Let 
be a surface energy density function such that the Wulff
shape
is a polygon having 
edges, called facets in this paper,
of lengths 
as is traversed in a
positive (counter-clockwise) direction. The directions of the unit
normals 
, common to
both the surface at time 
, 
, and ,
are defined as pointing away from the region which we will denote
as ``solid.''
(We will use the terminology ``surface'' in spite of
the fact that we are dealing with 1-dimensional polygonal curves.)
Initial data is given as the set, 
, which is the area covered by the
crystal.
The surface 
is a collection of
non-intersecting polygons which are the boundary of
such that each ``facet'' (edge) 
is parallel
to some facet of , and that if
and 
are consecutive facets in , they have normal
directions that are adjacent, consecutive in either direction, in
. We shall see that this ``adjacency'' condition is preserved.
is oriented so that the ``solid''
region is to the left as the interface is traversed in the positive
direction.
Note that some of the polygons may be completely contained within
other polygons (indicating the existence of `holes' in the solid),
and the surface may be quite complicated, having a very large number
of facets some of which have zero length.
We will focus on one of these polygons, a simply closed
surface with 
facets with positive
finite lengths 
.
In the usual case, motion of each facet is parallel to itself,
with velocity 
, that
is positive if it is in the direction of 
.
Sometimes, a single facet will break into two or more facets
by the insertion of one or more steps
(short facets of neighboring orientations).
will always
be polygonal, but the number of facets may change due to
disappearance of facets or insertion of steps
according to rules
that we will discuss in Section 3.
The weighted curvature 
of facet
is inversely proportional to 
[4]:
where 
is a ``convexity'' factor that is
, 
, or 
depending on how abuts its neighbors: Fig. 1.
If both ends of are regular, i.e. the corner is a left
turn
proceeding around the surface counter-clockwise, then 
since
the change in energy by moving
in the direction of 
is positive.
If both are inverse (both right turns), then 
.
If one is regular
and the other inverse, then 
; because of the adjacency
condition, the energy gained at
one end by the motion of is exactly compensated by
a loss at the other end. Note that the surface energy factor in this
equation
is 
instead of 
because the
change
in surface energy with volume due to motion of the facet is
which also happens to be the formula to compute .)