  {\em Riemannian cubics} are curves in a manifold $M$ that satisfy a
variational condition appropriate for interpolation problems. When $M$ is the
rotation group SO(3), Riemannian cubics are track-summands of {\em Riemannian
cubic splines}, used for motion planning of rigid bodies. Partial integrability
results are known for Riemannian cubics, and the asymptotics of Riemannian
cubics in SO(3) are reasonably well understood. The mathematical properties and
medium-term behaviour of Riemannian cubics in SO(3) are known to be be
extremely rich, but there are numerical methods for calculating Riemannian
cubic splines in practice. What is missing is an understanding of the
short-term behaviour of Riemannian cubics, and it is this that is important for
applications. The present paper fills this gap by deriving approximations to
nearly geodesic Riemannian cubics in terms of elementary functions. The high
quality of these approximations depends on mathematical results that are
specific to Riemannian cubics.
