Approximating data with parametric B-splines for computer aided design

This thesis addresses some of the difficulties that can be encountered when approximation is used as a design tool. In many cases, it is most intuitive for a designer to specify points on or near a desired contour, and approximate them with a curve. Merely computing some approximation to a set of data points is not difficult; the difficulty lies in producing an approximation that has the same general shape as the desired contour, and behaves reasonably between the data points. The smoothness and shape of a parametric approximation is affected by many factors, including the parameterization of the data points, the number of knots used and their locations, and the metric that is minimized to produce the final approximation. The interrelationships of these factors, and their effect on the resulting approximation, will be discussed. Existing methods for estimating parameters and knots will be compared to newly developed methods, which have been designed to overcome limitations of previous methods. Finally, the synthesis of these methods into an integrated approach will be explored. An approximation process has been developed which automatically determines an appropriate parametrization and knot vector, so that, with minimal user input, a discrete set of data points can be transformed into a continuous parametric B-spline curve that has the desired shape.