Arrow graphs and their focal curves

TL;DR

This paper introduces and analyzes focal curves derived from arrow graphs of functions, exploring their relation to derivatives, projective geometry, and transformations, with illustrative examples and applications.

Contribution

It provides a detailed study of focal curves, linking them to derivatives and dual curves, and explores their behavior under transformations and compositions.

Findings

Focal curves relate to the derivative of the function.

Focal curves can be parametrized explicitly.

Focal curves correspond to dual curves in projective geometry.

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.