  We present a new methodology for calculating the electromagnetic radiation
from accelerated charged particles. Our formulation --- the `endpoint
formulation' --- combines numerous results developed in the literature in
relation to radiation arising from particle acceleration using a complete, and
completely general, treatment. We do this by describing particle motion via a
series of discrete, instantaneous acceleration events, or `endpoints', with
each such event being treated as a source of emission. This method implicitly
allows for particle creation/destruction, and is suited to direct numerical
implementation in either the time- or frequency-domains. In this paper, we
demonstrate the complete generality of our method for calculating the radiated
field from charged particle acceleration, and show how it reduces to the
classical named radiation processes such as synchrotron, Tamm's description of
Vavilov-Cherenkov, and transition radiation under appropriate limits. Using
this formulation, we are immediately able to answer outstanding questions
regarding the phenomenology of radio emission from ultra-high-energy particle
interactions in both the Earth's atmosphere and the Moon. In particular, our
formulation makes it apparent that the dominant emission component of the
Askaryan Effect (coherent radio-wave radiation from high-energy particle
cascades in dense media) comes from coherent `bremsstrahlung' from particle
acceleration, rather than coherent Vavilov-Cherenkov radiation.
