Tag Archives: Algebraic Geometry

Geometric Solution to the Angles-Only Initial Orbit Determination Problem [EPA]

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http://arxiv.org/abs/2304.02157


Initial orbit determination (IOD) from line-of-sight (i.e., bearing) measurements is a classical problem in astrodynamics. Indeed, there are many well-established methods for performing the IOD task when given three line-of-sight observations at known times. Interestingly, and in contrast to these existing methods, concepts from algebraic geometry may be used to produce a purely geometric solution. This idea is based on the fact that bearings from observers in general position may be used to directly recover the shape and orientation of a three-dimensional conic (e.g., a Keplerian orbit) without any need for knowledge of time. In general, it is shown that five bearings at unknown times are sufficient to recover the orbit — without the use of any type of initial guess and without the need to propagate the orbit. Three bearings are sufficient for purely geometric IOD if the orbit is known to be (approximately) circular. The method has been tested over different scenarios, including one where extra observations make the system of equations over-determined.

Read this paper on arXiv…

M. Mancini, T. Duff, A. Leykin, et. al.
Thu, 6 Apr 23
42/76

Comments: 31 pages excluding back matter, 14 figures

Coverage Area Determination for Conical Fields of View Considering an Oblate Earth [CL]

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http://arxiv.org/abs/1906.12318


This paper introduces a new analytical method for the determination of the coverage area modeling the Earth as an oblate ellipsoid of rotation. Starting from the knowledge of the satellite’s position vector and the direction of the navigation antenna line of sight, the surface generated by the intersection of the oblate ellipsoid and the assumed conical field of view is decomposed in many ellipses, obtained by cutting the Earth’s surface with every plane containing the navigation antenna line of sight. The geometrical parameters of each ellipse can be derived analytically together with the points intersection of the conical field of view with the ellipse itself by assuming a proper value of the half-aperture angle or the minimum elevation angle from which the satellite can be considered visible from the Earth’s surface. The method can be applied for different types of pointing (geocentric, geodetic and generic) according to the mission requirements. Finally, numerical simulations compare the classical spherical approach with the new ellipsoidal method in the determination of the coverage area, and also show the dependence of the coverage errors on some relevant orbital parameters.

Read this paper on arXiv…

M. Nugnes, C. Colombo and M. Tipaldi
Mon, 1 Jul 19
33/52

Comments: 32 pages, 16 figures, 5 tables, 21 references. Post-print version of the article published on the Journal of Guidance, Control, and Dynamics

Estimating the number of solutions equation of N-point gravitational lens algebraic geometry methods [CL]

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http://arxiv.org/abs/1809.05392


One of the main problems in the study of system of equations of the gravitational lens, is the computation of coordinates from the known position of the source. In the process of computing finds the solution of equations with two unknowns. The difficulty lies in the fact that, in general, is not known constructive or analytical algorithm for solving systems of polynomial equations In this connection, use numerical methods like the method of tracing. For the N-point gravitational lenses have a system of polynomial equations. Systems Research is advisable to start with an assessment of the number of solutions. This can be done by methods of algebraic geometry.

Read this paper on arXiv…

A. Kotvytskiy, S. Bronza and S. Vovk
Fri, 28 Sep 18
3/52

Comments: 5 pages