Rotation Formula Research Articles

Understanding quaternions

Quaternion multiplication can be applied to rotate vectors in 3-dimensions. Therefore in Computer Graphics, quaternions are sometimes used in place of matrices to represent rotations in 3-dimensions. Yet while the formal algebra of quaternions is well-known in the Graphics community, the derivations of the formulas for this algebra and the geometric principles underlying this algebra are not well understood. The goals of this paper are: i. To provide a fresh, geometric interpretation of quaternions, appropriate for contemporary Computer Graphics; ii. To derive the formula for quaternion multiplication from first principles; iii. To present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in 3-dimensions based on insights from the algebra and geometry of multiplication in the complex plane; iv. To develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection; v. To show how to apply sandwiching to compute perspective projections. In Part I of this paper, we investigate the algebra of quaternion multiplication and focus in particular on topics i and ii. In Part II we apply our insights from Part I to analyze the geometry of quaternion multiplication with special emphasis on topics iii, iv and v.

Expressing intrinsic volumes as rotational integrals

A new rotational formula of Crofton type is derived for intrinsic volumes of a compact subset X ⊂ R d of positive reach. The formula provides a functional defined on the section of X with a j-dimensional linear subspace with rotational average equal to the intrinsic volumes of X. Simplified forms of the functional are derived in special cases.

A new rotational integral formula for intrinsic volumes in space forms

A new rotational version of Crofton's formula is derived for the intrinsic volumes of a domain Y in a space form. More precisely, a functional is defined on the intersection between Y and a totally geodesic submanifold (plane) through a fixed point, such that the rotational average of this functional is equal to the intrinsic volumes of Y. Particular cases of interest in stereology are considered for the Euclidean case.

CosmologicalCPTviolating effect on CMB polarization

A dark energy scalar (or a function of the Ricci scalar) coupled with the derivative to the matter fields will violate the $CPT$ symmetry during the expansion of the Universe. This type of cosmological $CPT$ violation helps to generate the baryon number asymmetry and gives rise to the rotation of the photon polarization which can be measured in the astrophysical and cosmological observations, especially the experiments of the cosmic microwave background radiation. In this paper, we derive the rotation angle in a fully general relativistic way and present the rotation formulas used for the cosmic microwave background data analysis. Our formulas include the corrections from the spatial fluctuations of the scalar field. We also estimate the magnitude of these corrections in a class of dynamical dark energy models for quintessential baryo/leptogenesis.

A rotational integral formula for intrinsic volumes

A rotational version of the famous Crofton formula is derived. The motivation for deriving the formula comes from local stereology, a new branch of stereology based on sections through fixed reference points. The formula shows how rotational averages of intrinsic volumes measured on sections passing through fixed points are related to the geometry of the sectioned object. In particular it is shown how certain weighting factors, appearing in the rotational integral formula, can be expressed in terms of hypergeometric functions. Close connections to geometric tomography will be pointed out. Applications to stereological particle analysis are discussed.

Study of Mutual Coupling Between Circular Stacked-Patch Antennas on a Sphere

A rigorous mathematical analysis is given of spherical stacked-patch arrays with emphasis on the physical interpretation of mutual coupling mechanisms present in doubly- curved convex structures. The analysis method is based on electromagnetic field representation in terms of spherical harmonics where each harmonic has the same angular variation as the spectral source component. To obtain the spectral representation the vector-Legendre transformation is applied to currents and fields. A novel approach to the mutual coupling calculation within the method of moments analysis of spherical arrays is applied. By expressing the patch current in terms of two suitable potential-like auxiliary functions, it is possible to avoid the use of Euler's formulas for coordinate system rotation and the related lengthy integrations. Instead, the rotation of antenna elements and corresponding current distributions can be done in closed form with the help of Vilenkin's addition theorem for associated Legendre functions. It is shown that the new approach results in significant acceleration and improved accuracy of the analysis of spherical patch antenna arrays. The algorithm is successfully tested against a commercially available electromagnetic software and measurements performed on the developed laboratory model, confirming its accuracy for both input impedance and mutual coupling calculation and with only a small difference between the predicted and measured resonant frequencies, due to limitations in the experimental model. The influence of the structure parameters on mutual coupling level is extensively investigated, including all coupling mechanisms and leakage of energy due to curvature of the structure. It is shown that stacked-patch antennas can have reduced coupling level comparing to single patch antennas with possible deep nulls above the antenna resonant frequency.

