Interpolating an arbitrary number of joint B-spline curves by Loop surfaces

Abstract

In a recent paper (Ma and Wang, 2009), it was found that the limit curve corresponding to a regular edge path of a Loop subdivision surface reduces to a uniform cubic B-spline curve (CBSC) under a degeneration condition. One can thus define a Loop subdivision surface interpolating a set of input CBSCs with various topological structures that can be mapped to regular edge paths of the underlying surface. This paper presents a new solution for defining a Loop subdivision surface interpolating an arbitrary number of CBSCs meeting at an extraordinary point. The solution is based on the concept of a polygonal complex method previously used for Catmull–Clark surface interpolation and is built upon an extended set of constraints of the control vertices under which local edge paths meeting at an extraordinary point reduces to a set of endpoint interpolating CBSCs. As a result, the local subdivision rules near an extraordinary point can be modified such that the resulting Loop subdivision surface exactly interpolates a set of input endpoint interpolating CBSCs meeting at the extraordinary point. If the given endpoint interpolating CBSCs have a common tangent plane at the meeting point, the resulting Loop surface will be G¹ continuous. The proposed method of curve interpolation provides an important alternative solution in curve-based subdivision surface design.

Introduction

Subdivision surfaces are widely used in recent years due to their multiresolution property and their simplicity, uniformity and powerful ability in representing complex surfaces [1], [2]. They were initially proposed as a generalization of B-spline surfaces to model smooth surfaces of arbitrary topology [3], [4]. More and more subdivision schemes with various refining operators were subsequently designed for control meshes of different connectivity [5], [6], [7], [8], [9]. Using these schemes, people can produce various subdivision surfaces with different properties according to their design requirements and application settings.

On the other hand, people usually want to model smooth surfaces under some constraints, such as points, tangents, normals, curves, etc. Surface design under constraints of given curves thus becomes an important topic in the fields of geometric design and computer graphics. However, since subdivision surfaces are defined as limits of recursively subdivided control meshes, they usually have no ready global parametric expressions. It is thus difficult to handle curves on a subdivision surface or impose a subdivision surface to pass through given curves compared with spline-based modeling.

Surface design from a set of input curves is a classic topic in geometric design and has been widely studied in spline-based modeling [10], [11]. One is interested in creating a surface interpolating or best fitting the input curves. Typical operations include surface lofting, surface skinning, surface blending, N-sided hole-filling [12], [13], [14], [15], and various other operations for curve-based shape control and surface modification. In the literature, one can also find some examples on surface design from input curves using Doo–Sabin [4], Catmull–Clark [3] and recently Loop [8] subdivision surfaces. In a survey paper, Nasri and Sabin also discussed and classified various interpolation constraints including curves on subdivision surfaces [16].

In connection with Doo–Sabin subdivision [4], a generalization of uniform bi-quadratic B-spline surfaces, Nasri proposed methods for interpolating given points, normal vectors and quadratic B-spline curves [17], [18], [19]. Nasri also developed a method for constructing a Doo–Sabin subdivision surface interpolating a set of intersecting quadratic B-spline curves using a set of polygonal strip complexes whose limit curves converge to the input curves, respectively [16], [20]. To improve the quality of the final surfaces, Nasri et al. proposed a method to regularize the polygonal meshes based on Laplacian smoothing and mean curvature flow of the vertices [21].

For Catmull–Clark subdivision surfaces [3], a generalization of uniform bi-cubic B-spline surfaces, Nasri and Abbas presented a method in using polygonal complexes to generate Catmull–Clark surfaces interpolating CBSCs [22]. Abbas and Nasri further generalized the method to interpolate an unlimited number of curves meeting at a point [23], [24]. One can also find another approach that modifies the subdivision rules such that the limit surface will interpolate the given curves. The approach was first reported in [25], [26] as a combined subdivision scheme in a general setting, and was reported in [27] for Catmull–Clark subdivision surfaces. Hui and Lai also presented an intuitive approach recently for constructing free-form objects using Catmull–Clark surfaces from planar profile curves [28].

In a recent paper [29], Ma and Wang proposed a method for curve-based surface design using Loop subdivision surfaces [8]. The method was based on the degeneration condition in which a regular edge path of a Loop subdivision surface reduces to a CBSC. Combined with the interpolating condition, one can define a Loop subdivision surface interpolating a set of CBSCs with various topological structures that can be mapped to regular edge paths of the underlying surface. This paper presents a new solution for defining a Loop subdivision surface interpolating an arbitrary number of CBSCs meeting at an extraordinary point. Our solution is built upon an extended set of constraints of the control vertices under which local edge paths of a Loop subdivision surface meeting at an extraordinary point reduces to a set of endpoint interpolating CBSCs. The solution is also based on the concept of polygonal complexes used for interpolating Catmull–Clark surfaces [22], [23], [24]. The proposed method provides an important alternative operation for curve-based subdivision surface design.

The rest of the paper is organized as follows. We first provide a brief overview regarding the background of the proposed method in Section 2. Section 3 presents in detail the modeling method of Loop subdivision surfaces interpolating joint endpoint interpolating CBSCs. Section 4 provides a general framework for finding a Loop subdivision surface interpolating given joint CBSCs. Some modeling examples of Loop surfaces interpolating joint CBSCs are presented in Section 5 followed by a summary of the conclusions in Section 6.