Lunar Libration Geometry, Part I: An Order-Free Geometric Framework in SO(3)
Year: 2026
Type: article
Abstract: NOTICE — MATHEMATICAL REVIEW IN PROGRESS (October 17, 2026) This manuscript is temporarily restricted while a comprehensive mathematical review and revision are undertaken. The review addresses consistency among the rotation matrix, quaternion, and related representations, as well as boundary cases and the treatment of limiting geometries and numerical validation. The original publication record and DOI are being retained. A corrected and independently validated version will be published upon completion of the review. The currently restricted version should not be relied upon for numerical implementation. ******************** This preprint is Part I of a planned six-part monograph on lunar libration geometry and its applications to eclipse prediction, occultation analysis, lunar limb-profile modeling, and three-dimensional umbra visualization. Classical treatments of lunar libration commonly describe the Moon's orientation through ordered sequences of rotations in longitude, latitude, and position angle. Because three-dimensional rotations do not commute, such approaches introduce an artificial dependence on rotation order and can obscure the underlying geometric structure of the problem. This paper develops an order-free formulation of lunar libration directly within the rotation group SO(3). The construction begins with the physically observable constraints: the displaced lunar pole direction and the observed position angle of the projected lunar north direction. These constraints uniquely determine the lunar orientation without requiring an Euler-angle sequence. Equivalent quaternion, axis-angle, Rodrigues-vector, and matrix representations are derived and shown to be mutually consistent. Particular attention is given to local invertibility, continuity, coordinate degeneracies, and the absence of Euler-angle singularities. The resulting framework provides a geometrically transparent and numerically stable description of lunar attitude that is valid for arbitrary libration states and observer geometries. To support practical implementation, the paper includes a consolidated formula sheet and a series of worked numerical examples. These examples provide independent verification of the derived expressions, and establish consistency among the quaternion, matrix, Rodrigues-vector, and axis-angle formulations. This first installment establishes the mathematical foundation upon which later parts of the monograph will build. Subsequent volumes will address practical libration computation, topocentric geometry, lunar limb-profile modeling, eclipse and occultation applications, and related observational techniques. This version is released as a preprint. References, historical context, and supporting background material are being maintained separately in an unpublished introductory manuscript, and will be incorporated or cross-referenced in future revisions, or additional parts of the monograph are uploaded.
Author: McGlaun, Dan
Language: English
Subfield: Aerospace Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7165042413
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