  In many situations, the dual-purpose heater/thermometer wires used in the
three-omega method, one of the most precise and sensitive techniques for
measuring the thermal conductivity of thin films and interfaces, must include
bends and curves to avoid obstructions on the surface of a sample. Although the
three-omega analysis assumes that the heating wire is infinitely long and
straight, recent experimental work has demonstrated that in some cases
curved-wire geometries can be used without introducing detectable systematic
error. We describe a general numerical method that can be used to calculate the
temperature of three-omega heating wires with arbitrary wire geometries. This
method provides experimentalists with a simple quantitative procedure for
calculating how large the systematic error caused by a particular wire
asymmetry will be. We show calculations of two useful cases: a straight wire
with a single bend of arbitrary angle and a wire that forms a circle. We find
that the amplitude of the in-phase temperature oscillations near a wire that
forms a circle differs from the prediction using the analytic straight-line
source solution by $<$12%, provided that the thermal penetration depth is less
than ten times the radius of curvature of the wire path. The in-phase
temperature amplitude 1.5 wire widths away from a 90$^{\circ}$ bend in a wire
is within 11% of the straight-line source prediction for all penetration depths
greater than the wire width. Our calculations indicate that the straight-line
source solution breaks down significantly when the wire bend angle is less than
45$^{\circ}$.