Euler-Rodrigues and Cayley Formulae for Rotation of Elasticity Tensors

It is well known that rotation in three dimensions can be expressed as a quadratic in a skew symmetric matrix via the Euler-Rodrigues formula. A generalized Euler-Rodrigues polynomial of degree 2n in a skew symmetric generating matrix is derived for the rotation matrix of tensors of order n. The Euler-Rodrigues formula for rigid body rotation is recovered by n = 1. A Cayley form of the nth-order rotation tensor is also derived. The representations simplify if there exists some underlying symmetry, as is the case for elasticity tensors such as strain and the fourth-order tensor of elastic moduli. A new formula is presented for the transformation of elastic moduli under rotation: as a 21-vector with a rotation matrix given by a polynomial of degree 8. Explicit spectral representations are constructed from three vectors: the axis of rotation and two orthogonal bivectors. The tensor rotation formulae are related to Cartan decomposition of elastic moduli and projection onto hexagonal symmetry.

DGPS형 정밀위치시스템을 이용한 이동 로봇 위치보정

Nowadays, CPS is used widely, especially in cases which need more precise position information, such as car navigation systems and even in the mobile robot for position measuring in the outdoor environment. RTK (Real-Time Kinematics) and DGPS (Differential Global Positioning System) have more precise accuracy than the general-purposed GPS. However can't easily use them because of high prices and large size of equipments. In order fur the mobile robot to obtain precise position information it is important that CPS receiver has portability and low price. In this study, we introduce a new GPS data acquisition system that offers the precise position data using the DGPS mechanism and satisfying low cost and portability. In addition to this, we propose an improved data compensation algorithm that offers more accurate position information to the outdoor mobile robot by compensating the error rate of CPS data measured from the three points with geometrical rotation and distance formula. Proposed method is verified by comparing with the precise real position data obtained by RTK. Proposed method has more than 70% performance enhancement.

Simple Model Calculations of Spin and Quantized Alignment for the <i≯A</i≯ ∼ 60–90 Superdeformed Mass Region

We have used a simple model based on the rotational energy formula E(I, K) to study the structure of the superdeformed (SD) mass region 60–90. The higher order inertial parameters A and B of such model were determined by using the Marquardt method of nonlinear least-squares routines to fit the proposed transition energies with their observed values. A good agreement between the calculated and corresponding experimental transition energies of the SD bands is obtained which supports our proposed model. In addition, the frequency dependence of the dynamic, θ(2), and static, θ(1), moments of inertia is used to determine the lowest spin (If) and the K-value of the considered SD bands; namely, 58Ni(b1), 58Cu, 59Cu(b1), 61Zn, 62Zn, 65Zn, 68Zn, 84Zr, 86Zr(b1), 88Mo(b1, b2, b3) and 89Tc. As a result of the identity exist among some of the considered SD bands, we have studied the incremental alignment and also the angular momentum alignment.

