# Evolving geometry of a vortex triangle

## TL;DR

This paper reformulates the three-vortex problem in the plane using evolving geometric quantities like the circumcircle and triangle angles, providing new insights into vortex dynamics.

## Contribution

It develops equations of motion for the vortex triangle's circumcircle and angles, offering a novel geometric perspective on vortex interactions.

## Key findings

- Derived autonomous dynamical system for geometric quantities
- Revealed new geometric insights into vortex motion
- Connected known results to geometric evolution

## Abstract

The motion of three interacting point vortices in the plane can be thought of as the motion of three geometrical points endowed with a dynamics. This motion can therefore be re-formulated in terms of dynamically evolving geometric quantities, viz. the circle that circumscribes the vortex triangle and the angles of the vortex triangle. In this study, we develop the equations of motion for the center, $Z$, and radius, $R$, of this circumcircle, and for the angles of the vortex triangle, $A$, $B$, and $C$. The equations of motion for $R$, $A$, $B$ and $C$ form an autonomous dynamical system. A number of known results in the three-vortex problem follow readily from the equations, giving a new geometrical perspective on the problem.