Lunar Libration Geometry, Part I: An Order-Free Geometric Framework in SO(3)

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 References, historical context, and supporting background material are being maintained separately in an unpublished introductory manuscript, and will be incorporated or cross-referenced in future revisions, or additional parts of the monograph are uploaded.