Unified Representation of Generalized Four-dimensional Geometric Element Definitions and Interferences for Five-dimensional Homogeneous Processing

Abstract

In previous studies, we constructed a theoretical framework via 5-D homogeneous processing to enable the uniform visualization of 4-D information in an interactive environment and succeeded in imparting generality to 4-D graphics. Furthermore, we developed advanced methods for 4-D geometric element definitions and interference via 5-D homogeneous processing. In the 4-D projective space, 5-D homogeneous processing can express a variety of transformations using 5×5 matrices and 5×5 determinants, which can be applied to all points, including points at infinity. Based on this perspective, I consider unified representations of point, line, plane, and hyperplane using 5-D homogeneous processing while investigating the geometric relations between the geometric element definitions and interferences in 4-D space. Subsequently, I describe strict duality of 4-D projective space. The proposed scheme provided more uniform geometric calculations than those described in our previous study.