Blending surfaces in solid geometric modelling
Mechanical CAD/CAM (computer aided design/manufacturing) as a field research concerns itself with the algorithms and the mathematics necessary to simulate mechanical parts of the computer, that is to produce a computer model. Solid modelling is a subdiscipline in which the computer model accurately simulates volumetric, i.e. 'solid', properties of mechanical parts. This dissertation deals with a particular type of free-form surface, the blending surface, which is particularly well-suited for solid modelling. A blending surface is one which replaces creases and kinks in the original model with smooth surfaces. A fillet surface is a simple example. We introduce an intuitive paradigm for devising different types of blending forms. Using the paradigm, three forms are derived: the circular, the rolling-ball, and the super-elliptic forms. Important mathematical properties are investigated for the blending surfaces, e.g. continuity, smoothness, containment etc. Blending on blends is introduced as a notion which both extends the flexibility of blending surfaces and allows the blending of multiple surfaces. Blending on blends requires one to think about the way in which the defining functions act as a distance measure from a point in space to a surface. The function defining the super-elliptic blend is offered as an example or a poor distance measure. The zero surface of this function is then embedded within a function which provides an improved distance measure. Mathematical properties are derived for the new function. A weakness in the continuity properties of above blending form is rectified by defining another method to embed the super elliptic blend into a function with better distance properties. This is the displacement form. The concern with this form is its computational reliability which is, therefore, considered in more depth. In the process of integrating the blending surface geometry into a solid modelling environment so it was usable, it was discovered that three other formidable problems needed some type of resolution. These were the topological, the intersection and the display problems. We report on the problems, and solutions which we developed.