The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems

We review the general problem of finding a global rotation that transforms a given set of points and/or coordinate frames (the "test" data) into the best possible alignment with a corresponding set (the "reference" data). For 3D point data, this "orthogonal Procrustes problem" is often phrased in terms of minimizing a root-mean-square deviation or RMSD corresponding to a Euclidean distance measure relating the two sets of matched coordinates. We focus on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; we focus on these exact solutions to expose the structure of the entire eigensystem for the traditional 3D spatial alignment problem. We then explore the structure of the less-studied orientation data context, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. We conclude with a brief discussion of the combined 3D translation-orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices, and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. Supplementary Material covers extensions of quaternion methods to the 4D problem.Comment: This replaces an early draft that lacked a number of important references to previous work. There are also additional graphics elements. The extensions to 4D data and additional details are worked out in the Supplementary Material appended to the main tex

Exploring Visualization Methods for Complex Variables

Applications of complex variables and related manifolds appear throughout mathematics and science. Here we review a family of basic methods for applying visualization concepts to the study of complex variables and the properties of specific complex manifolds. We begin with an outline of the methods we can employ to directly visualize poles and branch cuts as complex functions of one complex variable. $CP^2$ polynomial methods and their higher analogs can then be exploited to produce visualizations of Calabi-Yau spaces such as those modeling the hypothesized hidden dimensions of string theory. Finally, we show how the study of N-boson scattering in dual model/string theory leads to novel cross-ratio-space methods for the treatment of analysis in two or more complex variables

Enhanced Graphics for Extended Scale Range

Enhanced Graphics for Extended Scale Range is a computer program for rendering fly-through views of scene models that include visible objects differing in size by large orders of magnitude. An example would be a scene showing a person in a park at night with the moon, stars, and galaxies in the background sky. Prior graphical computer programs exhibit arithmetic and other anomalies when rendering scenes containing objects that differ enormously in scale and distance from the viewer. The present program dynamically repartitions distance scales of objects in a scene during rendering to eliminate almost all such anomalies in a way compatible with implementation in other software and in hardware accelerators. By assigning depth ranges correspond ing to rendering precision requirements, either automatically or under program control, this program spaces out object scales to match the precision requirements of the rendering arithmetic. This action includes an intelligent partition of the depth buffer ranges to avoid known anomalies from this source. The program is written in C++, using OpenGL, GLUT, and GLUI standard libraries, and nVidia GEForce Vertex Shader extensions. The program has been shown to work on several computers running UNIX and Windows operating systems

Estimation of Directional Surface Wave Spectra from a Towed Research Catamaran

During the High-Resolution Remote Sensing Main Experiment (1993), wave height was estimated from a moving catamaran using pitch-rate and roll-rate sensors, a three-axis accelerometer, and a capacitive wave wire. The wave spectrum in the frequency band ranging roughly from 0.08 to 0.3 Hz was verified by independent buoy measurements. To estimate the directional frequency spectrum from a wave-wire array, the Data-Adaptive Spectral Estimator is extended to include the Doppler shifting effects of a moving platform. The method is applied to data obtained from a fixed platform during the Risø Air–Sea Experiment (1994) and to data obtained from a moving platform during the Coastal Ocean Processes Experiment (1995). Both results show that the propagation direction of the peak wind waves compares well with the measured wind direction. When swells and local wind waves are not aligned, the method can resolve the difference of propagation directions. Using the fixed platform data a numerical test is conducted that shows that the method is able to distinguish two wave systems propagating at the same frequency but in two different directions

The highly resolved electronic spectrum of the square planar CuCl₄²⁻ ion

The low temperature magnetic circular dichroism(MCD) and electron paramagnetic resonance(EPR)spectra of Cu(II) dopedCs₂ZrCl₆ are reported. The Cu(II) ion is incorporated as the square planar copper tetrachloride ion, CuCl₄²⁻, which substitutes at the Zr(IV) site in the Cs₂ZrCl₆ lattice, with a complete absence of axial coordination. Both the EPR and MCD show highly resolved spectra from which it is possible to determine the superhyperfine coupling constants and excited state geometries respectively. The Franck–Condon intensity patterns suggest that there is a substantial relaxation of the host lattice about the impurity ion. For the lowest energy ²B1g(x²-y²)→²B2g(xy) transition, both the magnetic dipole allowed electronic origin as well as vibronic false origins are observed. The high resolution of the spectra allowed the accurate determination of the odd parity vibrations that are active in the spectra. The opposite sign of the MCD of the two components of the ²Eg(xz,yz)excited state allows this splitting to be determined for the first time. Accurate and unambiguous spectral parameters for the CuCl₄²⁻ ion are important as it has become a benchmark transition metal complex for theoretical electronic structure calculations

Geometry of Discrete Quantum Computing

Conventional quantum computing entails a geometry based on the description of an n-qubit state using 2^{n} infinite precision complex numbers denoting a vector in a Hilbert space. Such numbers are in general uncomputable using any real-world resources, and, if we have the idea of physical law as some kind of computational algorithm of the universe, we would be compelled to alter our descriptions of physics to be consistent with computable numbers. Our purpose here is to examine the geometric implications of using finite fields Fp and finite complexified fields Fp^2 (based on primes p congruent to 3 mod{4}) as the basis for computations in a theory of discrete quantum computing, which would therefore become a computable theory. Because the states of a discrete n-qubit system are in principle enumerable, we are able to determine the proportions of entangled and unentangled states. In particular, we extend the Hopf fibration that defines the irreducible state space of conventional continuous n-qubit theories (which is the complex projective space CP{2^{n}-1}) to an analogous discrete geometry in which the Hopf circle for any n is found to be a discrete set of p+1 points. The tally of unit-length n-qubit states is given, and reduced via the generalized Hopf fibration to DCP{2^{n}-1}, the discrete analog of the complex projective space, which has p^{2^{n}-1} (p-1)\prod_{k=1}^{n-1} (p^{2^{k}}+1) irreducible states. Using a measure of entanglement, the purity, we explore the entanglement features of discrete quantum states and find that the n-qubit states based on the complexified field Fp^2 have p^{n} (p-1)^{n} unentangled states (the product of the tally for a single qubit) with purity 1, and they have p^{n+1}(p-1)(p+1)^{n-1} maximally entangled states with purity zero.Comment: 24 page

A Successful Portable Computer Lab Training Program

Penn State Cooperative Extension and the Pennsylvania Farm Credit System joined forces to fund a portable computer laboratory. A simplified lab management procedure allowed Extension agents to offer 33 computer operation workshops for 300 participants at minimal participant cost. Participants indicated their future use of computers would focus on farm financial, crop, and livestock management. Although considerable competence was gained, more than 50% viewed themselves with poor to moderate computer skills at the end of the workshops. The lab has enabled agents to contact a preciously under-served population as 54% of the participants had not attended any Extension workshops in the previous year