Arrow graphs and their focal curves

Abstract

The arrow graph of a function consists of two parallel axes, with arrows from input values to output values. The lines through these arrows envelop a curve which we named the focal curve. This paper studies these focal curves in detail. We show how the focal curve relates to the derivative of the function, and how this also provided a parametrization of the focal curve. This is illustrated through various examples, and can be experienced in linked GeoGebra-applications. Next, we provide an interpretation of focal curves in projective geometry, relating the focal curve to the dual curve of the graph of the function. Finally, we provide some results on the behavior of focal curves under transformations and compositions.