Cartesian Vector Research Articles

Functions of multivector variables.

As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. One key relationship that we discover is that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation. We also find a single formula that produces the square root, amplitude and inverse of a multivector over one, two and three dimensions. Finally, comparing the functions over different dimension we observe that provides a particularly versatile algebraic framework.

The Cartesian Vector Solutions for the N‐Dimensional Compressible Euler Equations

In this paper, based on matrix and curve integration theory, we theoretically show the existence of Cartesian vector solutions for the general N‐dimensional compressible Euler equations. Such solutions are global and can be explicitly expressed by an appropriate formulae. One merit of this approach is to transform analytically solving the Euler equations into algebraically constructing an appropriate matrix . Once the required matrix is chosen, the solution is directly obtained. Especially, we find an important solvable relation between the dimension of equations and pressure parameter, which avoid additional independent constraints on the dimension N in existing literatures. Special cases of our results also include some interesting conclusions: (1) If the velocity field is a linear transformation on , then the pressure p is a relevant quadratic form. (2) The compressible Euler equations admit the Cartesian solutions if is an antisymmetric matrix. (3) The pressure p possesses radial symmetric form if is an antisymmetrically orthogonal matrix.

DESIGNING OF CORDIC PROCESSOR IN VERILOG USING XILINX ISE SIMULATOR

In this paper, designing of CORDIC Processor in Verilog to determine the sine and cosine of a given argument, and extending this code to determine the Cartesian co-ordinates of a complex number represented in Euler’s form. The inputs given are the Cartesian vector and the input angle in 17- bit signed number representation. The outputs obtained are sine and cosine of the input angle. To determine the Cartesian co-ordinates of the complex number, the magnitude of the complex number is given as input along with the input angle. The language used for the designing of CORDIC Processor is Verilog. The software used for the simulation is Xilinx ISE Simulator.

Cartesian workspace optimization of Tricept parallel manipulator with machining application

SUMMARYIn this research work, the maximum Cartesian workspace of a Tricept parallel robot with two rotational and one translational degrees of freedom was investigated. Generally, the Cartesian workspace identifies the maximum size of a work-piece, specifying its cubicx,yandzdimensions, on which the milling machine could perform operations. However, the workspace of a robot can be considered in its task space, such as ψ × θ ×zfor the Tricept Parallel Kinematic Mechanism (PKM). A novel homogeneous Jacobian matrix which transforms joint space velocity vector into end-effector Cartesian velocity vector has been generated named as a Cartesian Jacobian matrix. Using the indices derived from the homogeneous Cartesian Jacobian matrix, i.e. the maximum singular values and local conditioning indices, the manipulator is designed to reach the Cartesian workspace with rapid positioning rates as well as with singularity avoidance.

Analytical Solutions to Two-Impulse Nondrifting Transfer Problems for Rendezvous in Elliptical Orbits

This paper develops the analytical solution to the nondrifting transfer problem for rendezvous in elliptical orbits. The transfer algorithm generates a nondrifting, passively safe trajectory between two spacecraft states by finding the appropriate maneuvers and transfer time. The nondrifting solution is derived first for Cartesian state vectors, which is useful for transferring to a terminal approach point that is defined in terms of such a Cartesian state vector. Next, the transfer problem is solved in terms of differential orbital elements. This second solution lends itself well to perform close-range rendezvous in elliptical orbits.

Vector and scalar spherical harmonic spectral equations for rapidly rotating anisotropic alpha-effect dynamos

