The precise electrical patterns that arise in the heart during rhythm disorders are incompletely understood. Mathematically, arrhythmia patterns are solutions of a set of coupled partial differential equations of the reaction-diffusion type, which represent the local excitation dynamics of the tissue and their spatial coupling. While usually a numerical approach is required, I show that it is possible to also derive generic equations of motion for the coherent structures (fronts, vortices, filaments) in the tissue. Their dynamics is driven by curvature and the resulting equations are universal to all excitable media, but with system-dependent coefficients. Basic bifurcations in terms of these coefficients, such as filament tension, are explained. Lastly, we look into the structure of vortex core in the linear-core regime, which is often observed in cardiac experiments and simulations. Recently, it was demonstrated that these cores are equivalent to a mathematical branch cut, rather than a phase singularity in the classical theory. Implications for the extension of the geometric dynamics to this case are discussed.