| Abstract [eng] |
This dissertation explores theoretical results about Dupin cyclides and their applications in Computer-Aided Geometric Design (CAGD) and architecture. Dupin cyclides are unique surfaces in Euclidean space characterized by the property that their curvature lines are either circles or straight lines. These surfaces are used in surface modeling, where Dupin cyclides can be smoothly blended along these curvature lines. Our contributions are organized into three interconnected categories: Recognition through implicit equations: Dupin cyclides were distinguished from the larger class of cyclides called Darboux cyclides through the coefficients of their implicit equation; the obtained algebraic conditions were used to blend two Dupin cyclides along a common circle. Dupin cyclides and Dupin cyclidic (DC) systems based on quaternionic representations: Dupin cyclide principal patches, which are quadrilateral patches bounded by 4 curvature lines, were parametrized by quaternionic bilinear Bézier patches; their 3d generalizations, known as DC systems, were similarly defined by trilinear quaternionic formulas and were classified from the perspective of Moebius geometry. Generalized cyclidic splines of arbitrary topology: The problem of filling holes that naturally occur in DC splines was addressed by introducing a filling method based on a ring of principal patches and a spherical or planar patch in the middle; the effectiveness of this method was demonstrated in several examples, including the approximation of Lawson surfaces of genus 2 and 3, as well as improvements of the existing CAD model of the Boy’s surface. Each category has contributed unique insights that, when considered together, essentially enhanced our understanding of Dupin cyclides and their modeling applications. |
