  A theoretical framework for emission originating from rapidly rotating oblate
compact objects is described in detail. By using a Hamilton-Jacobi formalism,
we show how the special relativistic rotational effects such as aberration of
angles, Doppler boosting, and time dilatation naturally emerge from the general
relativistic treatment of rotating compact objects. We use the
Butterworth-Ipser metric expanded up to the second order in rotation and hence
include effects of light bending, frame-dragging, and quadrupole deviations to
our geodesic calculations. We also give detailed descriptions of the numerical
algorithms used and provide an open source implementation of the numerical
framework called bender. As an application, we study spectral line profiles
(i.e., smearing kernels) from rapidly rotating oblate neutron stars. We find
that in this metric description the second order quadrupole effects are not
strong enough to produce narrow observable features in the spectral energy
distribution for almost any physically realistic parameter combination, and
hence, actually detecting them is unlikely. The full width at tenth-maximum and
full width at half-maximum of the rotation smearing kernels are also reported
for all viewing angles. These can be then used to quantitatively estimate the
effects of rotational smearing on the observed spectra. We also calculate
accurate pulse profiles and observer skymaps of emission from hot spots on
rapidly rotating accreting millisecond pulsars. These allow us to quantify the
strength of the pulse fractions one expects to observe from typical fast
spinning millisecond pulsars.
