Page 184 - Geometric Modeling and Algebraic Geometry

186 I. Ivrissimtzis and H.-P. Seidel
In all of the subdivision schemes we mentioned above, the eigenvalue of the con-
stant polygon C 0 is equal to one, the eigenvalues of the two regular affine polygons
C 1 ,C n−1 are real, positive and equal, while all the other eigenvalues have smaller
norms. That is,
1= λ 0 >λ 1 = λ n−1 > |λ i |, i =2, 3,...,n − 2 (10.7)

As the constant eigenpolygon corresponds to the eigenvalue with the largest
norm, it follows from Eq. 10.2 that in the limit the polygon converges to a single
point, which is its barycenter. The eigenpolygons with the next two largest eigenval-
ues λ 1 ,λ n−1 will determine the limit shape of the polygon. To define this limit shape
explicitly, we first assume that the barycenter of the polygon is the origin, eliminat-
ing thus the first component of the sum in Eq. 10.2 and then we scale the polygon by
a factor of 1/λ m to counter the shrinkage effect. Eq. 10.2 becomes
1
m m m
(10.8)
(λ 1 /λ 1 ) z 1 C 1 +(λ 2 /λ 1 ) z 2 C 2 + ··· +(λ n−1 /λ 1 ) z n−1 C n−1
giving
(10.9)
A = z 1 C 1 + z n−1 C n−1
as m →∞. Thus, the limit shape is the sum of two coplanar regular polygons with
opposite orientations. In particular, A is the affine image of a regular polygon and

can be inscribed in an ellipse with semi-axes |z 1 |+|z n−1 | and |z 1 |−|z n−1 | , see [5].

If the eigenvalues λ 1 ,λ n−1 are complex, as it is the case with the simplest scheme
proposed in [15], then we can still study the limit of Eq. 10.8, but in this case it might
not exist. In the literature, the simplest schemes is analyzed by combining two steps
to obtain a binary refinement step with real eigenvalues.
The geometric interpretation of Eq. 10.9 is simple and insightful, justifying, in
our opinion, the choice of complex rather than real eigenvectors. For example, we
can immediately see that A degenerates into a line when one of the two semi-axes
of the ellipse has zero length, that is, when the two regular affine components have
equal norms. In this case the subdivision surface will have a singularity at the point
of convergence of the polygon. More interestingly, we can detect a second type of
singularity by noticing that A has the same orientation as the component with the
largest norm, cf. Fig. 10.3. In fact, this second type of singularity has higher dimen-
sion than the first in the space of planar polygons, even though such badly shaped
non-convex polygons rarely appear in practical applications.
Notice that the comparison between the orientation of A and its two components
is possible because they are all coplanar polygons. In fact, they are all on the tan-
gent plane at the point of convergence of the initial polygon. In the case of planar
meshes, we can also compare between the orientation of the mesh, its faces and their
eigencomponents. In [7] it was shown an example of a polygon, which was part of
a planar, consistently oriented mesh without self-intersections, and the larger affine
regular component had orientation opposite to the mesh. That means that the limit
shape of the polygon also had orientation opposite to the mesh, inverting the direc-
tion of the normal at that point of the plane. Of course, these types of singularities