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arxiv: 1906.12318 · v1 · pith:PSOEKW3Fnew · submitted 2019-06-28 · ⚛️ physics.space-ph · astro-ph.EP· astro-ph.IM· math.AG

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

Marco Nugnes , Camilla Colombo , Massimo Tipaldi This is my paper

Pith reviewed 2026-05-25 13:18 UTC · model grok-4.3

classification ⚛️ physics.space-ph astro-ph.EPastro-ph.IMmath.AG
keywords satellite coverage areaoblate Earthconical field of viewanalytical intersectionellipsoid geometrynavigation antennaminimum elevation anglepointing direction
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The pith

A new analytical method finds satellite coverage areas on an oblate Earth by decomposing the conical view intersection into ellipses.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an analytical technique to compute the region on Earth's surface visible to a satellite antenna with a conical field of view. Earth is modeled as an oblate ellipsoid of rotation instead of a sphere. The intersection surface is cut by successive planes that all contain the antenna line of sight, producing a family of ellipses whose parameters and boundary points are obtained in closed form for any chosen half-aperture angle or minimum elevation. The procedure handles geocentric, geodetic, and arbitrary pointing directions and is tested against the classical spherical model through orbital simulations.

Core claim

Starting from the satellite position vector and navigation antenna line-of-sight direction, the surface generated by the intersection of the oblate ellipsoid and the conical field of view is decomposed into many ellipses obtained by cutting the Earth's surface with every plane containing the line of sight; the geometrical parameters of each ellipse together with the intersection points of the cone with that ellipse are then derived analytically for a chosen half-aperture angle or minimum elevation angle.

What carries the argument

Decomposition of the cone-ellipsoid intersection surface into ellipses via planes containing the line of sight, from which analytical expressions for ellipse parameters and cone intersection points are obtained.

If this is right

  • The same analytical procedure applies without modification to geocentric, geodetic, and generic antenna pointing.
  • Coverage-area differences between the ellipsoidal and spherical models grow with eccentricity of the orbit and with latitude of the ground track.
  • The method supplies explicit expressions for ellipse semi-axes, eccentricity, and the two intersection points on each ellipse.
  • Numerical simulations quantify how coverage error depends on orbital altitude, inclination, and argument of perigee.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be used to generate instantaneous visibility polygons for constellation design without resorting to numerical ray-tracing.
  • Because each ellipse is treated independently, the method lends itself to parallel evaluation when many satellites must be assessed simultaneously.
  • Extension to time-varying pointing or to non-conical sensor patterns would require only a change in the generating surface while retaining the planar slicing step.

Load-bearing premise

The intersection surface between the cone and the oblate ellipsoid can be decomposed into a set of ellipses by slicing with planes that all contain the line of sight, allowing closed-form geometry.

What would settle it

Compute the exact numerical intersection curve between a specific cone and the WGS84 ellipsoid for a chosen satellite position and half-angle, then compare the resulting boundary points and enclosed area against the values produced by the ellipse-decomposition formulas.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents an analytical method to compute the coverage area of a conical antenna field of view from a satellite, treating Earth as an oblate ellipsoid rather than a sphere. Starting from satellite position and line-of-sight direction, the intersection is decomposed into ellipses via planes containing the line of sight; geometrical parameters of these ellipses and their intersection points with the cone are derived analytically for a chosen half-aperture angle or minimum elevation. The approach accommodates geocentric, geodetic, and generic pointing, and is illustrated with numerical comparisons to the spherical model showing coverage errors as a function of orbital parameters.

Significance. The work offers a geometrically grounded analytical alternative to numerical coverage calculations for oblate-Earth models, which is useful for mission analysis in navigation and remote sensing. The explicit comparison to the spherical case and dependence on orbital elements provides concrete quantification of the approximation error. The decomposition into planar sections is a standard geometric technique that reduces the 3D intersection problem to solvable 2D quadratics, lending credibility to the analytical claim.

major comments (2)
  1. [Abstract and method section] Abstract and method section: the central claim that ellipse parameters and cone-ellipse intersections 'can be derived analytically' is load-bearing, yet the manuscript provides no explicit closed-form expressions (e.g., for semi-major/minor axes or intersection coordinates after the planar quadratic reduction) that would allow independent verification of the derivation for generic pointing.
  2. [Numerical simulations section] Numerical simulations section: the reported coverage-area differences between spherical and ellipsoidal models depend on the number of planes used in the decomposition; without a stated convergence test or error bound on the 'many ellipses' summation, it is unclear whether the quoted orbital-parameter sensitivities are free of discretization bias.
minor comments (2)
  1. Define the input parameters (half-aperture angle vs. minimum elevation angle) with consistent symbols and units in the first use.
  2. Add a schematic figure illustrating one plane cut, the resulting ellipse, and the two generator lines to make the decomposition concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and constructive comments. We address each major point below and will revise the manuscript to strengthen the presentation of the analytical derivations and numerical validation.

read point-by-point responses

  1. Referee: [Abstract and method section] Abstract and method section: the central claim that ellipse parameters and cone-ellipse intersections 'can be derived analytically' is load-bearing, yet the manuscript provides no explicit closed-form expressions (e.g., for semi-major/minor axes or intersection coordinates after the planar quadratic reduction) that would allow independent verification of the derivation for generic pointing.

    Authors: We agree that the manuscript outlines the analytical approach but does not furnish the full set of closed-form expressions for ellipse semi-axes, eccentricity, and cone-ellipse intersection coordinates, especially under generic pointing. In the revised version we will insert the complete derivations, including the quadratic solutions after planar sectioning and the resulting coordinate transformations, to permit direct verification. revision: yes

  2. Referee: [Numerical simulations section] Numerical simulations section: the reported coverage-area differences between spherical and ellipsoidal models depend on the number of planes used in the decomposition; without a stated convergence test or error bound on the 'many ellipses' summation, it is unclear whether the quoted orbital-parameter sensitivities are free of discretization bias.

    Authors: The observation is correct: the coverage-area results are obtained by summation over a finite number of planes and the manuscript does not report a convergence study. We will add a dedicated subsection that quantifies the truncation error as a function of the number of planes, demonstrates convergence of the area to within a stated tolerance (e.g., 0.1 %), and supplies an a-priori error bound derived from the angular spacing of the planes. revision: yes

Circularity Check

0 steps flagged

Direct geometric derivation with no circular elements

full rationale

The paper's central method decomposes the cone-ellipsoid intersection into planar sections, each yielding an ellipse whose parameters and line intersections are solved analytically via standard quadratic geometry. This follows directly from the definitions of cone and ellipsoid without fitting parameters, without invoking self-citations as load-bearing premises, and without renaming or re-deriving prior results as new predictions. Numerical comparisons are used only for validation against the spherical case, not as part of the derivation itself. The chain is therefore self-contained in classical analytic geometry.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The method is analytical and relies on standard geometric properties of ellipsoids and cones; no new entities postulated and no data-fitted parameters beyond mission inputs.

free parameters (1)
  • half-aperture angle or minimum elevation angle
    Chosen based on mission requirements as an input to determine intersection points.
axioms (2)
  • domain assumption Earth modeled as oblate ellipsoid of rotation
    Standard assumption in orbital mechanics for more accurate modeling than sphere.
  • domain assumption Intersection surface can be decomposed into ellipses via planes containing the line of sight
    Core geometric assumption of the method described in abstract.

pith-pipeline@v0.9.0 · 5720 in / 1188 out tokens · 28820 ms · 2026-05-25T13:18:20.703624+00:00 · methodology

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Reference graph

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