  Considerable attention has been devoted to microscopic heat engines in both
theoretical and experimental aspects. Notably, the fundamental limits
pertaining to power and efficiency, as well as the tradeoff relations between
these two quantities, have been intensively studied. This study aims to shed
further light on the ultimate limits of heat engines by exploring the
relationship between the geometric length along the path of cyclic heat engines
operating at arbitrary speeds and their power and efficiency. We establish a
tradeoff relation between power and efficiency using the geometric length and
the timescale of the heat engine. Remarkably, because the geometric quantity
comprises experimentally accessible terms in classical cases, this relation is
useful for the inference of thermodynamic efficiency. Moreover, we reveal that
the power of a heat engine is always upper bounded by the product of its
geometric length and the statistics of energy. Our results provide a geometric
characterization of the performance of cyclic heat engines, which is
universally applicable to both classical and quantum heat engines operating far
from equilibrium.
