Page 192 - Geometric Modeling and Algebraic Geometry

194 I. Ivrissimtzis and H.-P. Seidel
Instead, we build a construction based on the decompositions of non-planar poly-
gons proposed in [2]. Recall that every non-planar n-gon can be written in the form
shown in Eq. 10.6. These polygons generally lie on different planes but C j and C n−j
are always coplanar.
Let A, B be the two n-gonal faces of the prism and let their decompositions be

A = a 0 C 0 + a 1 C 1 + ··· + a n−1 C n−1
(10.23)
B = b 0 C 0 + b 1 C 1 + ··· + b n−1 C n−1
Notice that the copies of C i used in the decomposition of A and B generally lie
on different planes of the 3-dimensional space. By concatenating the two polygonal
decompositions we get a prism decomposition
[a 0 C 0 ,b 0 C 0 ]+[a 1 C 1 ,b 1 C 1 ]+ ··· +[a n−1 C n−1 ,b n−1 C n−1 ] (10.24)

For simplicity we only deal with the components C 1 ,C n−1 , as all the other conjugate
pairs, as well as the single components C 0 and C n/2 (for n even) can be treated
similarly.
Let E A be the plane of the components a 0 C 0 ,a k−1 C k−1 lie and let E B be the
plane of the components b 0 C 0 ,b k−1 C k−1 . Working first on the E A plane we write
the components [a 1 C 1 , 0] and [a n−1 C n−1 , 0] as a linear combination of the four
eigenprisms of Fig. 10.8, i.e. as

x 1 [C 1 ,C 1 ]+x n−1 [C n−1 ,C n−1 ]+x n+1 [C 1 , −C 1 ]+x 2n−1 [C n−1 , −C n−1 ] (10.25)
We get x 1 C 1 + x n+1 C 1 = a 1 C 1

x 1 C 1 − x n+1 C 1 =0 · C 1
x n−1 C n−1 + x 2n−1 C n−1 = a n−1 C n−1
(10.26)
x n−1 C n−1 + x 2n−1 C n−1 =0 · C n−1
giving
x 1 = x n+1 = a 1 x n−1 = x 2n−1 = a n−1 (10.27)
2 2
Similarly, working with the components [0,b 1 C 1 ] and [0,b n−1 C n−1 ] we get four
more eigenprisms, this time on the E B plane, with

x 1 = x n+1 = b 1 x n−1 = x 2n−1 = b n−1 (10.28)
2 2
We notice that the obtained decomposition is quite heavy as we use eight eigenprisms
for just four polygonal components. However, the eigenvalue λ 1 corresponding to the
[C 1 ,C 1 ] and [C n−1 ,C n−1 ] components is usually larger than the eigenvalue of the
[C 1 , −C 1 ] and [C n−1 , −C n−1 ] components. Thus, the limit shape of the prism will
be determined by fewer than eight components.
Indeed, this is the case with the eigenvalues corresponding to the tensor product
of the Doo-Sabin subdivision rule. Under this subdivision scheme, the limit shape