Visualizing P-flats in N-space using parallel coordinates
John Scott Eickemeyer · 1992
A new parallel coordinate-based representation of p-flats (p-dimensional linear manifolds) in $R\sp{N}$ by $(N-p)\*(p+2)$ indexed points in the projective plane $P\sp2$ is found. Starting with points (0-flats), the representation is constructed recursively on the dimensionality of the flat. A collinearity algorithm enables the recursive construction of the representation of p-flats from their $(p-1)$-dimensional components. Based on this representation, the coefficients of the $(N-p)$ equations defining the p-flat are visible, and algorithms for translations, rotations, hyperplane intersections, and point membership queries are obtained. Algorithms for detection of coplanarity and multiple coplanarity of all orders in N-dimensional data (with or without noise) are given. The p-flat representation provides the foundation for an approach to representing polytopes and hypersurfaces. Algorithms in three and four dimensions for conflict detection and resolution in air traffic control are also obtained, based on the representation of certain quadric developable surfaces from their tangent planes.