Splines are piecewise polynomial functions of a given order of smoothness r on a triangulated region (or polyhedrally subdivided region) of Rd. The set of splines
of degree at most k forms a vector space Crk() Moreover, a nice way to study
Cr
k()is to embed n Rd+1, and form the cone b of with the origin. It turns
out that the set of splines on b is a graded module Cr b() over the polynomial ring
R[x1; : : : ; xd+1], and the dimension of Cr
k() is the dimension o
This dissertation follows the works of Billera and Rose, as well as Schenck and
Stillman, who each approached the study of splines from the viewpoint of homological
and commutative algebra. They both defined chain complexes of modules such that
Cr(b) appeared as the top homology module.
First, we analyze the effects of gluing planar simplicial complexes. Suppose
1, 2, and = 1 [ 2 are all planar simplicial complexes which triangulate
pseudomanifolds. When 1 \ 2 is also a planar simplicial complex, we use the
Mayer-Vietoris sequence to obtain a natural relationship between the spline modules
Cr(b), Cr (c1), Cr(c2), and Cr( \ 1 \ 2).
Next, given a simplicial complex , we study splines which also vanish on the
boundary of. The set of all such splines is denoted by Cr(b). In this case, we will
discover a formula relating the Hilbert polynomials of Cr(cb) and Cr (b).
Finally, we consider splines which are defined on a polygonally subdivided region
of the plane. By adding only edges to to form a simplicial subdivision , we will
be able to find bounds for the dimensions of the vector spaces Cr
k() for k 0. In
particular, these bounds will be given in terms of the dimensions of the vector spaces
Cr
k() and geometrical data of both and .
This dissertation concludes with some thoughts on future research questions and
an appendix describing the Macaulay2 package SplineCode, which allows the study
of the Hilbert polynomials of the spline modules.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/3915 |
| Date | 16 August 2006 |
| Creators | McDonald, Terry Lynn |
| Contributors | Schenck, Henry |
| Publisher | Texas A&M University |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | 229212 bytes, electronic, application/pdf, born digital |
Page generated in 0.002 seconds