分层T网格上的多项式样条：高效建模与几何处理

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本文主要探讨了"分层T网格上的多项式样条"（Polynomial Splines over Hierarchical T-meshes，简称PHT样条），这是一种新型的几何建模工具，用于在复杂的层次T网格结构上创建精确的曲面模型。T网格是一种特殊的多分辨率网格，它能够自适应地表示几何复杂度，而PHT样条则是B样条在T网格上的扩展，具有B样条的关键特性，如非负性、局部支持和统一分配。 PHT样条的基础构造过程详细阐述了如何在T网格上定义这些样条函数，确保它们在保持多项式性质的同时，能够灵活适应几何变化。与传统的NURBS（Non-Uniform Rational B-Splines）相比，PHT样条的一大优势在于它们是多项式而非有理数，这使得计算更为简洁。同时，PHT样条的交叉插入和交叉去除操作被设计为局部且简单，便于曲面模型的修改和优化。通过PHT样条，建模者可以高效地构建开放或封闭的网格模型，只需要解决少数线性方程组，大大降低了计算复杂度。这种方法特别适用于减少NURBS表面的冗余控制点，实现更紧凑的表示。此外，PHT样条还支持多种几何处理，如将PHT样条转化为张量积样条（Tensor Product Splines）的集合，以及在粗粒度的T网格上简化形状网络，从而简化几何操作。 PHT样条不仅保留了Sederberg T样条的适应性和局部性优点，还在连续性方面有所改进，实现了CI（Continuous Insertion）连续性，这对于保持曲面光滑度和细节至关重要。分层T网格上的多项式样条提供了一种强大的工具，可以简化几何建模流程，提高精度和效率，尤其适用于需要精细控制和高效计算的应用领域。

bottleneck of T-splines in practical applications. Compared

with T-splines and hierarchical B-splines, PHT-splines are

only C

continuous. However, PHT splines have a set of ba-

sis functions, which is a necessity in some theoretical anal-

ysis and applications, while hierarchical B-splines have a

redundant set of ‘basis functions’. On the other hand, hier-

archical B-splines require a very special hierarchical T-

mesh structure due to their reﬁnement scheme, while

PHT-splines work over arbitrary hierarchical T-meshes.

The remainder of the paper is organized as follows. In

Section 2, we introduce polynomial spline spaces over T-

meshes and the dimension formula proved in [2]. Section

3 describes in detail the construction of the basis functions

of a spline space over a hierarchical T-mesh. The properties

of the basis functions are discussed and PHT-spline sur-

faces are introduced. Section 4 presents two important

operations—cross insertion and removal in PHT-spline the-

ory. In Section 5, we propose a surface ﬁtting scheme to ﬁt

open and closed meshes of genus-zero with PHT-splines. In

Section 6, some geometry processing algorithms such as

shape conversion and shape simpliﬁcation are discussed.

Section 7 concludes the paper with a summary and some

future work.

2. Polynomial splines over T-meshes

In this section, we brieﬂy review the deﬁnition of T-

meshes, and then introduce polynomial spline spaces over

T-meshes. The dimension formula of the spline space is

given.

2.1. T-meshes

Given a rectangular domain, a T-mesh is a partition of

the domain and it is basically a rectangular grid that allows

T-junctions [2,21]. It is assumed that the end points of each

grid line in the T-mesh must be on two other grid lines, and

each cell or facet in the grid must be a rectangle. Fig. 3

shows an example of a T-mesh. A grid point in a T-mesh

is also called a vertex of the T-mesh. If a vertex is on the

boundary of the domain, then is called a boundary vertex.

Otherwise, it is called an interior vertex. For example, b

i ¼ 1; ...; 10, in Fig. 3 are boundary vertices, while all the

other vertices v

, i ¼ 1; ...; 5, are interior vertices. Interior

vertices have two types. One is crossing, for example, v

in Fig. 3; and the other is T-junctional, for example, v

Fig. 3. They are called crossing vertices and T-vertices,

respectively. The line segment connecting two adjacent

vertices on a grid line is called an edge of the T-mesh.

2.2. Hierarchical T-meshes

Instead of considering general T-meshes, we restrict our

attention to hierarchical T-meshes in the paper, since such

meshes do not lose the main property—adaptivity of gen-

eral T-meshes.

A hierarchical T-mesh is a special type of T-mesh which

has a natural level structure. It is deﬁned in a recursive

fashion. One generally starts from a TP mesh (level 0).

From level k to level k þ 1, one subdivide a cell at level k

into four subcells which are cells at level k þ 1. For simplic-

ity, we subdivide each cell by connecting the middle points

of the opposite edges with two straight lines. Fig. 4 illus-

trates the process of generating a hierarchical T-mesh.

Hierarchical T-meshes have appeared in many research

disciplines in computer science, computational mathemat-

ics, and so on. For example, adaptive ﬁnite elements [25,

Chapter 15] and hierarchical B-splines [7] are deﬁned over

hierarchical T-meshes.

2.3. Spline spaces over T-meshes

Given a T-mesh T, F denotes all the cells in T and X

the region occupied by all the cells in T. Deﬁne

Sðm; n; a; b; TÞ :¼fsðx; yÞ2C

a;b

ðXÞjsðx; yÞj

2 P

for any / 2 Fg;

where P

is the space of all the polynomials of bi-degree

ðm; nÞ, and C

a;b

ðXÞ is the space consisting of all the bivariate

functions which are continuous in X with order a along x

direction and with order b along y direction. It follows that

Sðm; n; a; b; TÞ is a linear space. It is called the spline space

over the given T-mesh T.

For a given T-mesh T, it is easy to see that T-splines (in

non-rational form) and hierarchial B-splines form a proper

subset of the spline space Sð3; 3; 2; 2; T

Þ in general, where

is a new T-mesh obtained by inserting some edges into

T. In this sense, the splines over a T-mesh are a generaliza-

tion of T-splines and hierarchical B-splines. This general-

ization makes full use of the current domain partition

and provides more ﬂexibility in geometric modeling than

T-splines and hierarchical B-splines in practice.

Theorem 4.2 in [2] provides a dimension formula for the

spline space Sðm; n; a; b; TÞ for m P 2a þ 1 and

n P 2b þ 1. Speciﬁcally, we have

dimSð3; 3; 1; 1; TÞ¼4ðV

þ V

Þ; ð1Þ

where V

and V

represent the number of boundary verti-

ces and interior crossing vertices, respectively. The current

paper focuses on the spline space Sð3; 3; 1; 1; TÞ for a hier-

archical T-mesh T, though the results are also valid over

general spline spaces Sð2a þ 1; 2b þ 1; a; b; TÞ for a, b P 1.

The dimension formula gives us a hint on how to con-

struct basis functions for the spline space, i.e., each bound-

ary vertex or interior crossing vertex associates with four

basis functions. This observation will be further explored

in the construction of the basis functions in the next

Fig. 3. An example of a T-mesh.

78 J. Deng et al. / Graphical Models 70 (2008) 76–86

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