Great Circle Research Articles

High-dimensional tennis balls

We show that there exist constants \(\alpha,\epsilon>0\) such that for every positive integer \(n\) there is a continuous odd function \(f:S^m\to S^n\), with \(m\geq \alpha n\), such that the \(\epsilon\)-expansion of the image of \(f\) does not contain a great circle. This result is motivated by a conjecture of Vitali Milman about well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to \(\ell_2^n\).Mathematics Subject Classifications: 46B09, 60C05Keywords: Anti-Ramsey, antipodal subsphere

Energy saving method for ship weather routing optimization

In this paper, an improved three-dimensional dynamic programming algorithm for ship route optimization is proposed. As a consequence of the weather conditions and constraints, the ship course and speed are planned, and the optimal solution of ship route planning is generated. Compared with the actual navigation data of the VLCC oil tanker, the calculation accuracy of the method is verified by referring to the great circle route. Under the objective function of minimum FOC (fuel oil consumption) and ETA (estimated time of arrival), the economic benefits of the VLCC oil tanker per voyage are increased by 1.59% and 1.33% respectively, and ETA is shortened by 14.63h and 17.79h. In addition, the method also has the ability of other objective function optimization, weather updating and route updating, which can provide sufficient support for ship operators in route planning.

Least variation in tool orientation control for 5-axis CNC machining

We address the problem of minimizing the variation of the tool axis orientation of a 5-axis CNC machine equipped with a ball-end cutter that is designed to maintain a constant cutting speed, which is ensured by a fixed angle ψ between the tool axis a(t) and the workpiece surface normal n(t) at each point of contact as the cutter moves along a specified toolpath. The configuration can be mathematically translated into a pair of curves a(t) and n(t) on the unit sphere with constant parametric distance ψ, and the least variation of the tool axis orientation is achieved when the overall length of the curve a(t) is minimized under certain auxiliary boundary conditions. Using the calculus of variation we show that the solution a(t) consists of arcs of great circles and offset curves of n(t) on the unit sphere. A numerical approximation scheme based on Euler's method of finite differences is also provided.

Exploring a Planet, Revisited

How should we place n great circles on a sphere to minimize the furthest distance between a point on the sphere and its nearest great circle? Fejes Tóth conjectured that the optimum is attained by placing n circles evenly spaced all passing through the north and south poles. This conjecture was recently proved by Jiang and Polyanskii. We present a short simplification of Ortega-Moreno’s alternate proof of this conjecture.

Non-existence of Yamabe Minimizers on Singular Spheres

We prove that a minimizer of the Yamabe functional does not exist for a sphere \(\mathbb S^n\) of dimension \(n\ge 3\), endowed with a standard edge-cone spherical metric of cone angle greater than or equal to \(4\pi \), along a great circle of codimension two. When the cone angle along the singularity is smaller than \(2\pi \), the corresponding metric is known to be a Yamabe metric, and we show that all Yamabe metrics in its conformal class are obtained from it by constant multiples and conformal diffeomorphisms preserving the singular set.

Great circle theorem and the application of the spherical cosine rule to estimate distances on a globe

This research paper explores my interest in the Great Circle Theorem and Non-Euclidean Geometry to apply its theory in Earth Sciences and explore navigating flight distances across the globe. I use three different equations, based on the situation of the two points on the map and their relation with each other, to obtain approximate great circle distances between the two points. The values acquired from the final equation, which is derived from the Spherical Law of Cosines, is compared with the actual great circle distances between the two points on the globe. A percentage error is calculated and the average is found, after which a correlation is drawn between the percentage error and the magnitude of the value of the great circle distance.

