Spatial join finds pairs of spatial objects having a specific spatial relationship in spatial database systems. A number of spatial join algorithms have recently been proposed in the literature. Most of them, however, perform the join in the original space. Joining in the original space has a drawback of dealing with sizes of objects and thus has difficulty in developing a formal algorithm that does not rely on heuristics. In this paper, we propose a spatial join algorithm based on the transformation technique. An object having a size in the two-dimensional original space is transformed into a point in the four-dimensional transform space, and the join is performed on these point objects. This can be easily extended to n-dimensional cases. We show the excellence of the proposed approach through analysis and extensive experiments. The results show that the proposed algorithm has a performance generally better than that of the R*-based algorithm proposed by Brinkhoff et al. This is a strong indicating that corner transformation preserves clustering among objects and that spatial operations can be performed better in the transform space than in the original space. This reverses the common belief that transformation will adversely affect clustering. We believe that our result will provide a new insight towards transformation-based spatial query processing.

Spatial access methods (SAMs) are often used as clustering indexes in spatial database systems. Therefore, a SAM should have the clustering property both in the index and in the data file. In this paper we argue that corner transformation preserves the clustering property such that objects having similar sizes and positions in the original space tend to be placed in the same region in the transform space. We then show that SAMs based on corner transformation are able to maintain clustering both in the index and in the data file for storage systems with fixed object positions and propose the MBR-MLGF as an example to implement such an index. Extensive experiments comparing with the R*-tree show that corner transformation indeed preserves the clustering property. This result reverses the common belief that transformation will adversely affect the clustering.

Ray tracing requires many ray-object intersection tests. A way of reducing the number of ray-object intersection tests is to subdivide the space occupied by objects into many nonoverlapping subregions, called voxels, and to construct an octree for the subdivided space. We propose the Octree-R, an octree-variant data structure for efficient ray tracing. The algorithm for constructing the Octree-R first estimates the number of ray-object intersection tests. Then, it partitions the space along the plane that minimizes the estimated number of ray-object intersection tests. We present the results of experiments for verifying the effectiveness of the Octree-R. In the experiment, the Octree-R provides a 4% to 47% performance gain over the conventional octree. The result shows the more skewed the object distribution (as is typical for real data), the more performance gain the Octree-R achieves.