For more than a century, rigid-body displacements have been viewed as affine transformations described as homogeneous transformation matrices wherein the linear part is a rotation matrix. In group-theoretic terms, this classical description makes rigid-body motions a semi-direct product. The distinction between a rigid-body displacement of Euclidean space and a change in pose from one reference frame to another is usually not articulated well in the literature. Here we show that, remarkably, when changes in pose are viewed from a space-fixed reference frame, the space of pose changes can be endowed with a direct product group structure, which is different from the semi-direct product structure of the space of motions. We then show how this new perspective can be applied more naturally to problems such as monitoring the state of aerial vehicles from the ground, or the cameras in a humanoid robot observing motions of its hands.

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