The least squares approximation of fixed-form and free-form B-spline surfaces and a smoothing technique using reflection

This dissertation focuses on constructing geometric CAD models from digitized surface points and on surface smoothing using reflection. The construction of surfaced CAD models from digitized data is part of the process of reverse engineering while the surface smoothing is an integral part of the design and tool production processes.; The dissertation develops least-squares techniques to fit fixed-form non-uniform rational B-spline (NURBs) surfaces, in particular, surfaces of revolution, cylinders and cones. A method is also developed to determine the axis of rotation from random data points on the surface. After determining the axis of rotation, the axis information can be used in an interactive environment to obtain cross-section data points required by the surface fitting method. An approximation method is also developed to fit free-form B-spline surfaces.; In this dissertation, techniques are also developed to smooth planar curves and surfaces. In the smoothing technique for planar curves, plots of normal vector ratios are used to determine the convex, concave and flat regions of the curve. Smoothing, based on the least-squares method, is accomplished by using interactively corrected ratio plots. In surface smoothing, reflections of a fixed-position light source from different points on the surface are created on a reflection plane producing a reflection pattern. Convex, concave and flat spots on the surface produce characteristic reflection patterns. Thus, non-smooth spots on a surface can be identified by recognizing the specific patterns. A surface smoothing technique is developed to smooth out surface irregularities using the user corrected reflection pattern. The least-squares method is employed in this smoothing technique.; Both the curve and surface smoothing techniques developed are intuitive and simple to apply. Only 2-D ratio plots or reflection patterns are required and are computationally simple, involving only first derivatives. Although B-splines are used in the smoothing techniques, the techniques are applicable to all types of curves and surfaces.