0306385 Farin, Gerald Arizona State University
Splines are used to describe geometric objects by breaking down into individual parts such that the overall smoothness of thes pieces is guaranteed. Splines are widely used in the automotive, aerospace, and other engineering disciplines. They work very well when the geometry has a rectilinear struture (such as the roof of a car), but need considerable effort when this is not the case. For that reason, splines are not much used in biological or medical applications, where shapes tend to be highly irregular. This research introduces a new class of splines having all the smoothness properties of standard ones but also being able to cope with complex and unstructured shapes. This research explores a new class splines, those that are defined over an unstructured set of 2D points - i.e., no regular structure is assumed. The most fundamental algorithm in the theory of B-spline curves is the recursive de Boor algorithm. This algorithm is generalized to a recursive algorithm involving the construction of a hierarchy of Voronoi diagrams in the plane. The resulting surfaces are of arbitrary degree and are piecewise smooth. They are capable of reproducing polynomials of any order. Since the de Boor Algorithm is fundamental to all spline theory, a proper reformulation for unstructured point sets is a significant breakthrough and constitutes the main intellectual merit of this research. The existence of splines defined over unstructured point sets opens up the possibility of handling complex shapes. This is particularly important for organic shapes, which tend to be highly irregular. Thus the broader impact of this research is the possibility to successfully describe non-engineering shapes using techniques similar to standard splines.
