Properties of normalized radial visualizations

This paper defines and establishes properties of a class of normalized radial visualizations (NRVs) that includes the RadViz mapping onto the two-dimensional unit disk. A NRV normalizes data to map highdimensional records into lower dimensional space, where records’ images are convex combinations of points called dimensional anchors. NRVs are radial visualizations because dimensional anchors are specified using hyperspherical coordinates. As RadViz and related radial visualizations have evolved, conjectures have been proposed for invariants, such as lines map to lines, convex sets map to convex sets, and spheres map to ellipses. We establish several properties for NRVs and experimentally illustrate them using RadViz. For example, a NRV mapping can be viewed as a composition of a perspective and an affine transformation, which yields a projective transformation. A number of properties emerge from these characterizations. In addition to lines mapping to lines, point-line incidence, points of intersection, and convexity are preserved, hyperspheres and ellipsoids map to ellipsoids, and betweenness holds. These insights suggest directions for future NRV work. We explore one idea: rotational preprocessing to separate certain data in RadViz. This is important for cluster visualization because if two clusters are separated by a hyperplane in data space, then we would like their images to remain separated in the radial visualization. Because the properties we establish hold regardless of whether or not the dimensional anchors lie on the RadViz unit circle in image space, future heuristics may exploit this freedom to enhance cluster separation with guarantees from the transformational characterization.