Algebraic Curves And Surfaces Design And Modeling

An important topic of Computer Aided Geometric Design and Computer Graphics is geometric modeling, whose major components are free-form curves and surfaces design and modeling. According to the representations, modeling technique can be classified into two categories: parametric methods and implicit methods. The parametric methods have been widely investigated, while the implicit methods have received increasing attention in recent years. Compared with parameter surfaces, implicit (especially algebraic) surfaces play an important role in constructing objects with complex topology.The thesis concentrates on algebraic curves and surfaces design and modeling, specifically on the analysis and applications of different forms of algebraic curves and surfaces, and determining free parameters in piecewise algebraic surface blending.For curve design, we first introduce a technique to construct an algebraic curve from the given point set with weights based on the barycentric coordinates of the polygon defined by the point set, and apply it in constructing interpolating curves; then based on a special kind of functional splines, we propose a G~2 cubic A-spline curves; we finally present a method to construct algebraic curves using the guidance vectors of the control polygon, derive their expressions and analyze their properties.For surface modeling and controlling, we propose an approach to construct algebraic surfaces based on guidance field. In addition, we give a set of options for determining the free parameters which usually appear in piecewise algebraic surface blending. By minimizing the surface energy and adding some point restrictions, we can select the free parameters such that the blending surface has a reasonable shape.