Generalized Andrews Curves as Linear Operators: Classification of Isometries, Distortion Calculus, and a Controlled Comparison of Attribute-Preserving Visualizations

Generalized Andrews curves turn a table of measurements into a family of functions, and the choice of functional basis has largely been made by taste. This paper treats the transformation as a linear operator and proves a classification theorem: the isometric transformations into a fixed subspace form exactly the orbit of a single orthonormal basis under the orthogonal group, and parallel coordinates fall out of the same family as its piecewise linear member. On that footing we build a distortion calculus that prices a design choice before anything is drawn. The multiplicative distortion of distances is governed by the condition number of a small metric matrix; any weighting that changes the picture necessarily deforms the metric; and the distance-preservation error becomes a predicted quantity with a closed-form bound. For parallel coordinates the distortion proves independent of both the data and the dimension, fixed by the choice of axis convention alone. An empirical study on nine public datasets compares seven bases and, within the same class of techniques, parallel coordinates, RadViz and Star Coordinates. No technique dominates: we separate the apparent advantages that are measured from those that follow structurally from the design, and report the ordering of bases descriptively, since a benchmark of this size cannot separate the isometric candidates.