Physically, it seems possible that different parts of an edge could move at different speeds. Since a moving edge is actually composed of atomic ledges, the situation could arise that many ledges gather in one place, effectively splitting an edge in two on a large scale. Because all edges must retain fixed normal directions, we refer to this process as stepping in which a single edge splits into two edges having the same normal direction connected by a small edge having a different normal. Below, we rigourously define a stepping, and define conditions under which such steps would be introduced.
A stepping
as a point at which a vanishingly small facet 
is introduced
in the middle of a facet 
to produce
three facets 
, and 
such that 
have the same normal direction as the
initial edge. This zero-length facet will have 
and
, while 
. 
will be
determined by the sense of the stepping
(i.e. whether the
direction of precedes or follows the direction of in the
Wulff shape), and the types of corners at the ends of (i.e.
whether the corners are regular or inverse).
We are interested in computing motions that are stable, in the sense
that small perturbations of the surface at some time should not result in
differences that grow in time. In our crystalline model, perturbations
are accomplished by steppings, and we need to ensure that any stepping
that will grow will be made.
We say that a particular configuration of zero and non-zero-length
edges at time 
has its associated potential (defined according to
the equations of Section 2) stable if no zero-length facet has
infinite velocity, if all zero-length facets become of
positive length for some immediately subsequent interval of time
when motion is given by the equations of Section 2, and if
no additional zero-length edges can be inserted while maintaining
these conditions. This last statement is not one that is easy to
check, however. We give below criteria which are easy to check, and
which imply the statements above if it is true (as we believe) that
there is a unique collection of steppings and resulting potential
which is stable for any initial polygon with no zero-length edges.
We distinguish two types of stepping events for motions which are
governed at all times by stable potentials:
those events which take place at topological times (including
the initial time 
) where
the structure of the interfaces changes in some topological way and thereby
those that take place at regular times where one or more facets
find it advantageous to step in to several pieces.
An algorithm for finding a stepping which results in a stable
potential at topological times is described in section 4.
At both types of times, in order for the velocity of a zero-length
edge to be
finite, Equation 6 implies that 
and Equation 10 implies that 
(since
is of order 
).
Also, at both types of times, the zero-length edge is required to grow
in time.
If such a stable point exists, then it is also required that
the facet grow in order for to appear.
At a regular time 
, there is a potential 
which is the limit
of the potentials as time comes up to .
We assert that the potential should be continuous at - that the
potentials computed after inserting a zero-length step should be the
same as that inherited from . Thus
a stepping should be considered at
any point where the potential is zero, and where a sense can be
chosen so that the velocities
computed for the resulting neighbors 
and 
(using
the appropriate values for their average potentials) are the
same as that of the unstepped edge. Alternatively, this condition
could be considered as the neighboring edges having the correct
average potentials when they are assumed to have the potential as that
inherited from the unstepped edge. This continuity of potential over
regular times follows from the assumption of stability at previous
times: if a step can be inserted at a point where 
and where
the velocities of the neighbors become different and related to each
other in such a way that the step will grow, then such a step would
also have been favorable at some previous time, and not inserting it
then would violate the hypothesis of the surface and its potential
being stable at all times.
Thus the criterion for stepping is:
From the current , create steps
on facet at the zeros of , and for each sense for the steps,
determine if the averages
of 
on the resulting facets have the
correct values given by their weighted mean curvatures. For example,
if there is one zero of and it occurs at 
, then check
whether
If this is true, then check whether
the length of the inserted segments will grow in time, and if so, make
that stepping. On the other hand, if 
and
then a stepping there would have facets 
and 
being
driven apart with an initially positive
rather than zero velocity, and so one must be at a topological time.
In this case, the potential cannot
be continuously converted into a stable potential, and an entirely new
potential must be computed. If 
and
then facets 
and 
would be driven back together and
that stepping should not be made. Equivalent (and consistent)
statements hold in the case that 
. In case
and the sense of the stepping is such that
, then such a stepping can only be
good if 
or 
(and then only if it will grow in time, an
additional condition to check as always).
An equivalent and clearer approach is to use
the Cahn-Hoffman
vector formulation. should have the properties that
for each corner 
of the surface, the value of is the vector which is the corner of
the Wulff shape 
corresponding to the orientations
and 
, and so that 
along
each facet.
Because the potential computed according to Section 2 has the property
that 
, where 
is
the length of the facet in with normal 
, the so
computed is single-valued and continuous.
We see that is thus uniquely determined by the surface (including
any zero-length edges) and its associated .
In particular, note that the value of at the two endpoints of a
facet is the same.
Critical points
of will always occur at the zeroes of , since is the
derivative of .
If this has the property that except on corners of
the surface is always within
the interior of a facet of , then in particular any critical point
of the is, and this implies that the potential
is stable.
The value of at a zero of
being in the interior of a facet of means that
the velocities of the segments computed by making a stepping at the
zero and the correct averages for the potentials
would be different from each other and would be such as to drive the two
segments back together).
If ever extends beyond the ends of the
appropriate facet, then the is definitely not stable; this is a
case where the velocities computed from the correct averages of the
potentials would drive the two
facets apart with positive velocity and mean that a topological
change has just occurred (since otherwise the previously computed
potentials could not have been stable).
Such bad- potentials are in fact
an indication that steppings
must be made at those topological times and the potential recomputed.
If a
critical point occurs at value of which is on a corner of
, one should consider a stepping at that point, and again, the stepping should be kept only if the zero-length segment becomes of nonzero length
as time progresses.
The vector is particularly easy to visualize, since its
dependence of position along a facet is cubic (being the integral of
the quadratic potential).
Motion by surface diffusion should also be gradient flow
in the 
inner product [9].
Given functions 
,
with 
,
their 
inner product 
is
, where 
is the inverse
Laplacian of - that is, satisfies 
.
In our case, is a set of velocities 
for facets,
and 
is the condition 
;
is the potential , computed so that the
continuity conditions hold.
(Such a potential is determined only up
to a constant 
, but the constant does not matter for the minimization
since 
.)
If is motion by gradient flow in this inner product, it should
minimize 
[10].
For the case of
crystalline curves being considered in this paper, this means that
over all possible velocities and steps we should minimize
If minimizes this quantity, and if we use
to mean velocity is 1 on segment 
and 0 on
other segments, then for each ,
This reduces to the previous condition 13:
On the other hand, if by introducing additional zero-length
steps and recomputing
we can reduce 21 further, we should do it.
In particular, introducing further steps when one has a potential
which is not stable does so reduce the above sum.
The
optimal stepping has been achieved if and only if the sum cannot be further
reduced.
To show that this 
gradient flow
approach is the same as the previous two, it would
be sufficient to prove that there is a unique stable potential for
the boundary of any given region bounded by a crystalline curve.