Visualization of High-Dimensional Systems via Geometric Modeling with Homogeneous Coordinates

Abstract

A framework is presented for the visualization of high-dimensional systems, particularly multicomponent phase diagrams. It is based on geometric modeling with homogeneous coordinates of the transformations among geometric varieties in high-dimensional space. By reduction of the dimensionality of the system through such transforms, the resulting linear cuts and projections, in their aggregate, provide a mental picture of the system in its entirety. A general way to perform cuts and projections is presented, including how to calculate the transforms and how to define the projective space. Specifically, a procedure is presented for the development of a set of canonical coordinates for the projective space with reduced dimensions. Also, it is shown that the seemingly different transformed coordinates for reactive systems in the literature can be unified under our framework. Examples are provided to highlight the procedure and to demonstrate the visualization of phase diagrams for nonreactive and reactive systems.