Lightning evolution and VLF perturbations associated with category 5 TC Yasa in the South Pacific Region

In this paper, we present the D-region ionospheric response during the lifespan (10–19 December 2020) of a severe category 5 tropical cyclone (TC) Yasa in the South Pacific by using the very low frequency (VLF, 3–30 kHz) signals from NPM, NLK, and JJI transmitters recorded at Suva, Fiji. Results indicate enhanced lightning and convective activity in all three regions (eyewall, inner rainbands, and outer rainbands) during the TC Yasa that are also linked to the wave-sensitive zones of these transmitter–receiver great circle paths. Of the three regions, the outer rainbands showed the maximum lightning occurrence; hence convective activity. Prominent eyewall lightning was observed just before the TC started to weaken following its peak intensity. Analysis of VLF signals amplitude showed both negative and positive perturbations (amplitudes exceeding ± 3σ mark) lasting for more than 2 h with maximum change in the daytime and nighttime signal amplitudes of − 4.9 dB (NPM) and − 19.8 dB (NLK), respectively. The signal perturbations were wave-like, exhibiting periods of oscillations between ~ 2.2 and 5.5 h as revealed by the Morlet wavelet analysis. Additionally, the LWPC modeling of the signal perturbations indicated a 10 km increase in the daytime D-region reference height, H′, and a 12 km decrease in the nighttime D-region H′ during TC Yasa. The D-region density gradients (sharpness), β, showed small perturbations of 0.01–0.14 km−1 from its normal values. We suggest that the observed changes to the D-region parameters are due to the enhanced convection during TC Yasa which excites atmospheric gravity waves producing traveling ionospheric disturbances to the D-region.Graphical

Геометрические места точек, равноотстоящих от двух заданных геометрических фигур. Часть 5: геометрические места точек, равноудаленных от сферы и плоскости

In this paper have been investigated the loci equidistant from sphere and plane, and properties of obtained surfaces have been studied. Four options for possible mutual arrangement of plane and sphere have been considered: the plane passes through the center of the sphere; the plane intersects the sphere; the plane is tangent to the sphere; the plane passes outside the sphere.
 In all options of the mutual arrangement of the sphere and the plane, the loci are two surfaces - two coaxial confocal paraboloids of revolution. The general properties of the obtained paraboloids of revolution have been studied: foci and vertices positions, axes of rotation, the distance from the sphere center to the vertices of the paraboloids, the distance between the vertices of the paraboloids, and the position of the directorial planes have been defined.
 Have been derived equations for the surfaces of the loci equidistant from the sphere and the plane: various paraboloids of revolution.
 The loci in each of the four options for the possible mutual arrangement of the plane and the sphere are as follows. 1. The original plane passes through the sphere center – two coaxial confocal multidirectional paraboloids of revolution symmetric relative to the original plane. 2. The initial plane intersects the sphere – two coaxial confocal multidirectional but not symmetrical paraboloids of revolution, since the circle of intersection of the plane and the sphere does not coincide with the diameter of the sphere great circle. 3. The plane is tangent to the sphere – a paraboloid of revolution and a straight line (more precisely, a second order zero-quadric – a cylindrical surface with zero radius) passing through the tangency point of the plane and the sphere and the sphere center. 4. The plane passes outside the sphere – the equidistant loci will be two coaxial confocal unidirectional paraboloids of revolution.

Mapping the edge of subducted slabs in the lower mantle beneath southern Asia

SUMMARYWe investigate the presence of seismic structures in the Earth's mantle by searching for seismic signals that travel off the great circle path direction and are reflected or scattered off structures in the lower mantle. We focus on areas of current and past subduction beneath Eurasia by using events from Indonesia and Japan recorded at the broad-band stations in Germany, Morocco and Namibia. Applying seismic array techniques, we measure the direction and traveltime of the out-of-plane arrivals and backtrace them to their location of reflection/scattering. We backtrace the signals as P-to-P and S-to-P waves and extend the methodology to P-to-S waves. There seems to be a low number of reflection points in the regions beneath Eurasia in our study. Investigating possible causes, we find that the focal mechanism influences the presence of out-of-plane reflected waves. However, the potential coverage with our data set is large and should allow detection, but there may potentially be few seismically visible structures in the region. Most of our backtraced reflectors are located beneath southern Asia and are found shallower than 1500 km depth. They correlate well with the edges of prominent high velocity anomalies in tomographic inversions beneath southern Asia, which have been interpreted as remnants of fossil slabs of the subduction of the Tethys Oceans. We also observe few reflectors deeper than 1600 km that are located away from subducting regions and their positions coincide with the eastern edge of the African low velocity anomaly. These observations suggest that the presence of reflectors in the mid-lower mantle is not exclusively related to current or past subducting regions, but widespread throughout the mantle.

