Right Triangle Research Articles

On Characterization of Absolute Geometries

Starting from a general absolute plane A = (P, L, α, ≡) in the sense of Karzel et al. (Einfuhrung in die Geometrie, p. 96, 1973), Karzel and Marchi introduced the notion of a Lambert–Saccheri quadrangle (L-S quadrangle) in Karzel and Marchi (Le Matematiche LXI:27–36, 2006): A quadruple (a, b, c, d) of points of P, no three collinear, is a L-S quadrangle, if \({\overline{a,d}\bot\overline{a,b}\bot\overline{b,c}\bot\overline{c,d}}\). Denoting the foot of a on the line \({\overline{c, d}}\) with \({a^{\prime}=\{a\bot\overline{c,d}\}\cap \overline{c,d}}\), the L-S quadrangle (a, b, c, d) is called rectangle, hyperbolic or elliptic quadrangle if \({a^{\prime}=d,\; a^{\prime}\,{\in}\, ]c,d[}\) or\({a^{\prime}\,{\notin}\, ]c,d[\cup \{d\}}\) respectively. Let LS be the set of all L-S quadrangles and LSr, LSh or LSe the subset of all rectangles, hyperbolic or elliptic L-S quadrangles respectively. In Karzel and Marchi (Le Matematiche LXI:27–36, 2006) it was claimed that either LS = LSr or LS = LSh or LS = LSe. To this classification we add five further classifications of general absolute planes by using “distance” [defined in Karzel and Marchi (Discrete Math 308:220–230, 2008)] or the notions of “interior” and “exterior” angle, introduced in Karzel et al. (Resultate Math 51:61–71, 2007) and considering besides Lambert–Saccheri quadrangles, also triangles in particular right-angled triangles. For Lambert–Saccheri quadrangles (a, b, c, d) the relations between distances of the diagonal points (a, c) and (b, d) or between the “midpoint” \({o:=\overline{a,c}\cap\overline{b,d}}\), and the corner points a, b, c, d give us possibilities for complete characterizations. Using triangles (a, b, c) and denoting by m and n the midpoints of (a, b) and (a, c) we classify the absolute planes by the relations between the distances |b, c| and 2|m, n|. All our main results are summarized at the end of the introduction.

THE RIGHT‐ANGLED MIXTURE TRIANGLE

Many problems in nutritional ecology concern the proportions of different dietary components, for example the proportions of food types or the balance of nutrients in the diet. Geometric analysis has proved a powerful way to analyze these multivariate relationships, but this approach has been applied mainly in laboratory studies. I recommend a technique, called the Right-angled Mixture Triangle (RMT), which is particularly well suited for geometric analysis in field-based nutritional studies. RMTs enable the relationships among three or more variables to be visualized within a two-dimensional plot, providing a powerful tool for disentangling complex nutritional relationships. One example of dietary nutrient balance explores the proportions of different categories of foods (graminoids, browse, and forbs) in the diets of four species of herbivores in the Nepalese Himalaya. Blue sheep in typical habitat. The photos: Four sympatric species of large herbivores in the Nepalese Himalaya. Analysis of the proportions of browse, forbs, and graminoids in the diet suggests a greater similarity between the diets of the blue sheep and the domestic sheep, and the least similarity between the goat and the yak. Domestic sheep driven along a mountain road. Goats being herded along a watercourse in Nepal. Yak grazing. These photographs illustrate the article “Toward a quantitative nutritional ecology: the right-angled mixture triangle” by David Raubenheimer, tentatively scheduled to appear in Ecological Monographs 81(3), August 2011. [doi:10.1890/10-1707.1]

Right unitarity triangles and tri-bimaximal mixing from discrete symmetries and unification

We propose new classes of models which predict both tri-bimaximal lepton mixing and a right-angled Cabibbo–Kobayashi–Maskawa (CKM) unitarity triangle, α ≈ 90 ° . The ingredients of the models include a supersymmetric (SUSY) unified gauge group such as SU ( 5 ) , a discrete family symmetry such as A 4 or S 4 , a shaping symmetry including products of Z 2 and Z 4 groups as well as spontaneous CP violation. We show how the vacuum alignment in such models allows a simple explanation of α ≈ 90 ° by a combination of purely real or purely imaginary vacuum expectation values (vevs) of the flavons responsible for family symmetry breaking. This leads to quark mass matrices with 1–3 texture zeros that satisfy the “phase sum rule” and lepton mass matrices that satisfy the “lepton mixing sum rule” together with a new prediction that the leptonic CP violating oscillation phase is close to either 0°, 90°, 180°, or 270° depending on the model, with neutrino masses being purely real (no complex Majorana phases). This leads to the possibility of having right-angled unitarity triangles in both the quark and lepton sectors.

