  This series of papers comprises a systematic exploration of the hypothesis
that the far ultraviolet radiation from star clusters and elliptical galaxies
originates from extremely hot horizontal-branch (HB) stars and their post-HB
progeny. This first paper presents an extensive grid of calculations of stellar
models from the Zero Age Horizontal Branch through to a point late in post-HB
evolution or a point on the white dwarf cooling track. We use the term `Extreme
Horizontal Branch' (EHB) to refer to HB sequences of constant mass that do not
reach the thermally-pulsing stage on the AGB. These models evolve after core
helium exhaustion
