Rotational symmetry field design on surfaces

Rotational symmetry fields appear in a wide range of applications in computer graphics, such as quadrangular and triangular remeshing, texture and geometry pattern synthesis, painterly rendering, pen-and-ink sketching, street network modeling, and computer architecture. In all of these applications, an orientation field is needed to guide the tiling of the surface (remeshing) and placement of graphics primitives (e.g., painting strokes, hatches, street segments, texture and geometry patterns). The continuity and smoothness of the input rotational symmetry field as well as its topology of the rotational symmetry field (singularities), can greatly influence the quality of the application results. In this course note, we will start with vector fields, a special kind of rotational symmetry fields that are important in their own right. We will cover the topology as well as smoothness of vector fields and provide algorithms for vector field design and analsysi. Next, we will extend the work to the general N-way rotational symmetry fields and provide algorithms for the representation, analysis, design, processing, and visualization of such fields.

The ground states encode the information of the topological phases of a 2-dimensional system, which makes them crucial in determining the associated topological quantum field theory (TQFT). Most numerical methods for detecting the TQFT relied on the use of minimum entanglement states (MESs), extracting the anyon mutual statistics and self statistics via overlaps and/or the entanglement spectra. The MESs are the eigenstates of the Wilson loop operators, and are labeled by the anyons corresponding to their eigenvalues. Here we revisit the definition of the Wilson loop operators and MESs. We derive the modular transformation of the ground states purely from the Wilson loop algebra, and as a result, the modular $S$ and $T$ matrices naturally show up in the overlap of MESs. Importantly, we show that due to the phase degree of freedom of the Wilson loop operators, the MES-anyon assignment is not unique. This ambiguity obstructs our attempt to detect the topological order, that is, there exist different TQFTs that cannot be distinguished solely by the overlap of MESs. In this paper, we provide the upper limit of the information one may obtain from the overlap of MESs without other additional structure. Finally, we show that if the phase is enriched by rotational symmetry, there may be additional TQFT information that can be extracted from overlap of MESs.

An optimization method is proposed for designing an LED freeform lens which produces a uniform circular pattern with high energy efficiency. This method is composed of three main aspects: design of the initial guess, parameterization of the freeform surface, and construction of the merit function. The initial guess is created by solving an ordinary differential equation numerically. An approach of selecting optimization points is introduced for parameterization of the freeform surface. The merit function is constructed by use of the irradiance uniformity and the efficiency of the lens. Design examples are given, and the results show that the irradiance distribution is well controlled with a maximum uniformity (the relative standard deviation of irradiance, RSD) of 0.0122 and a maximum efficiency of 93.88%. This optimization method can be generalized to design freeform lenses with different lighting patterns or without rotational symmetry.

Context. Zeeman-Doppler imaging (ZDI) is used to reconstruct the surface magnetic field of late-type stars from high-resolution spectropolarimetric observations. The results are usually described in terms of characteristics of the field topology, such as poloidality versus toroidality and axisymmetry versus non-axisymmetry, in addition to the field strength. Aims. In this study, we want to test how well these characteristics are preserved when applying the ZDI method to simulated data. We are particularly interested in how accurately the field topology is preserved and to what extent stellar parameters, such as projected rotation velocity and rotation axis inclination, influence the reconstruction. Methods. For these tests, we used published magnetic field vector data from direct numerical magnetohydrodynamic simulations taken near the surface of the simulation domain. These simulations have variable rotation rates and therefore represent different levels of activity of an otherwise Sun-like setup with a convective envelope of solar thickness. Our ZDI reconstruction is based on spherical harmonics expansion. By comparing the original values to those of the reconstructed images, we study the ability to reconstruct the surface magnetic field in terms of various characteristics of the field. Results. In general, the ZDI method works as expected. The main large-scale features are reasonably well recovered, but the strength of the recovered magnetic field is just a fraction of the original input. The quality of the reconstruction shows clear correlations with the data quality. Furthermore, there are some spurious dependencies between stellar parameters and the characteristics of the field. Conclusions. Our study uncovers some limits of ZDI. Firstly, the recovered field strength will generally be lower than the ‘real’ value, as smaller structures with opposite polarities will be blurred in the inversion. This is also seen in the relative distribution of magnetic energy in terms of the angular degree ℓ. Secondly, the axisymmetry is overestimated. The poloidality versus toroidality is better recovered. The reconstruction works better for a stronger field and faster rotation velocity. Still, the ZDI method works surprisingly well even for a weaker field and slow rotation provided the data have a high signal-to-noise ratio and good rotation phase coverage.