Equivalent lateral force procedure for the seismic story drift design of tall frames using a hand-calculated approach

The basic theory of the cantilever moment distribution method and the application of this method to conduct the equivalent lateral force procedure for the seismic design of tall frames are discussed in detail. Deflection–rotation formulas are introduced in this paper to determine the relative lateral story deflections and the joint rotations of laterally loaded rigid frames. An example is presented to conduct the seismic story drift design of a multistory, multi-bay, moment-resisting reinforced concrete frame using the cantilever moment distribution method and the deflection–rotation formulas developed in this paper. The hand-calculated approach presented in the example can be used as a rapid and accurate method to determine the story drift for any laterally loaded multistory, multi-bay rigid frames that are composed of identical single-bay symmetrical bents. The following are the advantages of using this rapid approach: (1) this approach can be carried out by using hand calculations only (without the use of computer computations); (2) the results obtained from this approach are as accurate as those derived from the traditional moment distribution and the slope deflection methods; and (3) the results obtained from this approach can be used to verify the accuracy of the results obtained from computer computations. Copyright © 2005 John Wiley & Sons, Ltd.

Unified treatment of overlap integrals with integer and noninteger n Slater-type orbitals using translational and rotational transformations for spherical harmonics

A unified treatment of two-center overlap integrals over Slater-type orbitals (STO) with integer and noninteger values of the principal quantum numbers is described. Using translation and rotation formulas for spherical harmonics, the overlap integrals with integer and noninteger n Slater-type orbitals are expressed through the basic overlap integrals and spherical harmonics. The basic overlap integrals are calculated using auxiliary functions Aσ and Bk. The analytical relations obtained in this work are especially useful for the calculation of overlap integrals for large integer and noninteger principal quantum numbers. The formulas established in this study for overlap integrals can be used for the construction of series expansions based on addition theorems. PACS Nos.: 31.15.–p, 31.20.Ej

On transformation between international celestial and terrestrial reference systems

Based on the current IAU hierarchy of the relativistic reference systems, practical formulae for the transformation between barycentric (BCRS) and geocentric (GCRS) celestial reference systems are derived. BCRS is used to refer to ICRS, International Celestial Reference System. This transformation is given in four versions, dependent on the time arguments used for BCRS (TCB or TDB) and for GCRS (TCG or TT). All quantities involved in these formulae have been tabulated with the use of the VSOP theories (IMCCE theories of motion of the major planets) � . In particular, these formulae may be applied to account for the indirect relativistic third-body perturbations in motion of Earth's satellites and Earth's rotation problem. We propose to use the SMART theory (IMCCE theory of Earth's rotation) in constructing the Newtonian three-dimensional spatial rotation transformation between GCRS and ITRS, the International Terrestrial Reference System. This transformation is compared with two other versions involving extra angular variables currently used by IERS, the International Earth Rotation Service. It is shown that the comparison of these three forms of the same transformation may be greatly simplified by using the proposed composite rotation formula �� .

The stochastic rotation problem: A comment

The stochastic Faustmann problem is solved by a simplified approach as compared with the rigorous analysis provided by Willassen (1998). The result is a closed-form rotation formula that simplifies to Faustmann's formula in the deterministic setting. The formula is discussed for two sets of assumptions: one where the rotation cost is exogenous and fixed, and the other where it is endogenous, and to be chosen optimally.

Machine vision system for the automatic identification of robot kinematic parameters

This paper presents an efficient, noncontact measurement technique for the automatic identification of the real kinematic parameters of an industrial robot. The technique is based on least-squares analysis and on the Hayati and Mirmirani kinematic modeling convention for closed kinematic chains. The measurement system consists of a single camera mounted on the robot's wrist. The camera measures position and orientation of a passive target in six degrees of freedom. Target position is evaluated by applying least-squares analysis on an overdetermined system of equations based on the quaternion representation of the finite rotation formula. To enhance the accuracy of the measurement, a variety of image processing functions including subpixel interpolation are applied.

Calculation of overlap integrals over Slater-type orbitals using translational and rotational transformations for spherical harmonics

Using translation and rotation formulas for spherical harmonics the finite sums through the basic overlap integrals and spherical harmonics are derived for the arbitrary overlap integrals over Slater-type orbitals (STOs). The recurrence relations for the evaluation of basic overlap integrals have been established recently [Guseinov II, Mamedov BA (1999) J Mol Struct (THEOCHEM) 465:1]. By the use of the derived expressions the overlap integrals can be calculated most efficiently and accurately, especially for large quantum numbers of STOs.

