A method of spatial reasoning based on qualitative trigonometry

Due to the lack of exact quantitative information or the difficulty associated with obtaining or processing such information, qualitative spatial knowledge representation and reasoning often become an essential means for solving spatial constraint problems as found in science and engineering. This paper presents a computational approach to representing and reasoning about spatial constraints in two-dimensional Euclidean space, where the a priori spatial information is not precisely expressed in quantitative terms. The spatial quantities considered in this work are qualitative distances and qualitative orientation angles. Here, we explicitly define the semantics of these quantities and thereafter formulate a representation of qualitative trigonometry (QTRIG). The resulting QTRIG formalism provides the necessary inference rules for qualitative spatial reasoning. In the paper, we illustrate how the QTRIG relationships can be employed in generating qualitative spatial descriptions in two-dimensional Euclidean geometric problems, and furthermore, how the derived qualitative spatial descriptions can be used to guide a simulated-annealing-based exact quantitative value assignment. Finally, we discuss an application of the proposed spatial reasoning method to the kinematic constraint analysis in computer-aided pre-parametric mechanism design.