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\begin{document}

\title{Mathematical Fundamentals of Spherical Kinematics of Plate Tectonics in
Terms of Quaternions dedicated to Xavier Le Pichon on the occasion of
the 50th anniversary of publication of ``Plate Tectonics'' (1973) with
Jean Francheteau and Jean Bonnin}



\author[1]{Helmut Schaeben}%
\author[1]{Uwe Kroner}%
\author[2]{Tobias Stephan}%
\affil[1]{TU Bergakademie Freiberg}%
\affil[2]{University of Calgary Department of Geoscience}%


\vspace{-1em}



  
  \date{May 22, 2024}


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\begin{abstract}
To be a quantitative and testable tectonic model, plate tectonics
requires spherical geometry and spherical kinematics in terms of finite
rotations conveniently parametrized by their angle and axis and
described by unit quaternions. In treatises on 'Plate Tectonics'
infinitesimal, instantaneous, and finite rotations, absolute and
relative rotations are said to be applied to model the motion of
tectonic plates. Even though these terms are strictly defined in
mathematics, they are often casually used in geosciences. Here their
definitions are recalled and clarified as well as the terms rotation,
orientation, and location on the sphere. For instance, infinitesimal
rotations refer to a mathematical limit, when the angle of rotation
converges to zero. Their rules do not apply to finite rotations, no
matter how small their finite angles of rotation are. Mathematical
approaches applying appropriate and feasible assumptions to model
spherical motion of tectonic plates over geological times of hundreds of
millions of years are derived including (i) sequences of incremental
finite rotations, (ii) sequences of accumulating successive
concatenations of finite rotations, (iii) continuous rotations in terms
of fully transient quaternions. The incremental and the accumulating
approach provide complementary views. While the relative Euler pole
appears to migrate in the latter, it appears fixed in the former. Path,
mean and instantaneous velocity of the migrating Euler pole are derived
as well as the angular and trajectoral velocity of the rotational motion
about it. The approaches are illustrated by a geological example with
actual data and a numerical yet geologically inspired example with
artificial data. The former revisits the three plates scenario with
stationary axes of two ``absolute'' rotations implying transient
``relative'' rotations about a migrating Euler pole and employs a proper
plate circuit argument to determine them numerically without resuming to
approximations. The latter applies an involved interplay of incremental
and accumulating modeling inducing split-join cycles to approximate
sinusoidal trajectories as reported to record plates' motion during the
Gondwana breakup.%
\end{abstract}%



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