Partition of unity parametrics: a framework for meta-modeling

We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains affine invariance. PUPs replace the weighted basis functions of NURBS with arbitrary weight-functions (WFs). By choosing appropriate WFs, PUPs yield a comprehensive geometric modeling framework, accounting for a variety of beneficial properties, such as local support, specified smoothness, arbitrary sharp features and approximating or interpolating curves. Additionally, we consider interactive specification of WFs to fine-tune the character of curves and generate non-trivial effects. This serves as a basis for a system where users model the tools used for modeling, here weight-functions, in tandem with the model itself, which we dub a meta-modeling system. PUP curves and surfaces are considered in detail. Curves illustrate basic concepts that apply directly to surfaces. For surfaces, the advantages of PUPs are more pronounced; permitting non-tensor WFs and direct parameter space manipulations. These features allow us to address two difficult geometric modeling problems (sketching features onto surfaces and converting planar meshes into parametric surfaces) in a conceptually and computationally simple way.