Anomalies of cosmic anisotropy from holographic universality of great-circle variance

We examine all-sky cosmic microwave background temperature maps on large angular scales to compare their consistency with two scenarios: the standard inflationary quantum picture, and a distribution constrained to have a universal variance of primordial curvature perturbations on great circles. The latter symmetry is not a property of standard quantum inflation, but may be a symmetry of holographic models with causal quantum coherence on null surfaces. Since the variation of great-circle variance is dominated by the largest angular scale modes, in the latter case the amplitude and direction of the unobserved intrinsic dipole (that is, the ℓ = 1 harmonics) can be estimated from measured ℓ = 2, 3 harmonics by minimizing the variance of great-circle variances including only ℓ = 1, 2, 3 modes. It is found that including the estimated intrinsic dipole leads to a nearly-null angular correlation function over a wide range of angles, in agreement with a null anti-hemispherical symmetry independently motivated by holographic causal arguments, but highly anomalous in standard cosmology. Simulations are used here to show that simultaneously imposing the constraints of universal great-circle variance and the vanishing of the angular correlation function over a wide range of angles tends to require patterns that are unusual in the standard picture, such as anomalously high sectorality of the ℓ = 3 components, and a close alignment of principal axes of ℓ = 2 and ℓ = 3 components, that have been previously noted on the actual sky. The precision of these results appears to be primarily limited by errors introduced by models of Galactic foregrounds.

Minimal differential lateral acceleration configurations for starshade stationkeeping in exoplanet direct imaging

Exoplanet imaging missions utilizing an external occulter (starshade) for starlight suppression require precise alignment between the telescope and starshade, necessitating maintenance of the starshade orbit during observations. Differential lateral acceleration between the two spacecraft serves as a proxy for fuel use and number of required thruster firings, which may interrupt the observation. Comparison against results from high fidelity simulations of stationkeeping validates the use of this easy-to-compute proxy. Among starshade positions constrained to the surface of a sphere centered about the telescope, minima of differential lateral acceleration lie on a great circle and its corresponding poles. We present a closed expression for telescope to star vectors requiring minimal stationkeeping for observation from a telescope at an arbitrary position. This proxy metric along with analytical knowledge of optimal observations facilitates computationally efficient design of exoplanet imaging missions employing a starshade.

A very high energy hadron collider on the Moon

The long-term prospect of building a hadron collider around the circumference of a great circle of the Moon is sketched. A circular collider on the Moon (CCM) of ∼11 000 km in circumference could reach a proton–proton center-of-mass collision energy of 14 PeV—a thousand times higher than the Large Hadron Collider at CERN—optimistically assuming a dipole magnetic field of 20 T. Several aspects of such a project are presented, including siting, construction, availability of necessary materials on the Moon, and powering, as well as a discussion of future studies and further information needed to determine the more concrete feasibility of each. Machine parameters and vacuum requirements are explored, and an injection scheme is delineated. Other unknowns are set down. Due to the strong interest from multiple organizations in establishing a permanent Moon presence, a CCM could be the (next-to-) next-to-next-generation discovery machine for high-energy particle physics and a natural successor to next-generation machines, such as the proposed future circular collider at CERN or a super proton–proton collider in China, and other future machines, such as a collider in the Sea, in the Gulf of Mexico. A CCM would serve as an important stepping stone toward a Planck-scale collider sited in our Solar System.

