  The Neutron Star Interior Composition Explorer (NICER) mission has provided a
unique opportunity to constrain the equation of state of neutron stars by using
the technique of pulse-profile modelling. This technique requires accurate and
efficient ray tracing, that in turn requires a robust representation of the
spacetime around a neutron star. Several exact and approximate metrics have
been proposed, and used, to perform ray tracing around neutron stars, with both
moderate and fast rotation. In this paper, we perform a comparison between
several of these metrics, when used for ray tracing. We calculate the shape of
the neutron star as seen by a distant observer using two different surface
formulae, the thermal spectrum and pulse profiles from circular and
crescent-shaped hotspots, for four configurations of pulsars with rotation
rates ranging from 622 to 1000 Hz, and using both a moderate and a stiff
equation of state to include realistic and extreme cases. We find small
differences between the metrics for rotation frequencies starting at ~700 Hz
that could theoretically be used for constraining the quadrupole moment or the
spacetime models. We also determine the practicality of use of each metric in
larger-scale applications such as pulse-profile
