## 1. Introduction

We demonstrate how creases can be added to Cashman's non-uniform rational B-spline (NURBS)-compatible subdivision [Cas10]. We produce creases that are mathematically similar to creases in NURBS surfaces, but with the additional feature that they can start and stop at arbitrary knots, fairing out smoothly to normal surface. Our creases are different in character from DeRose's creases [DKT98], which are for Catmull-Clark (degree 3) subdivision surfaces [CC78], and our creases are applicable to subdivision surfaces of any degree, not just those of degree three.

NURBS surfaces and subdivision surfaces are two of the principal surface representations used in industry. NURBS is industry-standard in CAD. Subdivision surfaces are widely used in 3D computer animation. Both can be considered to be different types of generalization of tensor-product uniform B-splines. NURBS generalizes to non-uniform knot spacing and to rational functions via weights, providing such features as perfect circular cross-sections, but maintains the constraint that the control mesh must be a rectangular array of control vertices. Subdivision surfaces generalize to arbitrary topology, but typically maintain the constraint that the underlying foundation is uniform. Practical subdivision surfaces are either quadratic (degree 2) or cubic (degree 3). NURBS surfaces can be of any degree, with commercial systems offering degrees up to above 20.

Cashman [CADS09; Cas10] developed *NURBS-compatible subdivision*, which provided a subdivision mechanism for a true superset of NURBS for arbitrary odd degree. That is, it is a generalization of non-uniform tensor-product B-splines. Therefore, any odd-degree NURBS can be represented in Cashman's formulation, with the advantage of arbitrary topology.

Our work considers creases in surfaces. Creases are an important modelling ingredient. Standard subdivision surfaces are smooth everywhere. It is known that using an everywhere smooth surface to model sharp features results in poor geometric fit and unwanted undulations [HDD*94].

We introduce a further generalization of Cashman's subdivision framework that supports modelling with creases (Figure 1). Our approach is based both on Cashman's subdivision framework [Cas10] and on arbitrary degree T-constructions, e.g. [Fin08; DLP13].

Traditional subdivision surfaces are a generalization of *uniform* tensor product B-splines [CC78]. Cashman's surfaces are a generalization of *non-uniform* tensor product B-splines [Cas10]. Creases in subdivision surfaces based on non-uniform tensor product B-splines can be achieved by three methods:

- by making control vertices coalesce,
- by modifying subdivision rules,
- by allowing knot lines to be multiple.

We investigate the third approach. It exposes limitations in Cashman's framework. Cashman's framework allows multiple knot lines, provided that the crease propagates across the entire mesh, from one boundary to another (or, equivalently, forms a closed loop). This is a significant limitation. Our construction, by contrast, allows multiple knot lines to terminate at any knot, rather than just at a boundary. The corresponding discontinuity in the surface then fairs out smoothly as shown in Figure 1. This extends the capability of Cashman's framework, enabling a wider range of surfaces to be modelled. Our construction yields subdivision surfaces of any degree.