Page 193 - Geometric Modeling and Algebraic Geometry

10 Cube Decompositions 195
of the prism is determined by five components corresponding to the second largest
eigenvalue λ 1 . These are the components [C 1 ,C 1 ] and [C n−1 ,C n−1 ]
These are the components [C 1 ,C 1 ] and [C n−1 ,C n−1 ] on the planes E A and E B ,
which are four in total, and the component corresponding to nth row of the matrix in
Eq. 10.22. The eigenvalue λ n of this component should also be equal to λ 1 . Other-
wise, the ratio between the height of the prism and the diameter of its base will tend
to0orto ∞, depending on whether λ n is smaller or larger than or λ 1 .
Eq.(10.25) gives

[ (a 1 + b 1 )C 1 +(a n−1 + b n−1 )C n−1 (a 1 + b 1 )C 1 +(a n−1 + b n−1 )C n−1 ]
,
2 2
(10.29)
where the use of the letter a or b in the coefficient also indicates the plane of the
component. We notice that A and B have the same limit shape. Moreover, the limit
shape of A, B is planar and thus, the limit shape of the prism is regular. Fig. 10.9
shows the evolution of a pentagonal prism under this subdivision scheme. An outline
for the explicit computations of A and B is shown in the Appendix.

10.6 Conclusion - Future Work

We studied decompositions of cubes and prisms by the eigenvectors of G-circulant
matrices. We concentrated on the geometric interpretations of these decompositions
and we studied the evolution of single cells under linear transformations. As an ap-
plication we obtained information about the singularities in quadratic n-dimensional
splines.
In the future we plan to extend our work to the study of evolutions of larger
configurations, instead of the single cells we are currently dealing with. Such a gen-
eralization will allow the study of singularities in higher degree splines and general
volume subdivision grids.

Fig. 10.9. The evolution of a pentagonal prism.