This paper defines a class of normalized radial visualizations (NRVs) that includes the RadViz mapping onto the two-dimensional unit disk. An NRV maps high-dimensional records into lower dimensional space, where records' images are convex combinations of the dimensions (called dimensional anchors) laid out in two dimensions as labels on a circle and in higher dimensions on the surface of a hypersphere. As radial visualizations have evolved, conjectures have been proposed for invariants, such as lines mapping to lines, and convex sets to convex sets. Some have been informally proven for RadViz. We formally establish these properties for all NRVs and illustrate them using RadViz. An extensive theory of Parallel Coordinates has been developed elsewhere with great benefit to the visualization community. Our theory should provide similar benefits for radial visualization users. We show that an NRV is the composition of a perspective and an affine transformation. This projective transformation characterization leads to a number of properties including line, point ordering and convexity invariance. Knowledge of these properties suggests that the visual existence of structure in the data can guide a visualization researcher in further productive exploration of the data. We show the established properties hold regardless of whether or not the dimensional anchors lie on the circle or the hypersphere. These insights also suggest directions for future NRV work, such as rotational preprocessing to separate data in RadViz and NRVs for better cluster visualization.
