  A straightforward and computationally efficient Consecutive Cubic Spline
(CCS) iterative algorithm is proposed for positioning the planar interface of
the unstructured geometrical Volume-of-Fluid method in arbitrarily-shaped
cells. The CCS algorithm is a two-point root-finding algorithm specifically
designed for the VOF interface positioning problem, where the volume fraction
function has diminishing derivatives at the ends of the search interval. As a
two-point iterative algorithm, CCS re-uses function values and derivatives from
previous iterations and does not rely on interval bracketing. The CCS algorithm
only requires only two iterations on average to position the interface with a
tolerance of $10^{-12}$, even with numerically very challenging volume fraction
values, e.g. near $10^{-9}$ or $1-10^{-9}$.
  The proposed CCS algorithm is very straightforward to implement because its
input is already calculated by every geometrical VOF method. It builds upon and
significantly improves the predictive Newton method and is independent of the
cell's geometrical model and related intersection algorithm. Geometrical
parametrizations of truncated volumes used by other contemporary methods are
completely avoided. The computational efficiency is comparable in terms of the
number of iterations to the fastest methods reported so far. References are
provided in the results section to the open-source implementation of the CCS
algorithm and the performance measurement data.
