Partitioning using coordinate bisection is susceptible to the orientation
of the mesh, a simple rotation of the mesh would result in different
partition. Inertia bisection [54]
remedies this by using a procedure
that is invariant to rotation. It finds the principal axis of
the communication graph. Here the coordinates of the vertices of the
communication graph are defined as explained in Section 3.1.1.
If one assumes that each vertex has a unit mass associated with it, then
the principal axis is the axis that minimizes the
angular momentum when the graph rotates around it.

Consider a graph of vertices associated with a 3D mesh.
Let the principal axis be
with a point in space,
real numbers and
a unit vector. Then any vertex with coordinates
has an angular
momentum equal to the square of its distance to the axis, or

The principal axis minimizes the sum of these momentums:

subject to . It is possible to prove, using the necessary
condition of constrained optimization, that
and that is the eigenvector that
corresponds to the largest eigenvalue of the matrix

(1)

Once this eigenvector has been found,
every vertex can be sorted based on its
projection onto the principal axis, that is, the value
,
or simply .

The main computational cost of the inertia algorithm is the formation
of the inertia matrix (1), which requires of the order of
operations. The algorithm is easy to
parallelize [18,74].