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

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 G ¹ continuous. The proposed method of curve interpolation provides an important alternative solution in curve-based subdivision surface design. © 2012 Elsevier Ltd. All rights reserved.