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Abstract:
In this paper, the very fundamental geometrical characteristics of the Mylar balloon like the profile curve, height, volume, arclength, surface area, crimping factor, etc. are recognized as geometrical moments ℐ

Keywords:
Mylar balloons, geometrical moments, elliptic integrals, beta and gamma functions, recursive relations, crimping factor, lemniscate constant

Abstract:
Here we compare the geometric descriptions of the deformed sphere (i.e., the so-called λ-sphere) and the standard spheroid (namely, World Geodetic System 1984’s reference ellipsoid of revolution). Among the main geometric characteristics of those two surfaces of revolution embedded into the three-dimensional Euclidean space we consider the semi-major (equatorial) and semi-minor (polar) axes, quartermeridian length, surface area, volume, sphericity index, and tipping (bifurcation) point for geodesics. Next, the RMS (Root Mean Square) error is defined as the square-rooted arithmetic mean of the squared relative errors for the individual pairs of the discussed six main geometric characteristics. As a result of the process of minimization of the RMS error, we have obtained the proposition of the optimized value of the deformation parameter of the λ-sphere, for which we have calculated the absolute and relative errors for the individual pairs of the discussed main geometric characteristics of λ-sphere and standard spheroid (the relative errors are of the order of 10−6 – 10−9). Among others, it turns out that the value of the flattening factor of the spheroid is quite a good approximation for the corresponding value of the deformation parameter of the λ-sphere (the relative error is of the order of 10−4).

Keywords:
deformed sphere, standard spheroid, sphericity index, tipping (bifurcation) point for geodesics, elliptic integrals and functions

Abstract:
Starting with identifications of the very fundamental geometric characteristics of a Mylar balloon such as the profile curve, height, volume, arclength, surface area, crimping factor, etc., using the geometrical moments In(x) and In, we present explicit formulas for them and those of the mechanical moments of both solid and hollow balloons of arbitrary order. This is achieved by relying on the recursive relationships among elliptic integrals and the final results are expressed via the fundamental mathematical constants such as π, lemniscate constant ˜ω, and Gauss’s constant G. An interesting periodicity modulo 4 was detected and accounted for in the final formulas for the moments. The principal results are illustrated by two tables, a few graphics, and some direct relationships with other fundamental areas in mathematics, physics and geometry are pointed out.

Keywords:
solid and hollow Mylar balloons, crimping factor, geometro-mechanical moments, recursive relations, elliptic integrals and functions, gamma functions, Gauss’s and lemniscate constants

Abstract:
In this article, we consider deformed spheres as a new reference model for the geoid, alternatively to the classical ellipsoidal one. The parametrization of deformed spheres is furnished through the incomplete elliptic integrals. From the other side, the solutions for geodesics on those surfaces are given entirely via elementary analytical functions, contrary to the case of ellipsoids of revolution. We explicitly described algorithms (all necessary computational steps) for the solution of the direct and inverse geodetic problems on the deformed spheres. Finally, we presented a few illustrative numerical solutions of the inverse geodetic problems for two conceptual cases of near and far points. It had turned out that even in the non-optimized case we obtained the good agreement with the predictions of the World Geodetic System 1984’s ellipsoidal reference model.

Keywords:
deformed spheres, direct and inverse geodetic problems, geodesy and navigation problems, reference models for geoid, World Geodetic System 1984

Abstract:
We propose here a new reference model approximating the actual shape of the Earth. It is based on

Keywords:
closed and open geodesics, deformed spheres, geodesy and mapping problems, geoid's reference models, geometric flattening parameter, incomplete elliptic integrals

Abstract:
The considerations of the mechanics of incompressible test bodies moving on spheres are generalized to the case of λ-spheres. Both surfaces are examples of Riemannian manifolds realized as two-dimensional manifolds with strictly positive Gaussian curvature under the condition λ < 1/3 for the λ-spheres. A convenient parametrization of λ-spheres embedded as two-dimensional surfaces of revolution into the three-dimensional Euclidean space is derived. The so-obtained parametrization is expressed in a concise form via the elliptic integrals of the first, second, and third kind. Next, the geodetic motion is considered. The explicit solutions for two branches of the incompressible motion are obtained in the parametric form. For the special case of geodetics corresponding to meridians taken as geodesics their analysis is performed in detail. In the latter case, the so-obtained geodetic solutions are reduced to the incomplete elliptic integrals of the first, second, and third kind.

