Browse Theses

If Computer Aided Geometric Design is to deliver on its promise to rationalize and speed up the process of designing physical objects, it must master the construction of free-form surfaces. A key challenge is to control smoothness while matching given data and keeping the surface representation simple. This thesis both derives and applies new techniques to meet the competing requirements.In particular, the thesis develops several equivalent notions of first- and second-order smoothness between patches, based on geometric invariants and of higher-order smoothness, based on the chain rule. Further, it exhibits the necessary and sufficient constraints on the data that allow enclosing a vertex by a $C\sp{1}$-complex of patches, or, for the symmetric case, by a $C\sp{k}$-complex of patches, and derives four techniques to overcome the enclosure problem. By classifying techniques for smoothly connecting patches and for enclosing vertices, a large number of algorithms in the literature are characterized. Variations of five new, implemented algorithms are described, each representing a different approach to supplying levels of smoothness for different types of data. For example, a method employing triangular cubic patches is shown to interpolate a mesh of curves with regular oriented tangent plane continuity and a surface assembled from tensor product patches of degree 2$k$ + 2 yields $k\sp{\rm th}$ -order smoothness while matching data of the same order along patch boundaries.