The power of parallel projection

Abstract

It is shown that from d projections of a k-flat into ( k+1)-dimensional linear subspaces one can still reconstruct the k-flat. Furthermore, we show that a k-flat and a j-flat intersect in d-space if and only if they intersect in ( d k+ j+1 ) linear subspaces of dimension ( k+ j+1) (and which are independent, of the k-flat and the j-flat). An application of these projection results is k-dimensional simplex searching for a set I of n points in d-dimensional space with a structure of size O( n k+1+ ε ) and O(log n) query time, for arbitrarily small positive ε. A second application is ray shooting in axis-parallel boxes in d-dimensional space, with a structure of size O( n 2+ ε ) and O(log n) query time.