Many algorithms in texture synthesis, nonphotorealistic rendering (hatching), or remeshing require to define the orientation of some features (texture, hatches, or edges) at each point of a surface. In early works, tangent vector (or tensor) fields were used to define the orientation of these features. Extrapolating and smoothing such fields is usually performed by minimizing an energy composed of a smoothness term and of a data fitting term. More recently, dedicated structures ( N -RoSy and N -symmetry direction fields ) were introduced in order to unify the manipulation of these fields, and provide control over the field's topology (singularities). On the one hand, controlling the topology makes it possible to have few singularities, even in the presence of high frequencies (fine details) in the surface geometry. On the other hand, the user has to explicitly specify all singularities, which can be a tedious task. It would be better to let them emerge naturally from the direction extrapolation and smoothing. This article introduces an intermediate representation that still allows the intuitive design operations such as smoothing and directional constraints, but restates the objective function in a way that avoids the singularities yielded by smaller geometric details. The resulting design tool is intuitive, simple, and allows to create fields with simple topology, even in the presence of high geometric frequencies. The generated field can be used to steer global parameterization methods (e.g., QuadCover).

Bolt joints are common in machine tools and have a direct impact on the overall performance of the tools. Therefore, the understanding of bolt joint characteristics is essential for improving machine design and assembly. Firstly, According to the experimental data obtained from the experiment, the stiffness curve formula was fitted. Secondly, a finite element model of unit bolt joints such as bolt flange joints, bolt head joints, and thread joints was constructed, and lastly the stiffness parameters of joint surfaces were implemented in the model by the secondary development of ABAQUS. The finite element model of the bolt joint established by this method can simulate the contact state very well.

Magnetized, cold, dense molecular cloud cores provide the birth environment for stars, discs, and planets. The multi-scale scenario of low-mass star formation occurs via the formation of two quasi-hydrostatic cores. Furthermore, the conservation of angular momentum can lead to the formation of a disc around the second core (i.e.~the forming protostar). During these early stages of star formation, magnetically driven outflows and jets can be launched from the first and second cores, respectively. Star, disc, and outflow formation involve complex physical processes, which require a robust, self-consistent numerical treatment. In this thesis, we use numerical simulations to probe the gravitational collapse scenario that involves the transition of an isolated molecular cloud core to a hydrostatic core with a surrounding disc. We use the \emph{PLUTO} code to perform radiation (magneto-)hydrodynamic (MHD) collapse simulations, using one- and two-dimensional (2D) grids. We include the effects of self-gravity and a grey flux-limited diffusion approximation for the radiative transfer. Additionally, we use for the gas equation of state density- and temperature-dependent thermodynamic quantities to account for the dissociation, ionisation, and molecular vibrations and rotations. Our spherically symmetric simulations span seven orders of magnitude in spatial scale. We survey a wide range of initial low- to high-mass (0.5 -- 100~$M_{\odot}$) molecular cloud cores, yielding the largest parameter scan so far. Our results highlight the dependence of the first and second hydrostatic core properties on the initial cloud core properties. These simulations indicate that in the high-mass regime, the first hydrostatic cores do not have enough time to form due to large accretion rates. We further expand our studies to three different sets of 2D simulations using axial and midplane symmetry. First, we perform 2D simulations for non-rotating molecular cloud cores with masses of 1~$M_{\odot}$, 5~$M_{\odot}$, 10~$M_{\odot}$, and 20~$M_{\odot}$. For each of these cases, we use an unprecedented resolution to model the evolution of the second core for $\geq$~100 years after its formation. For the first time, we demonstrate that convection is generated in the outer layers of the second core. This supports the intriguing possibility that dynamo-driven magnetic fields may be generated during the earliest phases of star formation. Following which, for the 1~$M_{\odot}$ case, we analyse the effects of solid-body rotation on the properties of the hydrostatic cores and disc formation. In this model, the first hydrostatic core evolves into a more oblate, pseudo-disc like structure and a sub-au disc starts forming after the formation of the second core. Finally, we explore the effects of ideal and non-ideal (including Ohmic resistivity) MHD during the collapse of rotating molecular cloud cores. We investigate the dependence of molecular outflows and disc formation on the initial cloud core mass, rotation, resistivity, and magnetic field strength. We find the presence of magnetically driven outflows launched from both first and second cores in the resistive models. We also reveal ongoing disc formation in some of our resistive simulations. In conclusion, we use detailed thermodynamical modelling to quantify the properties of the hydrostatic cores, outflows, and discs for collapse scenarios with a wide range of initial cloud core properties. The models presented herein will serve as the foundation for follow-up studies that link these theoretical insights with observational signatures.

