Local Linear Scale Factors in Map Projections of an Ellipsoid

The main problem in cartography is that it is not possible to map/project/transform a spherical or ellipsoidal surface into a plane without distortions. The distortions of areas, angles, and/or distances are immanent to all maps. It is known that scale changes from point to point, and at certain points, the scale usually depends on the direction. The local linear scale factor c is one of the most important indicators of distortion distribution in the theory of map projections. It is not possible to find out the values of the local linear scale factor c in directions of coordinate axes x and y immediately from the definition of c. To solve this problem, in this paper, we derive new formulae for the calculation of c for a rotational ellipsoid. In addition, we derive the formula for computing c in any direction defined by dx and dy. We also considered the position and magnitude of the extreme values of c and derived new formulae for a rotational ellipsoid.

Mathematical modeling of triode system on the basis of field emitter with ellipsoid shape

In this paper the mathematical modeling of the triode emission axially symmetric system on the basis of field emitter is considered. Emitter is an ellipsoid of revolution, anode is a confocal ellipsoidal surface of revolution. Modulator is a part of the ellipsoidal surface of revolution, confocal with the cathode and anode surfaces. The boundary-value problem for the Laplace's equation in the prolate spheroidal coordinates with the boundary conditions of the first kind is solved. The variable separation method is applied to calculate the axisymmetrical electrostatic potential distribution. The potential distribution is represented as the Legendre functions expansion. The expansion coefficients are the solution of the system of linear equations. All geometrical dimensions of the system are the parameters of the problem.

Mitigation of Geostationary Lightning Mapper Geolocation Errors

The geolocation of lightning flashes observed by spaceborne optical sensors depends upon a priori assumptions of the cloud-top height (or, more generally, the height of the radiant emitter) as observed by the satellite. Lightning observations from the Geostationary Lightning Mappers (GLMs) on Geostationary Operational Environmental Satellite 16 (GOES-16) and GOES-17 were originally geolocated by assuming that the global cloud-top height can be modeled as an ellipsoidal surface with an altitude of 16 km at the equator and sloping down to 6 km at the poles. This method produced parallax errors of 20–30 km or more near the limb, where GLM can detect side-cloud illumination or below-cloud lightning channels at lower altitudes than assumed by the ellipsoid. Based on analysis of GLM location accuracy using a suite of alternate lightning ellipsoids, a lower ellipsoid (14 km at the equator, 6 km at the poles) was implemented in October and December 2018 for GLM-16 and GLM-17, respectively. While the lower ellipsoid slightly improves overall GLM location accuracy, parallax-related errors remain, particularly near the limb. This study describes the identification of optimized assumed emitter heights, defined as those that produce the closest agreement with the ground-based reference networks. Derived using the first year of observations from GOES-East position, the optimal emitter height varies geographically and seasonally in a manner consistent with known meteorological regimes. Application of the optimal emitter height approximately doubles the fraction of area near the limb for which peak location errors are less than half a GLM pixel.

Continuos Parameterization of the Median Surface of an Ellipsoidal Shell and Its Geometric Parameters

When analyzing the stress-strain state of thin-walled structural elements that have the shape of an ellipsoid, it becomes necessary to calculate the geometric characteristics of the ellipsoidal surface. When using the canonical ellipsoid equation, regions of uncertainty appear in the Cartesian coordinate system at the intersection points of the ellipsoid surface with the horizontal coordinate plane. To exclude these areas of uncertainty, we propose an expression of the radius vector of an ellipsoidal surface whose projections are functions of two parametric representations in mutually perpendicular planes. One of the planes is the vertical plane XOZ, and the other plane is the plane perpendicular to the axis O at the point with the x coordinate. The parameter T of the ellipse obtained from the intersection of the ellipsoid with the XOZ plane was chosen as the argument of the first parametric function. The argument of the second parametric function t is the parameter of an ellipse formed as a result of the intersection of an ellipsoidal surface with a plane perpendicular to the abscissa axis at a distance of x from the origin. The proposed representation of the ellipsoidal surface allowed us to exclude uncertainties at the intersection points of the ellipsoid with the HOWE coordinate plane. By differentiating the proposed radius-vector expression at an arbitrary point on an ellipsoidal surface, we obtain relations for the basis vectors of an arbitrary point and their derivatives represented by components in the same local basis. These relations are necessary for the development of algorithms for numerical analysis of deformation processes of engineering structures that have ellipsoidal surfaces.