Spectral equations are derived for a mean field induction equation, with an α -effect, considered appropriate for rapid rotation, given by , where are Cartesian unit vectors, a 1(r, θ, φ), a 3(r, θ, φ) are scalar functions of position, (r, θ, φ) are spherical polar coordinates and R is the magnetic Reynolds number. The effect of rotation on convection for different boundaries and parameters is discussed. The effect of the flow structure on α for different geostrophic and near geostrophic models is analysed. The vector spherical harmonics where , , , the 2 × 3 matrix is a Wigner 3J coefficient and are scalar spherical harmonics, are used to derive the vector forms of the induction equation for this α -effect. The solenoidal condition is imposed by relating the formalism to the toroidal–poloidal harmonic formalism, and . The and components of the induction equation are thus derived in terms of , the components of F ; . These combined /, / vector spectral equations are then transformed into interaction type , , , and , , , equations for the isotropic and anisotropic components of α . As an application of the general spectral equations derived herein, the interaction equations can be specialised by restricting a 1 and a 3 to be proportional to r cos θ or cos θ, or restricting and α to be axisymmetric. These equations are then compared to those of previous works. The differences between the equations derived herein and those of past works provide corrections and account for, at least in part, the differences in numerical solutions of the past works.

On the physical interpretation of ab initio normal-mode coordinates for the three C–H stretching vibrations of methanol along the internal-rotation path

Graphical representations are presented for variation along the steepest-descent internal-rotation path in methanol of the normal mode frequencies and their associated eigenvectors in several sets of internal coordinates and in Cartesian atomic displacement vectors di(γ), as determined for the three CH stretching vibrations (ν3, ν2, and ν9) by projected-frequency calculations from the Gaussian suite of programs. The methyl-group CH stretching modes are interesting because the symmetry environment of each C–H bond changes significantly during the internal rotation, i.e., each of the methyl bonds takes turns passing (twice for a complete torsional revolution) through the plane of symmetry of the COH frame of the molecule. No accumulation of geometric phase is observed in any of these plots, and all quantities return to their original values after the internal rotation angle γ increases by 2π. A simple two-vibrational-state, three-parameter model, closely based on earlier models from the literature, can be used to understand nearly quantitatively much of the γ-variation observed in the Gaussian plots, including a number of cusp-like features. In particular, when the three parameters in the model are determined from a fit to the Gaussian projected frequencies for ν2 and ν9 at the top and bottom of the internal rotation path, it is found that the Renner–Teller-like torsion–vibration interaction term is slightly larger in magnitude than the Jahn–Teller-like term, which is consistent with no accumulation of geometric phase in the various plots. Finally, a highly simplified computation is presented to illustrate the changes that will be necessary in order to move from the usual diabatic torsion–vibration treatments in the literature to adiabatic treatments, in which the normal-mode Cartesian displacement vectors given at each point along the internal rotation path by the Gaussian projected frequency calculation are used directly in the torsion–vibration energy level calculation.

Internal‐to‐Cartesian back transformation of molecular geometry steps using high‐order geometric derivatives

In geometry optimizations and molecular dynamics calculations, it is often necessary to transform a geometry step that has been determined in internal coordinates to Cartesian coordinates. A new method for performing such transformations, the high-order path-expansion (HOPE) method, is here presented. The new method treats the nonlinear relation between internal and Cartesian coordinates by means of automatic differentiation. The method is reliable, applicable to any system of internal coordinates, and computationally more efficient than the traditional method of iterative back transformations. As a bonus, the HOPE method determines not just the Cartesian step vector but also a continuous step path expressed in the form of a polynomial, which is useful for determining reaction coordinates, for integrating trajectories, and for visualization.

Implantation of a customized toric intraocular lens for correction of post-keratoplasty astigmatism

