  Planar curves with periodically varying curvature arise in the natural
sciences as the result of a wide variety of periodic processes. The total
curvature of a periodic arc in such curves constrains their symmetry. It is
shown how the total curvature can be computed without reparameterizing the
curve to unit speed. The use of the total curvature of the periodic arcs is
demonstrated through a series of four examples from various branches of
science. Insights gained from these examples are applied to improve the
modeling of spiral wave meander.
