efficient algorithms for sorting geometrical shapes using trees

This paper proposes two new and efficient ways for sorting and querying simple geometrical shapessuch as intervals and circles. in both algorithms, we first introduce a map that converts each interval or acircle into a point by adding one dimension. then, an algorithm is presented for sorting intervals based onthe quad-tree and a new tree is introduced for searching circles. the advantage of the presented methodsin comparison with the conventional methods is that to build a tree, it is not necessary all objects havebeen defined at the first. with the article methods we can edit in the tree during the execution processes.