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# The Disappearing Second Derivative of Quadratics: Perceptual, Mathematical, and Statistical Properties of Judging Dependence on Visual Displays

When y is plotted against x to see how y depends on x, whether it is data or a function that is displayed, the aspect ratio (height/width) is a critical factor in our ability to visually decode information about the dependence. Dependence is in part decoded by judging the slopes of line segments: the local segments of a curve that is displayed or of a virtual curve that forms from the underlying pattern of displayed data points. A change in the aspect ratio changes the orientations of the physical slopes of the segments, which in turn changes our ability to visually decode slope to judge the rate of change of y with x.

Work by Cleveland and McGill showed that we can greatly enhance our ability to judge rate of change by "banking to 45 degrees": choosing the aspect ratio to center the absolute orientations of the segments on 45 degrees. R.A. Fisher, who founded modern statistics along with mathematical genetics, seems to have understood this result for a special case.

The question is a definition of centering. Banking algorithms have been put forward including some very interesting recent work of Heer and Agrawala.

Performance studies of the algorithms in the past have been purely empirical.

Segments are generated, algorithms applied, and the resulting distribution of segment orientations assessed.

We are studying the properties of banking algorithms theoretically using visual perception, geometry, and statistics. Mathematical descriptions of the curvature that the human visual system perceives elucidates why we see what we do on data displays such as a graph of a quadratic polynomial. We discovered that a geometrically motivated banking algorithm, resultant-vector banking, leads to simple formulas for the aspect ratio of a banked displayed that are tractable and enable mathematical and statistic investigations of properties.

Moving from the discrete case of past work on a finite set of line segments to the continuous case provides additional mathematical insights.

Joint work with Saptarshi Guha, Department of Statistics, Purdue University