Systematics of the Quadrupole–Quadrupole Interaction and Convergence Properties

Our main concern in this work is to show how higher shell admixtures affect the spectrum of a Q·Q interaction. We first review how, in the valence space, the familiar SU(3) result for the energy spectrum can be obtained using a coordinate space Q·Q interaction rather than the Elliott one which is symmetric in r and p. We then reemphasize that the Elliott spectrum goes as L(L+1) where L is the orbital angular momentum. While in many cases this is compatible with the rotational formula which involves I(I+1), where I is the total angular momentum, there are cases, e.g., odd–odd nuclei, where there is disagreement. Finally, we consider higher shell admixtures and devise a scheme so as to obtain results, with the Q·Q interaction, which converge as the model spaces are increased. We consider not only ground state rotational bands but also those that involve intruder states.

Generalized correlations of quasiband energies in nuclei

A remarkable set of correlations of yrast and excited quasiband energies in collective nonrotor and rotor nuclei is discussed. These correlations are essentially independent of the nature of the states (e.g., yrast, quasi-$\ensuremath{\gamma}$-vibrational, superdeformed), of mass region, and of the type of nucleus (even-even, odd-$A,$ or odd-odd), and indicate a universal behavior which apparently describes all collective low energy structures. The results also show that, for most cases in odd-$A$ deformed nuclei, the band energies do not follow the rotational formula. The nature of spherical-deformed phase/shape transitions is also discussed and related to the seemingly (but not actually) different behavior of the neighboring even-even core nuclei. Finally, theoretical treatments that reproduce these correlations in even-even and odd-A nuclei are discussed, and new calculations with the geometric collective model are presented.

Nuclear rotations

An analysis of the gamma-ray spectra produced using the quantum mechanical rotational energy formula is presented for nuclei with large angular momentum. This analysis is suitable for quantum mechanics, modern physics, or nuclear physics courses. (AIP)

Parametrization in laminate design for optimal compliance

In this paper we consider the maximal stiffness design of laminated plates subjected to single and multiple loads. The stiffness of the laminates are parametrized in terms of the so-called lamination parameters. These express the relation between the material parameters for the laminate and the laminate lay-up and are given as moments of the trigonometric functions that appear in the well-known rotation formulae for stiffness matrices. These relations are here given in a form suitable for optimization studies. The conditions for the laminate itself to be orthotropic are also given directly in terms of the lamination parameters.The design problem is analyzed by performing a reformulation to an equivalent problem which is local in character and it is shown how this, together with an enlargement of the design space to allow for out of plane chattering designs, leads to a significant simplification of the problem. Thus, the number of variables is reduced to only four for the stiffness problem at hand, even in the general case with coupling stiffnesses and multiple loads. Moreover, in the special case of in-plane loads, the optimal solution for each design element of the plate can be realized as a single rotated ply of material or in special strain situations by two plies. A computational solution procedure for the simplified problem is described and several numerical examples illustrate basic features of the design approach.

Electronic transition moments for the Herzberg I bands of O2

Recently published extensive high-resolution measurements of absolute integrated photoabsorption cross sections for rotational lines of the (ν′ = 4–11, ν″ = 0) bands of the O2 Herzberg I system have been fitted using general rotational line-strength formulae for [Formula: see text] transitions. Good fits were obtained using only three independent electronic transition-moment parameters that accounted for transition strength borrowed from electric-dipole-allowed transitions through spin-orbit and orbit-rotation interactions involving both upper and lower states of the transition. Absolute values of transition-moment parameters have been obtained, corresponding to R-centroids from 1.29 to 1.32 Å (1 Å = 10−10 m). Band oscillator strengths derived from the calculated integrated line strengths are in good agreement with most experimental measurements. Principal electronic matrix elements have been estimated by assuming that strength is borrowed from only two electric-dipole-allowed transitions.