the solids produced by extending the planes of the faces of the basic polyhedron, except in the case of the augmented octahedron which is Kepler's famous stella octangula. A particularly attractive model is the pseudo-stellated icosahedron. Models of an infinite subclass of deltahedra, the spiral (twisted) type, can be constructed from strips of equilateral triangles (FIGURE 1) flexing about the common sides. Any desired lengths of these strips can be formed easily by the cardboard panel-rubber band method. Models of spiral deltahedra can be started by joining the sides of the left-most angles of three strips to form a trihedral angle. The construction proceeds by joining sides of triangles of adjacent strips that have an end in common. (Differently colored strips emphasize the spiral nature of the deltahedron.) The three strips may be chosen from the Hand L-patterns in four ways, namely: HHH, LLL, HHL, and LLH. The HHH form appears to twist to the right (see FIGURE 2), and the LLL form appears to twist to the left. However, in general, each form consists of a pile of regular octahedra capped top and bottom with a

A uniform vector distribution for accurate spatial integration

A method is developed for the design of exactly uniform high-density vectorial distributions inside the 1/48th symmetry-irreducible trihedral angle of highest cubic symmetry. The first application led to a particular 489-vector distribution in the basic solid angle which extends, by permutations, to 23,472 uniformly-distributed vectors in the full 4π steradians. This density ( N = 489) meets a criterion of about one part in 10 5 set for the accuracy requirement of three-dimensional integration of functions in crystal lattice dynamics and in elastic wave theory. The dependence of the accuracy of numerical calculations on vector density is determined by comparison with exact integrals of certain octahedral functions. It is found to depend significantly on the three-dimensional anisotropy of the surfaces of these functions. A ten-decimal table of the direction cosines of the 489-vector distribution is supplied as an appendix.

Three-dimensional running waves in a barotropic gas: PMM vol. 33, no. 1, 1969, pp. 169–174

The equations of potential triple waves in a barotropic gas with an arbitrary equation of state are obtained. The properties of the solutions for contiguous flows of the double- and triple-wave type are investigated. The solutions of certain three- dimensional self-similar problems of three pistons are solved in the case of a “heavy” gas with a high initial velocity of sound. These problems concern three planes forming an infinite trihedral angle within which the gas is at rest at the instant t = 0, whereupon the planes begin to retract from the gas at high constant velocities.