Geometric Research Based On Algebraic B-spline Curves

Curves and surfaces modeling are important topics in Computer Aided Geometric Design (CAGD) and Computer Graphics (CG). The curve modeling is the base of the surface modeling. An algebraic B-spline curve is a piecewise continuous algebraic curve which has advantages of low degree, computational efficiency and piecewise smoothness. This thesis addresses the following topics of the curve modeling with algebraic B-spline curves which can be extended to the surface modeling with algebraic B-spline surfaces in the future: (1) Representation: Representing polynomial curves by using the algebraic B-spline curves. (2) Fitting: Curve reconstruction with the algebraic B-spline curves. (3) Editing: Algebraic B-spline curve modification and deformation. The thesis is organized as follows:In Chapter 1, the state of arts of the curves and surfaces modeling are introduced. Two types of the piecewise algebraic surfaces, i.e. Bernstein-Bezier(B-B) piecewise algebraic surfaces and algebraic B-spline surfaces, are described. At last the contributions of the thesis are given.In Chapter 2, we discuss how to transform an algebraic polynomial curve into an algebraic B-spline curve. Since the quadratic algebraic curves are important for CAD, the special attentions are paid on their transformation and a method for determining the transformation domain is given.In Chapter 3, an algebraic B-spline curve fitting algorithm based on the signed distance field is proposed. The algorithm fits the signed distance field of the source point set with an algebraic B-spline function and the zero set of the algebraic B-spline function is the constructed curve. It is an algebraic B-spline curve. Besides of the high quality fitting curve, the algebraic B-spline function also represents the signed distance field of the source point set. Meanwhile, the proposed method can avoids the unwanted branches which often occur in the algebraic fitting methods.In Chapter 5, how to edit an algebraic B-spline curve is discussed. Three editing approaches are proposed: firstly, the curve is edited by modifying their control coefficients;secondly, inspired by direct manipulation of FFD, the curve can be directly manipulated through solving an optimal function;lastly, considered thesmoothness of result curve in the range influenced by object points, the curve is directly manipulated to pass object points. ■ Finally, conclusions are drawn and some future works are discussed.