  We present a numerical method for computing the evolution of a planar,
star-shaped curve under a broad class of curvature-driven geometric flows,
which we refer to as the Andrews-Bloore flows. This family of flows has two
parameters that control one constant and one curvature-dependent component for
the velocity in the direction of the normal to the curve. The Andrews-Bloore
flow includes as special cases the well known Eikonal, curve-shortening and
affine shortening flows, and for positive parameter values its evolution
shrinks the area enclosed by the curve to zero in finite time. A question of
key interest has been how the shape itself, as well as various shape
descriptors evolve as this limit is approached. Star-like curves can be
represented by a periodic scalar polar distance function $r(\varphi)$ measured
from a reference point. An important question is how the numbers and the
trajectories of critical points of the distance function $r(\varphi)$ and of
the curvature $\kappa(\varphi)$ evolve under the Andrews-Bloore flows for
different choices of the parameters. Our method is specifically designed to
meet the challenge of computing accurate trajectories of these critical points
up to the vicinity of a limiting shape. Each curve is represented by a
piecewise polynomial periodic distance function, as determined by a chosen
mesh; different types of meshes and mesh adaptation can be chosen to ensure a
good balance between accuracy and computational cost. As we demonstrate with
benchmark tests and two longer case studies, our method allows one to perform
numerical investigations into subtle questions of planar curve evolution. More
specifically we provide illustrations of some known results, numerical evidence
for two stated conjectures, as well as new insights and observations regarding
the limit shapes of initial curves with different types of symmetry.
