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Dimension And Local Bases Of Homogeneous Spline Spaces
"... Recently, we have introduced spaces of splines defined on triangulations lying on the sphere or on spherelike surfaces. These spaces arose out of a new kind of BernsteinB'ezier theory on such surfaces. The purpose of this paper is to contribute to the development of a constructive theory for ..."
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Cited by 28 (13 self)
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Recently, we have introduced spaces of splines defined on triangulations lying on the sphere or on spherelike surfaces. These spaces arose out of a new kind of BernsteinB'ezier theory on such surfaces. The purpose of this paper is to contribute to the development of a constructive theory for such spline spaces analogous to the wellknown theory of polynomial splines on planar triangulations. Rather than working with splines on spherelike surfaces directly, we instead investigate more general spaces of homogeneous splines in R³. In particular, we present formulae for the dimensions of such spline spaces, and construct locally supported bases for them.
Locally Linearly Independent Basis for C¹ Bivariate Splines of Degree q >= 5
, 1998
"... We construct a locally linearly independent basis for the space S 1 q (\Delta) (q 5). Bases with this property were available only for some subspaces of smooth bivariate splines. ..."
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Cited by 7 (7 self)
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We construct a locally linearly independent basis for the space S 1 q (\Delta) (q 5). Bases with this property were available only for some subspaces of smooth bivariate splines.
Graphs, Syzygies and Multivariate Splines
, 2004
"... The module of splines on a polyhedral complex can be viewed as the syzygy module of its dual graph with edges weighted by powers of linear forms. When the assignment of linear forms to edges meets certain conditions, we can decompose the graph into disjoint cycles without changing the isomorphism cl ..."
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Cited by 2 (0 self)
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The module of splines on a polyhedral complex can be viewed as the syzygy module of its dual graph with edges weighted by powers of linear forms. When the assignment of linear forms to edges meets certain conditions, we can decompose the graph into disjoint cycles without changing the isomorphism class of the syzygy module. Thus we can use this decomposition to compute the homological dimension and the Hilbert series of the module. We provide alternate proofs of some results of Schenck and Stillman, extending those results to the polyhedral case. We also provide examples which illustrate the role that geometry plays in determining the syzygy module. 1
On Singularity of Spline Space Over MorganScott’s Type Partition
, 2006
"... Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is impossible to avoid in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the ge ..."
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Cited by 1 (1 self)
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Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is impossible to avoid in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over MorganScott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines (or ponits), and Characteristic number of algebraic curve. Using these concepts and the relevant results, a polished necessary and sufficient conditions to the singularity of spline space Sµµ+1(∆ µ MS) are geometrically given for any smoothness µ by recursion. Moreover, the famous Pascal’s theorem is generalized to algebraic plane curves of degree n ≥ 3.
Geometric Condition of Singularity of)(
"... The aim of this paper is to investigate the geometric condition of singularity of) ( 223 MSS Δ. The algebraic of singularityof) ( 223 MSS Δ is obtained in (Luo and Chen, 2005). The result of this paper will be useful to further study the geometric condition of singularity of)3)((1>Δ+ μμμμ MSS. ..."
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The aim of this paper is to investigate the geometric condition of singularity of) ( 223 MSS Δ. The algebraic of singularityof) ( 223 MSS Δ is obtained in (Luo and Chen, 2005). The result of this paper will be useful to further study the geometric condition of singularity of)3)((1>Δ+ μμμμ MSS.
AN INVARIANT FROM THE PASCAL THEOREM
, 2007
"... In 1640’s, Blaise Pascal discovered a remarkable property of a hexagon inscribed in a conic Pascal Theorem, which gave birth of the projective geometry. In this paper, a new geometric invariant of algebraic curves is discovered by a different comprehension to Pascals mystic hexagram or to the Pas ..."
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In 1640’s, Blaise Pascal discovered a remarkable property of a hexagon inscribed in a conic Pascal Theorem, which gave birth of the projective geometry. In this paper, a new geometric invariant of algebraic curves is discovered by a different comprehension to Pascals mystic hexagram or to the Pascal theorem. Using this invariant, the Pascal theorem can be generalized to the case of cubic (even to algebraic curves of higher degree), that is, For any given 9 intersections between a cubic Γ3 and any three lines a, b, c with no common zero, none of them is a component of Γ3, then the six points consisting of the three points determined by the Pascal mapping applied to any six points (no three points of which are collinear) among those 9 intersections as well as the remaining three points of those 9 intersections must lie on a conic. This generalization differs quite a bit and is much simpler than Chasles’s theorem and CayleyBacharach theorems.
APPROXIMATION © 2004 SpringerVerlag New York, LLC
"... Abstract. Given a rectangular box which has been split into 24 tetrahedra, we show how to construct a C1 macroelement using polynomial pieces of degree 6. 1. ..."
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Abstract. Given a rectangular box which has been split into 24 tetrahedra, we show how to construct a C1 macroelement using polynomial pieces of degree 6. 1.