A heuristic for the circle packing problem with a variety of containers

In this paper we present a heuristic algorithm based on the formulation space search method to solve the circle packing problem. The circle packing problem is the problem of finding the maximum radius of a specified number of identical circles that can be fitted, without overlaps, into a two-dimensional container of fixed size. In this paper we consider a variety of containers: the unit circle, unit square, rectangle, isosceles right-angled triangle and semicircle. The problem is formulated as a nonlinear optimization problem involving both Cartesian and polar coordinate systems. Formulation space search consists of switching between different formulations of the same problem, each formulation potentially having different properties in terms of nonlinear optimization. As a component of our heuristic we solve a nonlinear optimization problem using the solver SNOPT. Our heuristic improves on previous results based on formulation space search presented in the literature. For a number of the containers we improve on the best result previously known. Our heuristic is also a computationally effective approach (when balancing quality of result obtained against computation time required) when compared with other work presented in the literature.

Some nice relations between right-angled triangles and the Golden Section

The international debate about experimental approaches to the teaching and learning mathematics is very current. While number theory lends itself naturally to such approaches, elementary geometry can also provide interesting starting points for creative work in class. This article shows how simple considerations about right triangles and the golden section can be a starting point for a mathematical didactic laboratory.

Rays in a right triangle

Rays in a right triangle

Heat transfer comparison between a vertical rectangular cavity and an isosceles right-angled triangular cavity of equal cross-sectional area

This paper addresses the heat transfer performance of natural convection flows in three different, (but related) cavities in the form of: a square, isosceles right-angled triangle, and vertical rectangle with aspect ratio 2:1. The isosceles right-angled triangular cavity is derived from a square cavity when cut in half diagonally, whereas the vertical rectangular cavity is derived from a square cavity when cut in half vertically. In the three cavities, the left vertical wall is the common wall heated. The buoyant air flow is characterized by height-based Rayleigh numbers ranging from a conduction-dominant to up to 106 for the laminar natural convection regime. Employing the finite volume method, the velocity and temperature fields as well as the mean convective coefficients evaluated at the common heated vertical wall are numerically determined for the isosceles right-angled triangular cavity. For this cavity, flow streamlines and temperature contours are presented in graphical form and some numerical results are validated against published experimental measurements. A one-to-one comparison for the heat transfer performance of the three interconnected cavities is reported in tabulated form.

반사판이 있는 스피드론 프랙탈 원형 편파 안테나

본 논문에서는 스피드론 프랙탈(Spidron fractal)을 이용하여 원형 편파를 발생시키는 마이크로스트립 슬롯 안테나를 설계하였다. 설계된 안테나의 접지면에는 직각삼각형 7개가 연속적으로 붙어있는 형태의 슬롯이 형성되어 있다. 이 슬롯은 원형 편파 발생을 위한 구조이고, 설계된 안테나는 단일 급전 구조로 원형 편파를 발생시킨다. 설계된 안테나의 이득을 높이기 위해, 마이크로스트립 급전 라인의 아랫 부분에 반사판을 배치하였다. 시뮬레이션을 이용하여 첫 번째 삼각형의 높이, 빗변과 높이 사이의 각도, 마이크로스트립 급전 라인의 위치, 기판과 반사판 사이의 간격 등을 조절하였고, 최적화 설계하였다. 제안된 안테나는 Taconic사의 RF-35 기판을 사용하여 제작하였다. 제작된 안테나의 크기는 <TEX>$40{\times}40{\times}18.6mm^3$</TEX>이고, 반사판은 접지면으로부터 18.6 mm 아래에 위치한다. 측정된 안테나의 이득은 4.3 GHz에서 6.7 dBi이고, -10 dB 반사계수 대역폭은 41 %, 3 dB 축비 대역폭은 7.4%이다. In this paper, design of a microstrip-fed, Spidron fractal-shaped slot antenna with circular polarization is represented. The ground plane of the designed antenna has a slot that comprises seven right-angled triangle next to each other in a low. A reflector is placed at the bottom on microstrip feeding line to enhance the antenna gain. The optimized design was conducted by varying a length of the first right-angled triangle, location of feeding line, space between a substrate and a reflector. The proposed antenna was fabricated on a Taconic-RF35 substrate. The entire dimension of the fabricated antenna is <TEX>$40{\times}40{\times}18.6mm^3$</TEX> and the reflector is 18.6 mm away from the ground plane. The measured gain of the fabricated antenna is 6.7 dBi at 4.3 GHz. The measured bandwidths of -10 dB reflection and 3 dB axial ratio are 41 % and 7.4 %, respectively.