Calculating geodesics:

Introduction: The article is the product of the research “Connections on Semi-Riemannian Geometry and Christoffel Coefficients – Towards the study of the computational calculation of geodesics”, developed at the Pascual Bravo University Institution in the year 2021. Problem: Based on solutions of the Euler-Lagrange equations, the explicit calculation of geodesics on certain manifolds is possible. However, there are several cases in which it is impossible to continue calculating analytically and we have to resort to a numerical calculation. In this sense, several geometric and dynamic characteristics of geodesics, unexpectedly emerge. Objective: The objective of the research is to calculate geodesics of a Riemannian or semi-Riemannian manifold using SageMath as software to more easily go beyond what intuition provides. Methodology: First, some simple examples of characterizations of geodesics on certain manifolds, based on solutions of the Euler-Lagrange equations, are presented. Then, an ellipsoid is selected as a test subject with which to numerically calculate geodesics, observing how it changes depending on whether it is defined within a Spherical, Triaxial or Mercator coordinate system. Results: With the flexibility of software like SageMath, an explicit expression of the differential equations was made possible along with, from numeric solutions for these equations, their corresponding simulations depending on the selected parameters. Conclusion: These simulations confirm that great circles are not the only geodesics existing on the ellipsoid, but rather there are many other types of geodesic curves, some of which can be dense curves on the surface and others can be closed curves. At the same time, this shows a relationship between the existence of certain types of geodesic curves and the parameterization of the surface.

A center compact scheme for the Shallow Water equations on the sphere

We consider the Shallow Water equations (SW) on a rotating sphere and their approximation by a finite difference scheme. The discrete unknowns are located at the vertices of the equiangular Cubed Sphere grid, (Croisille, 2013; 2015). The standard fourth order Hermitian difference derivative Lele (1991) is used along a set of suitable great circles. No one sided difference formula is used at any point. All differential operators on the sphere (gradient, divergence and curl) are approximated in a centered fashion. The approximation procedure is close in spirit to the one of compact schemes used in Computational Aeroacoustics. Numerical results on a series of numerical test cases for SW on the sphere are presented. A particular attention is devoted to the temporal scheme. In particular two Rosenbrock exponential time schemes are used and compared to the RK4 scheme.The results demonstrate the interest of the present approach in a variety of situations of interest in numerical climatology.

Mathematical Aspects of a New Spherical Musical Instrument: Sphereharmonic

We consider the mathematical properties of a new musical instrument referred to as a SphereHarmonic. The SphereHarmonic is designed based on the basic divisions of a temari, that is, a spherical surface with congruent triangles defined by great circles. Each triangle on the sphere corresponds to a tone, and each vertex, which is connected to multiple triangles, represents a chord composed of the corresponding tones. Different chords are created by different combinations of the triangles, which are obtained by rotating the hemispheres. We describe four types of SphereHar- monics characterized by their divisions: Sn, C6, C8, and C10. These four SphereHarmonics have different mathematical characteristics, including the numbers of triangles, the degrees of the verti- ces, and the numbers of axes. We summarize these properties and also calculate the combinations for the possible vertices.

Broken Circle: Premature Deindustrialization, Chinese Capital Exports, and the Stumbling Development of New Territorial Industrial Complexes

AbstractWhile inherited models of industrial development and the role of Foreign Direct Investment (FDI) in the developmental process associate urbanization and rising industrial output with the industrialization of employment, suggesting that the structural shift to a service-based job market is a sign of developmental “maturity,” such models fail to explain the secular tendency toward “premature deindustrialization” that has become increasingly evident in poor countries worldwide. These models cannot account for the deep bifurcations between output and employment, formality and informality, and industrialization and urbanization observable on the ground in the world's fastest growing cities. Meanwhile, the alternative models of critical development theorists tend to focus on failures of industrial takeoff and classic relations of dependency, none of which adequately account for the phenomenon being observed today. This paper explores an alternate explanation for premature deindustrialization, drawing from Marxian theories of technical change and the secular tendencies of capitalist development. The argument is illustrated with examples taken from the author's field work in Dar es Salaam, Tanzania, and the Pearl River Delta in China. Ultimately, the phenomenon of premature deindustrialization suggests that the great circle of development may collapse under its own contradictions before industrialization circumnavigates the globe.