The Hawking mass for an ellipsoidal surface

The mass of a system in general relativity cannot be defined locally. Thus, one defines mass at quasilocal level. There are many definitions of quasilocal mass. One of them is the Hawking mass. In this paper, we determine the Hawking mass for ellipsoidal 2-surface for a non-Schwarzschild spacetime. In order to do this, we first determine a null tetrad and then compute the Hawking mass. The results are presented graphically also.

Design and Modeling of a Radiofrequency Coil Derived from a Helmholtz Structure

This paper focuses on radiofrequency (RF) coils that can produce a high electromagnetic field homogeneity to be used for magnetic resonance imaging (MRI) applications. The proposed structure is composed of four wire loops symmetrically located on an ellipsoidal surface. The main objective of this work is to improve field homogeneity compared to a standard Helmholtz coil. Numerical simulation was carried out to assess the RF electromagnetic field behavior of the proposed coil. Different electrical modeling and simulations were investigated, particularly the study of the whole modeling of the proposed structure taking into account all the couplings between the loops. The proposed coil was evaluated and compared with the standard Helmholtz coil. Simulation and experimental results confirmed the good performance of the developed coil in terms of electromagnetic field homogeneity, efficiency, sensitivity, and quality factor.

Three-Dimensional Cloud Volume Reconstruction from the Multi-angle Imaging SpectroRadiometer

Characterizing 3-D structure of clouds is needed for a more complete understanding of the Earth’s radiative and latent heat fluxes. Here we develop and explore a ray casting algorithm applied to data from the Multi-angle Imaging SpectroRadiometer (MISR) onboard the Terra satellite, in order to reconstruct 3-D cloud volumes of observed clouds. The ray casting algorithm is first applied to geometrically simple synthetic clouds to show that, under the assumption of perfect, clear-conservative cloud masks, the reconstruction method yields overestimation in the volume whose magnitude depends on the cloud geometry and the resolution of the reconstruction grid relative to the image pixel resolution. The method is then applied to two hand-picked MISR scenes, fully accounting for MISR’s viewing geometry for reconstructions over the Earth’s ellipsoidal surface. The MISR Radiometric Camera-by-camera Cloud Mask (RCCM) at 1.1-km resolution and the custom cloud mask at 275-m resolution independently derived from MISR’s red, green, and blue channels are used as input cloud masks. A wind correction method, termed cloud spreading, is applied to the cloud masks to offset potential cloud movements over short time intervals between the camera views of a scene. The MISR cloud-top height product is used as a constraint to reduce the overestimation at the cloud top. The results for the two selected scenes show that the wind correction using the cloud spreading method increases the reconstructed volume up to 4.7 times greater than without the wind correction, and that the reconstructed volume generated from the RCCM is up to 3.5 times greater than that from the higher-resolution custom cloud mask. Recommendations for improving the presented cloud volume reconstructions, as well as possible future passive remote sensing satellite missions, are discussed.

Three-Dimensional Cloud Volume Reconstruction from the Multi-Angle Imaging SpectroRadiometer

Characterization the 3-D structure of clouds is needed for a more complete understanding of the Earth's radiative and latent heat fluxes. Here we develop and explore a “ray casting” algorithm applied to the Multi-angle Imaging SpectroRadiometer (MISR) on board the Terra satellite, to reconstruct 3-D cloud volumes for observed clouds. The ray casting algorithm is first applied to geometrically simple synthetic clouds to show that, under the assumption of perfect, clear-conservative cloud masks, the reconstruction method yields overestimation whose magnitude depends on the cloud geometry and the resolution of the reconstruction grid relative to the image pixel resolution. The method is then applied to two select MISR scenes, fully accounting for MISR’s viewing geometry for reconstructions over the Earth’s ellipsoidal surface. The MISR Radiometric Camera-by-camera Cloud Masks at 1.1 km resolution and custom cloud masks at 275 m resolution independently derived from MISR RGB channels are used as input cloud masks. A wind correction method, termed “cloud spreading”, is devised and applied to the cloud masks to offset potential cloud movements over short time intervals (around 7 minutes at maximum) between the cameras. The MISR cloud top height product is used as a constraint to reduce the overestimation at the cloud top. The reconstruction results show that their uncertainty is significant when the wind correction is applied, and that they have more refined structures when the input cloud mask has a higher resolution. Recommendations for improving the presented cloud volume reconstructions as well as for future passive remote sensing satellite missions are discussed.