Periodic orbits close to grazing for an impact oscillator

We show how the geometric impact surface approach to the dynamics of an impact oscillator provides an immediate visualisation of the criteria that determine the existence of an impacting periodic orbit close to grazing. We recover the criteria set out earlier by A. Nordmark, and indicate how the geometric setting and singularity geometry may be exploited to yield appropriate criteria in degenerate situations where the Nordmark criteria would not appl