Optical and computed evaluation of keyhole diffractive imaging for lensless x-ray microscopy

X-ray microscopy without the need to focus the x-rays is obviously desirable. Thus, various ways of reconstructing the image from the far-field diffraction pattern have been demonstrated during the last decade. Here, the authors describe a variant of keyhole diffractive imaging in which the knowledge of the illuminating wavefront is used in the reconstruction. Unlike prior work in keyhole imaging in which a diverging beam was used, the authors employ a collimated beam such as emerges from a free electron x-ray source. From both computer simulations and scaled optical experiments, the authors found that the reconstruction is best when the shape is a right-angled triangle. For the authors’ samples, the best size of the triangle is one that covers about half of a rectangle 3000×3000 wavelengths. If the area is smaller, then the time to cover the entire sample is too long; if larger, then the reconstruction takes far longer and results in a noisy image. From the authors’ scaled experiment conducted under blind conditions (no a prior knowledge of the sample), it appears that a resolution of 10 nm should be readily achievable at an x-ray wavelength of 0.18 nm.

Going the distance: spatial scale of athletic experience affects the accuracy of path integration

Evidence suggests that athletically trained individuals are more accurate than untrained individuals in updating their spatial position through idiothetic cues. We assessed whether training at different spatial scales affects the accuracy of path integration. Groups of rugby players (large-scale training) and martial artists (small-scale training) participated in a triangle-completion task: they were led (blindfolded) along two sides of a right-angled triangle and were required to complete the hypotenuse by returning to the origin. The groups did not differ in their assessment of the distance to the origin, but rugby players were more accurate than martial artists in assessing the correct angle to turn (heading), and landed significantly closer to the origin. These data support evidence that distance and heading components can be dissociated. Furthermore, they suggest that the spatial scale at which an individual is trained may affect the accuracy of one component of path integration but not the other.

The evaluation of American options in a stochastic volatility model with jumps: An efficient finite element approach

We consider the problem of pricing American options in the framework of a well-known stochastic volatility model with jumps, the Bates model. According to this model the asset price is described by a jump-diffusion stochastic differential equation in which the jump term consists of a Lévy process of compound Poisson type, while the volatility is modeled as a CIR-type process correlated with the asset price. Pricing American options under the Bates model requires us to solve a partial integro-differential equation with the final condition and boundary conditions prescribed on a free boundary. In this paper a numerical method for solving such a problem is proposed. In particular, first of all, using a Richardson extrapolation technique, the problem is reduced to a problem with fixed boundary. Then the problem obtained is solved using an ad hoc finite element method which efficiently combines an implicit/explicit time stepping, an operator splitting technique, and a non-uniform mesh of right-angled triangles. Numerical experiments are presented showing that the option pricing algorithm developed in this paper is extremely accurate and fast. In particular it is significantly more efficient than other numerical methods that have recently been proposed for pricing American options under the Bates model.

Temperature‐independent vibration sensor with a fiber Bragg grating

AbstractA novel temperature‐independent vibration sensor based on a fiber Bragg grating (FBG) is demonstrated.The FBG was glued in a slanted direction onto the lateral side of a right‐angled triangle cantilever beam. Vertical vibration applied to the cantilever beam leads to a periodical variation of bending curvature along the beam length. As a result of beam bending, the FBG is chirped and its reflection bandwidth and optical power change with the deflection of the beam. Experimental results were compared with the data of vibration measurement of a conventional resistance strain gauge. Good agreement has been achieved. The sensor is temperature independent, owning to the temperature‐independence nature of the reflection bandwidth and optical power of the FBG. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52:2282–2285, 2010; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.25429

Pythagorean quantization, action(s) and the arrow of time

Searching for the first well-documented attempts of introducing some kind of "quantization" into the description of nature inevitably leads to the ancient Greeks, in particular Plato and Pythagoras. The question of finding the so-called Pythagorean triples, i.e., right-angled triangles with integer length of all three sides, is, surprisingly, connected with complex nonlinear Riccati equations that occur in time-dependent quantum mechanics. The complex Riccati equation together with the usual Newtonian equation of the system, leads to a dynamical invariant with the dimension of an action. The relation between this invariant and a conserved "angular momentum" for the motion in the complex plane will be determined. The "Pythagorean quantization" shows similarities with the quantum Hall effect and leads to an interpretation of Sommerfeld's fine structure constant that involves another quantum of action, the "least Coulombic action" e2/c. Since natural evolution is characterized by irreversibility and dissipation, the question of how these aspects can be incorporated into a quantum mechanical description arises. Two effective approaches that also both possess a dynamical invariant (like the one mentioned above) will be discussed. One uses an explicitly time-dependent (linear) Hamiltonian, whereas the other leads to a nonlinear Schrödinger equation with complex logarithmic nonlinearity. Both approaches can be transformed into each other via a non-unitary transformation that involves Schrödinger's original definition of a (complex) action via the wave function.

