Progressive refinement rendering of implicit surfaces

The visualisation of implicit surfaces can be an inefficient task when such surfaces are complex and highly detailed. Visualising a surface by first converting it to a polygon mesh may lead to an excessive polygon count. Visualising a surface by direct ray casting is often a slow procedure. In this paper we present a progressive refinement renderer for implicit surfaces that are Lipschitz continuous. The renderer first displays a low resolution estimate of what the final image is going to be and, as the computation progresses, increases the quality of this estimate at an interactive frame rate. This renderer provides a quick previewing facility that significantly reduces the design cycle of a new and complex implicit surface. The renderer is also capable of completing an image faster than a conventional implicit surface rendering algorithm based on ray casting

Aircraft geometry verification with enhanced computer generated displays

A method for visual verification of aerodynamic geometries using computer generated, color shaded images is described. The mathematical models representing aircraft geometries are created for use in theoretical aerodynamic analyses and in computer aided manufacturing. The aerodynamic shapes are defined using parametric bi-cubic splined patches. This mathematical representation is then used as input to an algorithm that generates a color shaded image of the geometry. A discussion of the techniques used in the mathematical representation of the geometry and in the rendering of the color shaded display is presented. The results include examples of color shaded displays, which are contrasted with wire frame type displays. The examples also show the use of mapped surface pressures in terms of color shaded images of V/STOL fighter/attack aircraft and advanced turboprop aircraft

A Comprehensive Scan for Heterotic SU(5) GUT models

Compactifications of heterotic theories on smooth Calabi-Yau manifolds remains one of the most promising approaches to string phenomenology. In two previous papers, http://arXiv.org/abs/arXiv:1106.4804 and http://arXiv.org/abs/arXiv:1202.1757, large classes of such vacua were constructed, using sums of line bundles over complete intersection Calabi-Yau manifolds in products of projective spaces that admit smooth quotients by finite groups. A total of 10^12 different vector bundles were investigated which led to 202 SU(5) Grand Unified Theory (GUT) models. With the addition of Wilson lines, these in turn led, by a conservative counting, to 2122 heterotic standard models. In the present paper, we extend the scope of this programme and perform an exhaustive scan over the same class of models. A total of 10^40 vector bundles are analysed leading to 35,000 SU(5) GUT models. All of these compactifications have the right field content to induce low-energy models with the matter spectrum of the supersymmetric standard model, with no exotics of any kind. The detailed analysis of the resulting vast number of heterotic standard models is a substantial and ongoing task in computational algebraic geometry.Comment: 33 pages, Late

Approximate and exact nodes of fermionic wavefunctions: coordinate transformations and topologies

A study of fermion nodes for spin-polarized states of a few-electron ions and molecules with $s,p,d$ one-particle orbitals is presented. We find exact nodes for some cases of two electron atomic and molecular states and also the first exact node for the three-electron atomic system in $^4S(p^3)$ state using appropriate coordinate maps and wavefunction symmetries. We analyze the cases of nodes for larger number of electrons in the Hartree-Fock approximation and for some cases we find transformations for projecting the high-dimensional node manifolds into 3D space. The node topologies and other properties are studied using these projections. We also propose a general coordinate transformation as an extension of Feynman-Cohen backflow coordinates to both simplify the nodal description and as a new variational freedom for quantum Monte Carlo trial wavefunctions.Comment: 7 pages, 7 figure

Nonaxisymmetric mathematical model of the cardiac left ventricle anatomy

We describe a mathematical model of the shape and fibre direction field of the cardiac left ventricle. The ventricle is composed of surfaces which model myocardial sheets. On each surface, we construct a set of curves corresponding to myocardial fibres. Tangents to these curves form the myofibres direction field. The fibres are made as images of semicircle chords parallel to its diameter. To specify the left ventricle shape, we use a special coordinate system where the left ventricle boundaries are coordinate surfaces. We propose an analytic mapping from the semicircle to the special coordinate system. The model is correlated with Torrent-Guasp’s concept of the unique muscular band and with Pettigrew’s idea of nested surfaces. Subsequently, two models of concrete normal canine and human left ventricles are constructed based on experimental Diffusion Tensor Magnetic Resonance Imaging data. The input data for the models is only the left ventricle shape. In a local coordinate system connected with the left ventricle meridional section, we calculate two fibre inclination angles, i.e. true fibre angle and helix angle. We obtained the angles found from the Diffusion Tensor Magnetic Resonance Imaging data and compared them with the model angles. We give the angle plots and show that the model adequately reproduces the fibre architecture in the majority of the left ventricle wall. Based on the mathematical model proposed, one can construct a numerical mesh that makes it possible to solve electrophysiological and mechanical left ventricle activity problems in norm and pathology. In the special coordinate system mentioned, the numerical scheme is written in a rectangular area and the boundary conditions can simply be written. By changing the model parameters, one can set a general or regional ventricular wall thickening or the left ventricle shape change, which is typical for certain cardiac pathologies

