  Quaternions are important for a wide variety of rotation-related problems in
computer graphics, machine vision, and robotics. We study the nontrivial
geometry of the relationship between quaternions and rotation matrices by
exploiting the adjugate matrix of the characteristic equation of a related
eigenvalue problem to obtain the manifold of the space of a quaternion
eigenvector. We argue that quaternions parameterized by their corresponding
rotation matrices cannot be expressed, for example, in machine learning tasks,
as single-valued functions: the quaternion solution must instead be treated as
a manifold, with different algebraic solutions for each of several
single-valued sectors represented by the adjugate matrix. We conclude with
novel constructions exploiting the quaternion adjugate variables to revisit
several classic pose estimation applications: 2D point-cloud matching, 2D
point-cloud-to-projection matching, 3D point-cloud matching, 3D orthographic
point-cloud-to-projection matching, and 3D perspective
point-cloud-to-projection matching. We find an exact solution to the 3D
orthographic least squares pose extraction problem, and apply it successfully
also to the perspective pose extraction problem with results that improve on
existing methods.
