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A Qualitative Calculus for Three-Dimensional Rotations
Authors:Azam Asl  Ernest Davis
Affiliation:1. Department of Computer Science , New York University , New York , NY , USA;2. Dept. of Computer Science , New York University , New York , NY , USA
Abstract:We have developed a qualitative calculus for three-dimensional directions and rotations. A direction is characterized in terms of the signs of its components relative to an absolute coordinate system. A rotation is characterized in terms of the signs of the components of the associated 3 × 3 rotation matrix.

A system has been implemented that can solve the following problems: 1. Given the signs of direction /></span> and rotation matrix <i>P</i>, find the possible signs of the image of <span class= /></span> under <i>P</i>. Moreover, for each possible sign vector of <span class= /></span> · <i>P</i>, generate numerical instantiations of <span class= /></span> and <i>P</i> that yields that result.</p> 2. Given the signs of rotation matrices <i>P</i> and <i>Q</i>, find the possible signs of the composition <i>P</i> · <i>Q</i>. Moreover, for each possible sign matrix for the composition, generate numerical instantiations of <i>P</i> and <i>Q</i> that yield that result.</p> </p> <p xmlns:mml=We have also proved some related complexity and expressivity results. The satisfiability problem for a qualitative rotation constraint network is NP-complete in two dimensions and NP-hard in three dimensions. In three dimensions, any two directions are distinguishable by a qualitative rotation constraint network.
Keywords:qualitative spatial reasoning  qualitative calculus  three-dimensional rotation
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