The aims of this research are the creation of new surface forms and the determination of geometric properties of surfaces. More particularly, under the first topic, rational curve and surface schemes will be developed and studied. Rational B-spline curves and surfaces are becoming the industry standard for Computer Aided Design/Computer Aided Manufacturing information. Moreover, although polynomial B-spline curves and surfaces have been studied for some time, their rational counterparts have received considerably less attention, particularly in connection with design. Thus there are both scientific and application motivations for studying this class of curves and surfaces. The determination of geometric properties of surfaces is called "Geometry Processing." For an already constructed curve or surface, geometric properties may be desired to understand the surface better or the geometric property may itself be desired for an application. Examples of geometric properties which help explain a surface are curvatures and reflection lines. Geometric properties of interest per se include surface/surface intersections, offset surfaces, and surfaces on surfaces. The full sweep from constructive mathematics through the implementation of algorithms and the interactive computer graphics display of surfaces is utilized in this research.