Image-Based Angular Distortion Metric of Map Projections by Using Surface Fitting for Noise Reduction

Measuring, analyzing, reducing, and optimizing distortions in map projections is important in cartography. In this study, we introduced a novel image-based angular distortion metric based on the previous spherical great circle arcs-based metric. Images with predefined patterns were used to generate distorted images using mapping software. The generated distorted images with known patterns were then exploited to calculate the proposed angular distortion metric. The mapping software performed the underlying transformation of map projections. Therefore, there was no direct explicit dependence on the forward equations of the map projections in our proposed metric. However, there were fairly large computation errors in the ordinary image-based approach without special correction. To reduce the error, we introduced surface-fitting-based noise reduction in our approach. We established and solved systems of linear equations based on bivariate polynomial functions in the process of noise reduction. Sufficient experiments were made to validate the proposed image-based metric and the accompanying noise reduction approach. In the experiment, the NASA G.Projector was employed as the mapping software for evaluating more than 200 map projections. Experimental results demonstrated that the proposed image-based approach and surface fitting-based noise reduction are feasible and practical for the evaluation of the angular distortion of map projections.

Emergent behaviors of the discrete thermodynamic Cucker–Smale model on complete Riemannian manifolds

We propose an intrinsic discrete-time counterpart of the abstract thermomechanical Cucker–Smale (TCS) model on connected, complete, and smooth Riemannian manifolds and study its emergent dynamics. Our proposed discrete model is expressed in terms of exponential map on the tangent bundle endowed with the Sasaki metric. Compared to projection-based discrete models on the manifold, it is embedding free and enjoys the same structural properties as the corresponding continuous models. For the proposed model, we provide a sufficient framework leading to asymptotic velocity alignment in which all particles’ velocity align when they lie in the same tangent plane via the parallel transport along the length-minimizing geodesic. For the unit-d sphere (Sd), we provide explicit representations of the Sasaki metric and the corresponding geodesics on TSd and show that the TCS model exhibits a dichotomy in asymptotic spatial patterns (either energy tends to zero or all particles move along a common geodesic on Sd, which is a great circle). We also provide several numerical examples and compare them with analytical results.

The Clustering of Orbital Poles Induced by the LMC: Hints for the Origin of Planes of Satellites

Abstract A significant fraction of Milky Way (MW) satellites exhibit phase-space properties consistent with a coherent orbital plane. Using tailored N-body simulations of a spherical MW halo that recently captured a massive (1.8 × 1011 M ⊙) LMC-like satellite, we identify the physical mechanisms that may enhance the clustering of orbital poles of objects orbiting the MW. The LMC deviates the orbital poles of MW dark matter particles from the present-day random distribution. Instead, the orbital poles of particles beyond R ≈ 50 kpc cluster near the present-day orbital pole of the LMC along a sinusoidal pattern across the sky. The density of orbital poles is enhanced near the LMC by a factor δ ρ max = 30% (50%) with respect to underdense regions and δ ρ iso = 15% (30%) relative to the isolated MW simulation (no LMC) between 50 and 150 kpc (150–300 kpc). The clustering appears after the LMC’s pericenter (≈50 Myr ago, 49 kpc) and lasts for at least 1 Gyr. Clustering occurs because of three effects: (1) the LMC shifts the velocity and position of the central density of the MW’s halo and disk; (2) the dark matter dynamical friction wake and collective response induced by the LMC change the kinematics of particles; (3) observations of particles selected within spatial planes suffer from a bias, such that measuring orbital poles in a great circle in the sky enhances the probability of their orbital poles being clustered. This scenario should be ubiquitous in hosts that recently captured a massive satellite (at least ≈1:10 mass ratio), causing the clustering of orbital poles of halo tracers.

Fibrations of ℝ3 by oriented lines

A fibration of $\mathbb{R}^3$ by oriented lines is given by a unit vector field $V : \mathbb{R}^3 \to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationships with great circle fibrations of $S^3$ and with tight contact structures on $\mathbb{R}^3$. Both geometric and topological classifications of the space of skew fibrations have appeared; these classifications rely on certain rigid geometric properties exhibited by skew fibrations. Here we study these properties for line fibrations which are not necessarily skew, and we offer some partial answers to the question: in what sense do nonskew fibrations look and behave like skew fibrations? We develop and utilize a technique, called the parallel plane pushoff, for studying nonskew fibrations. In addition, we summarize the known relationship between line fibrations and contact structures, and we extend these results to give a complete correspondence. Finally, we develop a technique for generating nonskew fibrations and offer a number of examples.