Geometries of four vortex relative equilibria

Four interacting point vortices can form rigidly moving patterns called relative equilibria, yet a complete understanding of the possible geometrical arrangements has been elusive. Using a parametric formulation and configuration matrices, we mapped the locations of the fourth vortex that form relative equilibria for a fixed triangular arrangement of the first three vortices. The results reveal bounded and unbounded continua of vortex positions, uncovering families of relative equilibria and rich bifurcation structures in the configuration space.