Keywords:
λ-sphere as a generalization of the usual sphere, elliptic integrals and elliptic functions, geodesic and geodetic equations of motion, incompressibility constraints, mechanics of infinitesimal test bodies

Abstract:
In this paper, we present a new reference model that approximates the actual shape of the Earth, based on the concept of the deformed spheres with the deformation parameter λ. These surfaces, which are called λ-spheres, were introduced in another setting by Faridi and Schucking as an alternative to the spheroids (i.e., ellipsoids of revolution). Using their explicit parametrizations that we have derived in our previous papers, here we have defined the corresponding isothermal (conformal) coordinates as well as obtained and solved the differential equation describing the loxodromes (or rhumb lines) on such surfaces. Next, the direct and inverse problems for loxodromes have been formulated and the explicit solutions for azimuths and arc lengths have been presented. Using these explicit solutions, we have assessed the value of the deformation parameter λ for our reference model on the basis of the values for the semi-major axis of the Earth a and the quarter-meridian mp (i.e., the distance between the Equator and the North or South Pole) for the current best ellipsoidal reference model for the geoid, i.e., WGS 84 (World Geodetic System 1984). The latter is designed for use as the reference system for the GPS (Global Positioning System). Finally, we have compared the results obtained with the use of the newly proposed reference model for the geoid with the corresponding results for the ellipsoidal (WGS 84) and spherical reference models used in the literature.

Keywords:
deformed spheres, incomplete elliptic integrals, geoid’s reference models, loxodromes or rhumb lines, azimuths and arc lengths, geodesy and navigation problems

Abstract:
In this paper we explore the mechanics of infinitesimal gyroscopes (test bodies with internal degrees of freedom) moving on an arbitrary member of the helicoid-catenoid family of minimal surfaces. As the configurational spaces within this family are far from being trivial manifolds, the problem of finding the geodesic and geodetic motions presents a real challenge. We have succeeded in finding the solutions of those motions in an explicit parametric form. It is shown that in both cases the solutions can be expressed through the elliptic functions and elliptic integrals, but in the geodetic case some appropriately chosen compatibility conditions for glueing together different branches of the solution are needed. Additionally, an action-angle analysis of the corresponding Hamilton-Jacobi equations is performed for external potentials that are well-suited to the geometry of the problem under consideration. As a result, five different sets of conditions between the three action variables and the total energy of the infinitesimal gyroscopes are obtained.

Keywords:
action-angle analysis, mechanics of infinitesimal gyroscopes, geodesic and geodetic equations of motion, helicoid-catenoid deformation family of minimal surfaces, elliptic integrals and elliptic functions

Abstract:
This paper starts with the derivation of the most general equations of motion for the infinitesimal rotators moving on arbitrary two‐dimensional surfaces of revolution. Both geodesic and geodetic (i.e., without any external potential) equations of motion on surfaces with nontrivial curvatures that are embedded into the three dimensional Euclidean space are discussed. The Mylar balloon as a concrete example for the application of the scheme was chosen. A new parameterization of this surface is presented, and the corresponding equations of motion for geodesics and geodetics are expressed in an analytical form through the elliptic functions and elliptic integrals. The so‐obtained results are also compared with those for the two‐dimensional sphere embedded into the three‐dimensional Euclidean space for which it can be shown that the geodesics and geodetics are plane curves realized as the great and small circles on the sphere, respectively.

Keywords:
elliptic integrals and elliptic functions, geodesics and geodetics, infinitesimal rotators, mylar balloons, non-Euclidean spaces, surfaces of revolution

Abstract:
Here, we apply the general scheme for description of mechanics of infinitesimal bodies in Riemannian spaces to the example of geodetic and non-geodetic (for two different model potentials) motions of infinitesimal rotators on the Mylar balloon. The structure of partial degeneracy is investigated with the help of the corresponding Hamilton-Jacobi equation and action-angle analysis. In all situations it was found that for any of the sixth disjoint regions in the phase space among the three action variables only two are essential for the description of our models at the level of the old quantum theory (according to the Bohr-Sommerfeld postulates). Moreover, in both non-geodetic models, the action variables were intertwined with the quantum number N corresponding to the quantized value of the radius r of the inflated Mylar balloon.

Keywords:
action-angle analysis, affinely-rigid bodies, Bohr-Sommerfeld quantum theory, Hamilton-Jacobi equation, infinitesimal gyroscopes, mylar balloons, residue analysis

Abstract:
This paper discusses the motion of infinitesimal gyroscopes along the two-dimensional surfaces of constant mean curvature embedded into the three-dimensional Euclidean space. We have considered the cases of unduloids, spheres, and cylinders for which the corresponding Hamilton-Jacobi equations are written and analyzed with the help of the action-angle variables. Spheres and cylinders are considered as limiting cases of unduloids and the residue analysis was performed which provides the connection between all three action variables and the energy. This has been traced also for the geodetic situations and for two additional classical model potentials.

Keywords:
action-angle variables, affinely-rigid bodies, d'Alembert mechanics, Delaunay surfaces, geometry of curves and surfaces, Hamilton-Jacobi equation, harmonic/anharmonic oscillator potential models, infinitesimal test bodies, residue analysis, Riemannian spaces