Designing rotational symmetries on surfaces is a necessary task for a wide variety of graphics applications, such as surface parameterization and remeshing, painterly rendering and pen-and-ink sketching, and texture synthesis. In these applications, the topology of a rotational symmetry field such as singularities and separatrices can have a direct impact on the quality of the results. In this paper, we present a design system that provides control over the topology of rotational symmetry fields on surfaces.

Rotational Symmetry and Design

In the present paper, we classify curves and surfaces in \({\mathbb {S}}^2\times \mathbb {R}\), which make constant angle with a rotational Killing vector field. We obtain the explicit parametrizations of such curves and surfaces, and we find examples in some particular cases. Finally, we give the complete classification of minimal constant angle surfaces in \({\mathbb {S}}^2\times \mathbb {R}\).

We draw connections between the field of contact topology (the study of totally non-integrable plane distributions) and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami fields (non-zero fields parallel to their non-zero curl). This immediately yields existence proofs for smooth, steady, fixed-point free solutions to the Euler equations on all 3-manifolds and all subdomains of 3 with torus boundaries. This correspondence yields a hydrodynamical reformulation of the Weinstein conjecture from symplectic topology, whose recent solution by Hofer (in several cases) implies the existence of closed orbits for all rotational Beltrami flows on S 3. This is the key step for a positive solution to a `hydrodynamical' Seifert conjecture: all steady flows of a perfect incompressible fluid on S 3 possess closed flowlines. In the case of spatially periodic Euler flows on 3, we give general conditions for closed flowlines derived from the algebraic topology of the vector field.

It can be shown that the stress produced by a spatially uniform dislocation density field in a body comprising a linear elastic material under no loads vanishes. We prove that the same result does not hold in general in the geometrically nonlinear case. This problem of mechanics establishes the purely geometrical result that the $\mathop{\mathrm{curl}}\nolimits $ of a sufficiently smooth two-dimensional rotation field cannot be a non-vanishing constant on a domain. It is classically known in continuum mechanics, stated first by the brothers Cosserat (SIAM J. Appl. Math. 25(3):483–491, 1973), that if a second order tensor field on a simply connected domain is at most a curl-free field of rotations, then the field is necessarily constant on the domain. It is shown here that, at least in dimension 2, this classical result is in fact a special case of a more general situation where the curl of the given rotation field is only known to be at most a constant.

