A Qualitative Calculus for Three-Dimensional Rotations

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 x 3 rotation matrix. A system has been implemented that can solve the following problems: Given the signs of direction (upsilon) over cap and rotation matrix P, find the possible signs of the image of (upsilon) over cap under P. Moreover, for each possible sign vector of (upsilon) over cap . P, generate numerical instantiations of (upsilon) over cap and P that yields that result. Given the signs of rotation matrices P and Q, find the possible signs of the composition P Q. Moreover, for each possible sign matrix for the composition, generate numerical instantiations of P and Q that yield that result. 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.