Low degree spline approximation of surfaces and functions
Splines (piecewise polynomial or, more generally, rational functions) are commonly used for numerical approximation of (arbitrary) functions in a variety of contexts. For example, the celebrated Bezier/NURBS curves and surfaces are ubiquitous in geometric design and computer graphics. Some of the most widely used frameworks involve tensor products of 1d cubic polynomials which lead to degree 9 polynomials in 3d. This high degree leads to excessive amounts of data and long evaluation times, which makes the utility of such approximations in real-time environments quite limited, even with the state of the art hardware. One of the ways to deal with this issue is to develop methods for low degree approximation of geometric data. There are, however, some geometric and topological obstructions to this. In my talk I will review the history of this subject, highlight some specific problems and ideas of how to address them.