To report visual and refractive outcomes, and endothelial cell loss following primary and secondary 'piggyback' toric intraocular lens (IOL) implantation in patients with high post-penetrating keratoplasty (PK) astigmatism. Prospective case series. Nine eyes of nine patients with post-PK astigmatism were consecutively recruited for implantation of a customized toric IOL. Six underwent simultaneous phacoemulsification (PE) and three pseudophakic eyes had a secondary 'piggyback' toric IOL implanted in the ciliary sulcus. Mean follow-up time was 17.2±7.7 months. Pre- and post-operative uncorrected (UDVA) and best-corrected (BDVA) distance visual acuities and refractive errors were collected for comparison. Cartesian astigmatic vectors were calculated to identify a change in the magnitude of astigmatism pre- compared to postoperatively. Pre- and post-operative endothelial cell counts were also collected for analysis. UDVA (logMAR) improved from 1.13±0.51 preoperatively to 0.48±0.24 postoperatively (P-value=0.003). There was no significant change in BDVA (P-value=0.905) from 0.31±0.27 to 0.26±0.19. Corneal astigmatism preoperatively was 6.57±4.40 diopters (D). Post-operative refractive cylinder was 0.83±1.09 D compared to 3.89±4.01 D preoperatively (P=0.039). Analysis of astigmatic Cartesian x and y coordinates found a significant reduction postoperatively compared to preoperatively (P=0.005 and P=0.002), respectively. Mean endothelial cell loss was 9.9%. Implantation of a customized primary or secondary 'piggyback' toric IOL serves as an effective modality in treating patients with high post-PK astigmatism.

Building an Orthonormal Basis from a 3D Unit Vector Without Normalization

Abstract I present two tools that save the computation of a dot product and a reciprocal square root in operations that are used frequently in the core of many rendering programs. The first tool is a formula for rotating a direction sampled around the z-axis to a direction sampled around an arbitrary unit vector. This is useful in Monte Carlo rendering techniques, such as path tracing, where directions are usually sampled in spherical coordinates and then transformed to a Cartesian unit vector in a local coordinate system where the zenith direction is the z-axis. The second tool is a more general result extracted from the first formula, namely a faster way of building an orthonormal basis from a 3D unit vector. These tools require fewer arithmetic operations than other methods I am aware of, and a performance test of the more general tool confirms that it is faster.

Assessment of Wake-Tail Interference Effects on the Dynamics of Flexible Aircraft

This work investigates the effect of aerodynamic interference in the coupled nonlinear aeroelasticity and flight mechanics of flexible lightweight aircraft at low speeds. For that purpose, a geometrically exact composite-beam formulation is used to model the vehicle flexible-body dynamics by means of an intuitive and easily linearizable representation based on the displacement and Cartesian rotation vectors. The aerodynamics are modeled using the unsteady vortex-lattice method, which captures the instantaneous shape of the lifting surfaces and the free inviscid wake, including large deformations and interference effects. This results in a framework for simulation of high aspect ratio planes that provides a medium-fidelity representation of flexible-aircraft dynamics with a modest computational cost. Previous independent studies on the structural-dynamics and aerodynamics modules are complemented here with the integrated simulation methodology, including vehicle trim, and linear and nonlinear time-domain solutions. A numerical investigation is next presented on a simple wing-fuselage-tail configuration, assessing the interference effects between wing wake and horizontal tail, and the downwash due to the proximity of the wake is shown to play a significant role in the longitudinal dynamics of the vehicle. Finally, a brief discussion of direct wake-tail encounters is included to show the limitations of the approach.