Mapping the human cortical surface by combining quantitative T(1) with retinotopy

We combined quantitative relaxation rate (R1= 1/T1) mapping-to measure local myelination-with fMRI-based retinotopy. Gray-white and pial surfaces were reconstructed and used to sample R1 at different cortical depths. Like myelination, R1 decreased from deeper to superficial layers. R1 decreased passing from V1 and MT, to immediately surrounding areas, then to the angular gyrus. High R1 was correlated across the cortex with convex local curvature so the data was first "de-curved". By overlaying R1 and retinotopic maps, we found that many visual area borders were associated with significant R1 increases including V1, V3A, MT, V6, V6A, V8/VO1, FST, and VIP. Surprisingly, retinotopic MT occupied only the posterior portion of an oval-shaped lateral occipital R1 maximum. R1 maps were reproducible within individuals and comparable between subjects without intensity normalization, enabling multi-center studies of development, aging, and disease progression, and structure/function mapping in other modalities

Polarization Aberrations

The analysis of the polarization characteristics displayed by optical systems can be divided into two categories: geometrical and physical. Geometrical analysis calculates the change in polarization of a wavefront between pupils in an optical instrument. Physical analysis propagates the polarized fields wherever the geometrical analysis is not valid, i.e., near the edges of stops, near images, in anisotropic media, etc. Polarization aberration theory provides a starting point for geometrical design and facilitates subsequent optimization. The polarization aberrations described arise from differences in the transmitted (or reflected) amplitudes and phases at interfaces. The polarization aberration matrix (PAM) is calculated for isotropic rotationally symmetric systems through fourth order and includes the interface phase, amplitude, linear diattenuation, and linear retardance aberrations. The exponential form of Jones matrices used are discussed. The PAM in Jones matrix is introduced. The exact calculation of polarization aberrations through polarization ray tracing is described. The report is divided into three sections: I. Rotationally Symmetric Optical Systems; II. Tilted and Decentered Optical Systems; and Polarization Analysis of LIDARs

Interactive visualization tools for topological exploration

Thesis (Ph.D.) - Indiana University, Computer Science, 1992This thesis concerns using computer graphics methods to visualize mathematical objects. Abstract mathematical concepts are extremely difficult to visualize, particularly when higher dimensions are involved; I therefore concentrate on subject areas such as the topology and geometry of four dimensions which provide a very challenging domain for visualization techniques. In the first stage of this research, I applied existing three-dimensional computer graphics techniques to visualize projected four-dimensional mathematical objects in an interactive manner. I carried out experiments with direct object manipulation and constraint-based interaction and implemented tools for visualizing mathematical transformations. As an application, I applied these techniques to visualizing the conjecture known as Fermat's Last Theorem. Four-dimensional objects would best be perceived through four-dimensional eyes. Even though we do not have four-dimensional eyes, we can use computer graphics techniques to simulate the effect of a virtual four-dimensional camera viewing a scene where four-dimensional objects are being illuminated by four-dimensional light sources. I extended standard three-dimensional lighting and shading methods to work in the fourth dimension. This involved replacing the standard "z-buffer" algorithm by a "w-buffer" algorithm for handling occlusion, and replacing the standard "scan-line" conversion method by a new "scan-plane" conversion method. Furthermore, I implemented a new "thickening" technique that made it possible to illuminate surfaces correctly in four dimensions. Our new techniques generate smoothly shaded, highlighted view-volume images of mathematical objects as they would appear from a four-dimensional viewpoint. These images reveal fascinating structures of mathematical objects that could not be seen with standard 3D computer graphics techniques. As applications, we generated still images and animation sequences for mathematical objects such as the Steiner surface, the four-dimensional torus, and a knotted 2-sphere. The images of surfaces embedded in 4D that have been generated using our methods are unique in the history of mathematical visualization. Finally, I adapted these techniques to visualize volumetric data (3D scalar fields) generated by other scientific applications. Compared to other volume visualization techniques, this method provides a new approach that researchers can use to look at and manipulate certain classes of volume data