In spite of the large efforts spent so far and the large body of observational data available, no satisfactory explanation has been found for the origin of the envelopes around Be type stars. This is due, among other reasons, to the fact that the spectrum of the central star is contaminated by the radiation of the envelope. Therefore, it is mandatory to separate the combined spectrum into its two components: central star + envelope. Moreover, as these objects show large photometric and spectroscopic variability, the combination of data from different epochs leads to uncertain conclusions. With this ideas in mind, we have developed an observational program where simultaneous uvbybeta photometry and Balmer line spectroscopy of Be stars in open clusters are combined to yield general properties of Be stars, as a class, and physical properties of their circumstellar envelopes, by means of a new method to decouple the combined spectrum. The main results of this research can be summarized as follows: A uvbybeta calibration has been developed to obtain the intrinsic radiative parameters of the underlying B stars. Linear relationships between the equivalent widths of the Balmer emission lines and the photometric anomalies are presented. It is independently confirmed that, on the average, Be stars are 0.3 mag brighter than B stars of the same spectral type due to the circumstellar contribution to MV. In the (b-y)0-MV plane, Be stars deviate to redder positions with respect to the non-emission B stars due to additional reddening caused by hydrogen free-free recombination in the envelope. In the c0-MV plane we have found two different behaviors: Be stars earlier than B5 V show smaller (bluer) c0 values due to free-bound transitions in the envelope, which reduce the Balmer jump, while late-type Be stars show normal or larger c0 values (in spite of the emission levels at Hα ) indicating absorption at the Balmer discontinuity of circumstellar origin. Balmer-line spectroscopy supports the picture of an envelope with axial symmetry, probably of disk-like geometry, with a quasi-Keplerian rotation field, already deduced for Be field stars. I have found evidence however, that the bulk of the Balmer line emission arises in a relatively small fraction of the envelope. Average physical properties of the envelopes are inferred from a study of Balmer decrements and can be resumed in the following table: The spread in the Balmer decrements can be ascribed partially to different temperatures of the circumstellar material which may range between Te ~ (2.0-1.0)x 104 K. For the latest spectral types, the temperature of the envelope could be slightly lower. I find that this temperature is not clearly correlated with the temperature of the central star but with the size of the envelope. I conclude that, on the average, the smallest envelopes are the hotter, due to luminosity conservation. Finally some clues on the evolutionary status of Be stars are presented. I conclude that a Be star cannot be a very young object. My results would point towards the interpretation of Be objects as the post mass-transfer phase of close binary systems.

The main objective of this study is to develop a semi-analytical formulation for the radiation problem of a fully immersed spheroid in a liquid field of infinite depth. The term “spheroid” refers herein to the oblate geometry of arbitrary eccentricity and to the axisymmetric case, where the axis of symmetry is normal to the free surface. The proposed numerical approach is based on the method of image singularities, and it enables the accurate and fast calculation of the hydrodynamic coefficients for the translational degrees of freedom of the oblate spheroid. The excellent agreement of the results, with those of other investigators for the limiting case of the sphere and with those obtained using a respected boundary integral equation code, demonstrates the accuracy of the proposed methodology. Finally, extensive calculations are presented, illustrating the direct impact of the immersion depth and the slenderness of the spheroid on the hydrodynamic coefficients.

The primary scientific target of the CMB polarization experiments that are\ncurrently being built and proposed is the detection of primordial tensor\nperturbations. As a byproduct, these instruments will significantly improve\nconstraints on cosmic birefringence, or the rotation of the CMB polarization\nplane. If convincingly detected, cosmic birefringence would be a dramatic\nmanifestation of physics beyond the standard models of particle physics and\ncosmology. We forecast the bounds on the cosmic polarization rotation (CPR)\nfrom the upcoming ground-based Simons Observatory (SO) and the space-based\nLiteBIRD experiments, as well as a "fourth generation" ground-based CMB\nexperiment like CMB-S4 and the mid-cost space mission PICO. We examine the\ndetectability of both a stochastic anisotropic rotation field and an isotropic\nrotation by a constant angle. CPR induces new correlations of CMB observables,\nincluding spectra of parity-odd type in the case of isotropic CPR, and\nmode-coupling correlations in the anisotropic rotation case. We find that\nLiteBIRD and SO will reduce the 1$\\sigma$ bound on the isotropic CPR from the\ncurrent value of 30 arcmin to 1.5 and 0.6 arcmin, respectively, while\nCMB-S4-like and PICO will reduce it to $\\sim 0.1$ arcmin. The bounds on the\namplitude of a scale-invariant CPR spectrum will be reduced by 1, 2 and 3\norders of magnitude by LiteBIRD, SO and CMB-S4-like/PICO, respectively. We\ndiscuss implications of the forecasted CPR bounds for pseudoscalar fields,\nprimordial magnetic fields (PMF), and violations of Lorentz invariance. We find\nthat CMB-S4-like and PICO can reduce the 1$\\sigma$ bound on the amplitude of\nthe scale-invariant PMF from 1 nG to 0.1 nG, while also probing the magnetic\nfield of the Milky Way. They will also significantly improve bounds on the\naxion-photon coupling, placing stringent constraints on the string theory\naxions.\n