3D math primer for graphics and game development

Cartesian Coordinate Systems 1D Mathematics 2D Cartesian Space 3D Cartesian Space Odds and ends Vectors Vector - mathematical definition and other boring stuff Vector - a geometric definition Specifying vectors using Cartesian coordinates Vectors vs. points Negating a vector Vector multiplication by a scalar Vector addition and subtraction Vector magnitude (length) Unit vectors The distance formula Vector dot product Vector cross product Linear algebra identities Multiple Coordinate Spaces Why multiple coordinate spaces? Some useful coordinate spaces Coordinate space transformations Nested coordinate spaces In defense of upright space Introduction to Matrices Matrix - a mathematical definition Matrix - a geometric interpretation The bigger picture of linear algebra Matrices and Linear Transformations Rotation Scale Orthographic projection Reection Shearing Combining transformations Classes of transformations More on Matrices Determinant of a matrix Inverse of a matrix Orthogonal matrices 4 x 4 homogeneous matrices 4 x 4 matrices and perspective projection Polar Coordinate Systems 2D Polar Space Why would anybody use Polar coordinates? 3D Polar Space Using polar coordinates to specify vectors Rotation in Three Dimensions What exactly is orientation? Matrix form Euler angles Axis-angle and exponential map representations Quaternions Comparison of methods Converting between representations Geometric Primitives Representation techniques Lines and rays Spheres and circles Bounding boxes Planes Triangles Polygons Mathematical Topics from 3D Graphics How graphics works Viewing in 3D Coordinate spaces Polygon meshes Texture mapping The standard local lighting model Light sources Skeletal animation Bump mapping The real-time graphics pipeline Some HLSL examples Further reading Mechanics 1: Linear Kinematics and Calculus Overview and other expectation-reducing remarks Basic quantities and units Average velocity Instantaneous velocity and the derivative Acceleration Motion under constant acceleration Acceleration and the integral Uniform circular motion Mechanics 2: Linear and Rotational Dynamics Newton's three laws Some simple force laws Momentum Impulsive forces and collisions Rotational dynamics Real-time rigid body simulators Suggested reading Curves in 3D Parametric polynomial curves Polynomial interpolation Hermite curves Bezier curves Subdivision Splines Hermite and Bezier splines Continuity Automatic tangent control Afterword What next? Appendix A: Geometric Tests Appendix B: Answers to the Exercises Bibliography Index Exercises appear at the end of each chapter.

802.11a Compliant Spatial Diversity Receiver IC in ${\hbox{0.25-}}\mu{\hbox {m}}$ BiCMOS

This paper presents an integrated 802.11a compliant receiver integrated circuit (IC), which is capable of processing four antenna signals using RF multiple-input multiple-output (RF-MIMO) schemes. Following four low-noise amplifiers, the weighting is performed using Cartesian vector modulators, whose output signals are combined and down-converted. The baseband filters, variable gain amplifiers (VGAs), a quadrature voltage-controlled oscillator, and digital interfaces are integrated. Experimental results demonstrate the performance of the individual components, as well as the RF-MIMO capabilities of the whole chip. The IC achieves a receiver noise figure of 3.6 dB, a phase noise of -113 dBc/Hz at 1-MHz offset, and an equivalent Cartesian weighting precision of 6 × 6 bit. An RF input signal of up to -35.3 dBm (1-dB compression) can be processed, while the maximum RF path gain amounts to 28.6 dB. The baseband VGA provides further 9-61-dB gain in 64 steps.

Hamiltonians and variational principles for Alfvén simple waves

The evolution equations for the magnetic field induction B with the wave phase for Alfvén simple waves are expressed as variational principles and in the Hamiltonian form. The evolution of B with the phase (which is a function of the space and time variables) depends on the generalized Frenet–Serret equations, in which the wave normal n (which is a function of the phase) is taken to be tangent to a curve X, in a 3D Cartesian geometry vector space. The physical variables (the gas density, fluid velocity, gas pressure and magnetic field induction) in the wave depend only on the phase. Three approaches are developed. One approach exploits the fact that the Frenet equations may be written as a 3D Hamiltonian system, which can be described using the Nambu bracket. It is shown that B as a function of the phase satisfies a modified version of the Frenet equations, and hence the magnetic field evolution equations can be expressed in the Hamiltonian form. A second approach develops an Euler–Poincaré variational formulation. A third approach uses the Frenet frame formulation, in which the hodograph of B moves on a sphere of constant radius and uses a stereographic projection transformation due to Darboux. The equations for the projected field components reduce to a complex Riccati equation. By using a Cole–Hopf transformation, the Riccati equation reduces to a linear second order differential equation for the new variable. A Hamiltonian formulation of the second order differential equation then allows the system to be written in the Hamiltonian form. Alignment dynamics equations for Alfvén simple waves give rise to a complex Riccati equation or, equivalently, to a quaternionic Riccati equation, which can be mapped onto the Riccati equation obtained by stereographic projection.

