Akihiro SUGIMOTO
Projective Invariants of Intersections of Hyperplanes
in the n-dimensional Projective Space
Abstract:We often treat information that was projected into a subspace from a space where the original information exists. For example, visual information is information that was projected onto the retina from the 3-dimensional Euclidean space. Because there is a deficiency of information caused by the projection, we can not uniquely recover the original information in general. Therefore, it is definitely important to find properties, if any, that essentially connect the original information with the projected information. When a class of admissible transformations to which the original information is subject is specified, projective invariants, which are real-valued functions in terms of the projected information and which are unaffected by the class of admissible transformations, provide an essential relationship between the original information and the projected one. This paper is a study on projective invariants under the condition that the n-dimensional projective space is projected into the (n-1)-dimensional projective space by the projection of a certain class; and that the class of admissible transformations involves projective transformations in the n-dimensional projective space. It is shown that, for given integers i and j such that 1≦i≦j≦n-i, we have a projective invariant derived from (n+i+j) subspaces of (n-2) dimensions, where the (n+i+j) subspaces are the intersections of the adjacent hyperplanes of (n+i+j+1) hyperplanes arranged in the letter H. The nonsingularity condition, i.e., the condition under which the invariant is nonsingular, is also given.
Key Words: projective invariants, admissible transformations, interpretation vector, intersections
of hyperplanes, nonsingularity condition.