  Geometric phase has historically been defined using closed cycles of
polarization states, often derived using differential geometry on the Poincare
sphere. Using the recently-developed wave model of geometric phase, we show
that it is better to define geometric phase more generally, allowing every
polarized wave to have a well-defined value at any point in its path. Using
several example systems, we show how this approach provides more insight into
the wave's behavior. Moreover, by tracking the continuous evolution of
geometric phase as a wave propagates through an optical system, we encounter a
natural explanation of why the conventional Poincare sphere solid angle method
uses geodesic paths rather than physical paths of the polarization state.
