Research On Approximation And Convergence Problems In Curves And Surfaces Modeling

Computer Aided Geometric Design (CAGD) is a subject which emerged with the development of modern industry and computer science. Free-form curves and surfaces modeling is one of the most important tasks in CAGD. This dissertation focuses on solving geometric approximation and convergence problems in curves and surfaces modeling. The major contributions of this dissertation are summarized as follows.1. Constant radius offsetting for curves and surfaces is one of the most important geometric operations in CAD/CAM due to its immediate application to NC machining. Due to the square root function in the denominator of unit normal vectors generally, the exact offset curves and surfaces are not rational. Therefore approximations are needed, often by using rational parameter curves and surfaces with low degree. This dissertation presents two new methods of offset approximation: (1) Bezier Approximation algorithm of offset curves. This algorithm firstly translates arbitrary parameter curves into the piecewise cubic-degree Bezier curves, then using the properties of Bezier curve we can get the tangent vector and the normal vector of every point on the approximation curve, and calculate the approximating offset curve. (2) The approximating offset curve by interpolatory using spline curve. Spline curve and base curve are combined to generate a new rational curve by adding the weight. This curve approximates offset curve by interpolating some sample nodes on the offset curve. We analyze and compare the advantage and the weakness of these two algorithms, and apply the second method to the approximation of tensor product offset surface.2. In order to solve practical problems in real engineering applications, the classic definition of offset curve should be extended, which means the fixed distance and direction in the classic definition will not be a necessary request. This dissertation presents a definition of general offset curve, which has fixed offset distance, but variable offset direction. The offset direction is defined by the local coordinate system, which is formed of the tangent vector and the normal vector of every point on the curve. Based on this definition, the curvature,regular and integral properties of the general offset curve are discussed. As a result, J.steiner's celebrated theorem of the oval is developed.3. The algorithms of degree elevation and subdivision for Bezier curves,surfaces play an important role in geometric modeling. For the Bezier tensor product surface, recursive degree elevation and subdivision both generate a sequence of control meshes that converge to the underlying Bezier surface, and get the piecewise bilinear approximation of the original surface. This dissertation uniformly parameterizes the control nets, provides the definition of its discrete partial derivatives for arbitrary order and proves the smooth convergence property of these two algorithms. That is, the discrete partial derivatives of control nets convergent to its corresponding continuous partial derivatives. Ron Goldman introduced an alternative notion of rational Bezier curves defined in terms of the negative degree Bernstein blending functions. This dissertation proves the smooth convergence property of degree elevation for this kind of rational Bezier curve as well. It is significative for the smooth property of approximating curves and surfaces.4. Subdivision surfaces are powerful and useful technique in modeling free-form surfaces. The Catmull-Clark subdivision surface was designed to generalize the bi-cubic B-spline surface to the meshes of arbitrary topology. By introducing the concept of neighbor points and using the first-order difference of control points of Catmull-Clark surfaces, we obtain the rate of convergence of control meshes of Catmull-Clark surface. With the result of convergence we derive a computational formula of subdivision depth for Catmull-Clark surface.