A SLIC–VOF Method Based on Unstructured Grid

In this paper, an interface reconstruction technology similar with SLIC method on structured grid is developed for the unstructured grid. For a cell containing the interface, a straight line segment parallel with one edge of the triangle cell is set to the interface according to the VOF function values in the three neighbor cells and volume flux of the particular fluid is determined by the geometry relation between the interface line and volume flux on each edge. Finally, four classical cases are used to test this unstructured SLIC–VOF interface reconstruction method including advection test of a right-angled triangle and a hollowed circle, Zalesak slotted disk rotation test as a test of scalar advection, and single-vortex shearing flow test, which are effective to test interface capturing method. It was found that the interface transported back by the reversed velocity field is comparative with SLIC–VOF on structure grid. Through the comparison between two triangle grids generated by Delaunay Triangulation Method and Bubble Packing Method, it is found that the grid quality has great effect on the proposed unstructured SLIC–VOF method. More regular triangle cells will lead to better interface capturing result.

Quark mixing sum rules and the right unitarity triangle

In analogy with the recently proposed lepton mixing sum rules, we derive quark mixing sum rules for the case of hierarchical quark mass matrices with 1-3 texture zeros, in which the separate up and down-type 1-3 mixing angles are approximately zero, and ${V}_{ub}$ is generated from ${V}_{cb}$ as a result of 1-2 up-type quark mixing. Using the sum rules, we discuss the phenomenological viability of such textures, including up to four texture zeros, and show how the right-angled unitarity triangle, i.e., $\ensuremath{\alpha}\ensuremath{\approx}90\ifmmode^\circ\else\textdegree\fi{}$, can be accounted for by a remarkably simple scheme involving real mass matrices apart from a single element being purely imaginary. In the framework of grand unified theories, we show how the quark and lepton mixing sum rules may combine to yield an accurate prediction for the reactor angle.

Applications of the theorem of Pythagoras in R3

Three distinct points and with are taken, respectively on the x, y and the z-axes of a rectangular coordinate system in Using the converse of the theorem of Pythagoras, it is shown that the triangle can never be a right-angled triangle. The result seems to be intuitive, but nevertheless requires a proof. As an application, some intuitive results about a tetrahedron are confirmed.

Temperature-insensitive accelerometer based on a strain-chirped FBG

A novel accelerometer based on a strain-chirped optical fiber Bragg grating (FBG) is proposed. The FBG is glued in a slanted direction onto the lateral side of a right-angled triangle cantilever beam with a mass bonded on its free end. Vertical acceleration applied to the cantilever beam leads to a uniform bending along the beam length. As a result, the FBG is chirped and its reflection bandwidth changes linearly with the applied acceleration. A high sensitivity of 0.679 nm/g has been achieved in the experiment. This sensor is temperature insensitive, owning to the temperature-independence nature of reflection bandwidth of the FBG.

Delta lithography method to increase CD uniformity and throughput of SLM-based maskless lithography

To increase the CD uniformity and throughput of SLM-based maskless lithography, the delta lithography method (DLM) has been developed. The basic idea of the DLM is to create a delta configuration by using a rectangular point array. The DLM enables irradiation in a honeycomb structure with equilateral triangles or a square structure with right-angled isosceles triangles. To verify the DLM, lithography simulations and a lithography experiment to fabricate a photomask were performed. The pattern appeared to be well defined with extremely low LER. The potential of the DLM to increase the CD uniformity and throughput is confirmed.

The Calibration Algorithm of a 3D Color Measurement System based on the Line Feature

This paper describes a novel 3 dimensional color measurement system. After 3 kinds of geometrical features are analyzed, the line features were selected. A calibration board with right-angled triangle outline was designed to improve the calibration precision. For this system, two algorithms are presented. One is the calibration algorithm between 2 dimensional laser range finder (2D LRF), while the other is for 2D LRF and the color camera. The result parameters were obtained through solving the constrain equations by the correspond data between the 2D LRF and other two sensors. The 3D color reconstruction experiments of real data prove the effectiveness and the efficient of the system and the algorithms.

Effects of Geometry of Ring Spinning Triangle on Yarn Torque: Part II: Distribution of Fiber Tension within a Yarn and Its Effects on Yarn Residual Torque

This paper presents a theoretical analysis of fiber tension distribution within a ring spun yarn without relaxation. Yarn residual torque is determined on the basis of the translation of fiber tension at the spinning triangle, as presented in the first part of this series of papers, into the fiber tension within the yarn. The results from numerical simulations indicate that, with fiber buckling, the yarn exhibits a lower average fiber tension and, thus, a much reduced yarn residual torque than that without fiber buckling. Comparison with experiments confirmed that fiber buckling exists in ring yarns while the assumption of no fiber buckling is not realistic. Generally, a low yarn twist results in low average fiber tension in the yarn and, thus, a reduced yarn residual torque. A symmetrical spinning triangle leads to a slightly higher yarn residual torque than a right-angle spinning triangle when the yarn counts and twists are identical and fiber buckling occurs.