## Jet Subdivision

If you are reading this page, then perhaps you had passed the age to find Pixar's animated movies such as Toy Story II and A Bug's Life highly amusing. However, if you are into multiscale mathematics then your adrenaline level may rise a bit if you realize that the type of geometric modelling techiques -- known as subdivision methods -- used in the production of these films are intriguingly linked to things such as fractals and wavelets. Despite the box office successes in computer graphics, subdivision methods have not made a similar impact in the engineering community, where the requirements for surface quality are much more demanding than those of the entertainment and marketing communities. In particular, both the automobile industry (e.g. BMW) and the aircraft industry (e.g. Boeing) would like to have surfaces with good curvature properties, yet all existing subdivision surface schemes  generate surfaces that are either flat or have curvature singularities at extraordinary vertices. (Of course, the latter is by no means the only reason why the engineering design community has not picked up on subdivision methods.)

A new methodology: jet subdivision. Ordinary subdivision schemes start with an initial polyhedral surfaces, called a base mesh, and construct a sequence of refined meshes converging to a smooth surface. The initial mesh is determined by the positions of the vertices of the base mesh, while the later meshes are formed by a two-step process of subdivision involving a step of "topological refinement" followed by an averaging step. Jet subdivision schemes incorporate higher order data at each vertex of the base mesh. This higher order data may be viewed as an approximation of the r-th order Taylor approximation of the limiting surface - defined in an intrinsic way and is called an "r-jet". From this perspective, ordinary subdivision schemes use 0-th order Taylor approximations of the limiting surface, and thus a special case of a jet subdivision scheme with r=0.

Collaborators: Thomas Duchamp, Bin Han, Bruce Piper
Students: Matthew Ferrara, Kyle McDonald, Dr. Yonggang Xue
Optimization consultant: Michael Overton

#### Gallery

A Jet Subdvision Surface

Characteristics maps in B-net representations. The followings are the characteristic maps of the two-point interpolatory jet subdivision scheme constructed in the article Jet Subdivision Schemes on the k-regular complex. This subdivision scheme happens to have an intimate connection with the classical Powell-Sabin spline, as shown in our earlier article Multivariate Refinable Hermite interpolant. As such, the characteristic map of this particular jet subdivision scheme has a piecewise quadratic polynomial representation and (consequently) can be expressed in B-net representations. Not only are such B-net representations used to prove the fact that these characteristic maps are regular and injective for arbitrary valence k, they are also used to give explicit paramaterization of the surfaces produced by our subdivision schemes. You may notice another, somewhat similar, set of figures in the paper (Figure 11, first row  which shows how the characteristic map distorts the lines of the refined k-regular complex.  It is over a 'characteristic plane' that one can smoothly parametrize a subdivision surface in the vicinity of an extraordinary vertex -- an idea pioneered by Ulrich Reif about a decade ago. This is why characteristic maps have to be injective and regular, otherwise the 'characteristic planes' would not look like planes at all...

Another Jet Subdvision Surface: comparion with three variants of Loop surface
 Original Loop Scheme -- unbounded Curvature at k=3 A variant with bounded (but discontinuous) curvature A zero curvature variant Our flexible C2 JSS See [section 5.2]

The extrinsic and intrinsic views of differential geometry: The following picture is an outcome of my attempt to alleviate my headache when learning about geometry. I share it with you anyway.

Related Publications:
Jet Subdivision Schemes on the k-Regular Complex, Y. Xue, T. P.-Y. Yu and T. Duchamp (pdf)(bibtex entry)
Multivariate Refinable Hermite Interpolants, by B. Han, T. P.-Y. Yu and B. Piper (pdf) (ps) (bibtex entry)
Design of Hermite Subdivision Schemes aided by  Spectral Radius Optimization, by B. Han, M. Overton and T. P.-Y. Yu (pdf)(ps)(bibtex entry)
Non-Interpolatory Hermite Subdivision Schemes,  by B. Han, T. P.-Y. Yu and Y. Xue (pdf)(ps)(bibtex entry)
Face-based Hermite Subdivision Schemes, by B. Han and T. P.-Y. Yu (pdf)(ps)(bibtex entry)
Honeycomb and k-fold Hermite Subdivision Schemes, Y. Xue and T. P.-Y. Yu (pdf)(ps)(bibtex entry)

The Research presented at this Web Site has been sponsored by  a National Science Foundation CAREER award (CCR 9984501).
The PI is also grateful to a software donation by Alias