arXiv subjects

Falai Chen

Publications and source records attributed to Falai Chen.

21 records · Page 2Linked to original sources

arXiv2011

The Dimension of Spline Spaces with Highest Order Smoothness over Hierarchical T-meshes

This paper discusses the dimension of spline spaces with highest order smoothness over hierarchical T-meshes over certain type of hierarchical T-meshes. The major step is to set up a bijection between the spline space with highest order smoothness over a hierarchical T-mesh and a univariate spline space whose definition depends on the l-edges of the extended T-mesh. We decompose the univariate spline space into direct sums in the sense of isomorphism using the theory of the short exact sequence in homological algebra. According to the decomposition of the univariate spline space, the dimension formula of the spline space with highest order smoothness over certain type of hierarchical T-mesh is presented. A set of basis functions of the spline space is also constructed.

cs.CG↗

arXiv2010

Moving Planes and Singular Points of Rational Parametric Surfaces

In this paper we discuss the relationship between the moving planes of a rational parametric surface and the singular points on it. Firstly, the intersection multiplicity of several planar curves is introduced. Then we derive an equivalent definition for the order of a singular point on a rational parametric surface. Based on the new definition of singularity orders, we derive the relationship between the moving planes of a rational surface and the order of singular points. Especially, the relationship between the $μ$-basis and the order of a singular point is also discussed.

math.NA↗

arXiv2008

Dimensions of Biquadratic Spline Spaces over T-meshes

This paper discusses the dimensions of the spline spaces over T-meshes with lower degree. Two new concepts are proposed: extension of T-meshes and spline spaces with homogeneous boundary conditions. In the dimension analysis, the key strategy is linear space embedding with the operator of mixed partial derivative. The dimension of the original space equals the difference between the dimension of the image space and the rank of the constraints which ensuring any element in the image space has a preimage in the original space. Then the dimension formula and basis function construction of bilinear spline spaces of smoothness order zero over T-meshes are discussed in detail, and a dimension lower bound of biquadratic spline spaces over general T-meshes is provided. Furthermore, using level structure of hierarchical T-meshes, a dimension formula of biquadratic spline space over hierarchical T-meshes are proved. A topological explantation of the dimension formula is shown as well.

math.NA↗