Quasi-Newton parallel geometry optimization methods

Algorithms for parallel unconstrained minimization of molecular systems are examined. The overall framework of minimization is the same except for the choice of directions for updating the quasi-Newton Hessian. Ideally these directions are chosen so the updated Hessian gives steps that are same as using the Newton method. Three approaches to determine the directions for updating are presented: the straightforward approach of simply cycling through the Cartesian unit vectors (finite difference), a concurrent set of minimizations, and the Lanczos method. We show the importance of using preconditioning and a multiple secant update in these approaches. For the Lanczos algorithm, an initial set of directions is required to start the method, and a number of possibilities are explored. To test the methods we used the standard 50-dimensional analytic Rosenbrock function. Results are also reported for the histidine dipeptide, the isoleucine tripeptide, and cyclic adenosine monophosphate. All of these systems show a significant speed-up with the number of processors up to about eight processors.

A polar coordinate particle swarm optimiser

The Particle Swarm Optimisation (PSO) algorithm consists of a population (or swarm) of particles that are “flown” through an n-dimensional space in search of a global best solution to an optimisation problem. PSO operates in Cartesian space, producing Cartesian solution vectors. By making use of an appropriate mapping function the algorithm can be modified to search in polar space. This mapping function is used to convert the position vectors (now defined in polar space) to Cartesian space such that the fitness value of each particle can be calculated accordingly. This paper introduces the polar PSO algorithm that is able to search in polar space. This new algorithm is compared to its Cartesian counterpart on a number of benchmark functions. Experimental results show that the polar PSO outperforms the Cartesian PSO in low dimensions when both algorithms are applied to the search for eigenvectors of different n × n square matrices. Performance of the polar PSO on general unconstrained functions is not as good as the Cartesian PSO, which emphasizes the main conclusion of this paper, namely that the PSO is not an efficient search algorithm for general unconstrained optimisation problems defined in polar space.

Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors

In large strain elasticity the existence of minimizers is guaranteed if the variational functional to be minimized is sequentially weakly lower semicontinuous (s.w.l.s.) and coercive. Therefore, polyconvex functions which are always s.w.l.s. are usually considered. For isotropic as well as for transversely isotropic and orthotropic materials constitutive functions that are polyconvex already exist. The main goal of this contribution is to provide a new method for the construction of polyconvex hyperelastic models for more general anisotropy classes. The fundamental idea is the introduction of positive definite second-order structural tensors G = HH T encoding the anisotropies of the underlying crystal. These tensors can be viewed as a push-forward of a cartesian metric of a fictitious reference configuration to the real reference configuration. Here the driving transformations H in the push-forward operation are mappings of the cartesian base vectors of the fictitious configuration onto crystallographic motivated base vectors. Restrictions of this approach are based on the polyconvexity condition as well as on the usage of second-order structural tensors and pointed out in detail.

Comments on “A Preliminary Study of Severe Wind-Producing MCSs in Environments of Limited Moisture”

Corresponding author address: Dr. Charles A. Doswell III, Cooperative Institute for Mesoscale Meteorological Studies, 120 David L. Boren Blvd., Ste. 2100, Norman, OK 73072-7304. Email: cdoswell@gcn.ou.edu

Hodograph computation and bound estimation for rational B-spline curves

Abstract It is necessary to compute the derivative and estimate the bound of rational B-spline curves in design system, which has not been studied to date. To improve the function of computer aided design (CAD) system, and to enhance the efficiency of different algorithms of rational B-spline curves, the representation of scaled hodograph and bound of derivative magnitude of uniform planar rational B-spline curves are derived by applying Dir function, which indicates the direction of Cartesian vector between homogeneous points, discrete B-spline theory and the formula of translating the product into a summation of B-spline functions. As an application of the result above, upper bound of parametric distance between any two points in a uniform planar rational B-spline curve is further presented. *Supported by the National Program on Key Basic Research Projects (Grant No.2004CB719400), and National Natural Science Foundation of China (Grant Nos.60673031, 60333010)

Making the Complex Simple

This installment of computing prescriptions illustrates how complex arithmetic can simplify algorithms in two-dimensional Cartesian vector space as well as how to make difficult numerical integrals tractable. In other words, computer languages for scientific applications should support complex arithmetic.