Evolving geometry of a vortex triangle

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.