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Qualitative spatial reasoning forms an important part of the commonsense reasoning required for building intelligent Geographical Information Systems (GIS). Previous research has come up with models to capture cardinal direction relations for typical GIS data. In this paper, we target the problem of efficiently computing the cardinal direction relations between regions that are composed of sets of polygons and present two algorithms for this task. The first of the proposed algorithms is purely qualitative and computes, in linear time, the cardinal direction relations between the input regions. The second has a quantitative aspect and computes, also in linear time, the cardinal direction relations with percentages between the input regions. Our experimental evaluation indicates that the proposed algorithms outperform existing methodologies. The algorithms have been implemented and embedded in an actual system, CARDIRECT, that allows the user to 1) specify and annotate regions of interest in an image or a map, 2) compute cardinal direction relations between them, and 3) pose queries in order to retrieve combinations of interesting regions. Index Terms--Spatial databases and GIS, cardinal direction relations, computing spatial relations.