The Algorithm For Offset Curves And Surfaces Based On The Progressive Iteration Approximation

Offset curves and surfaces, also called potential difference or parallel curves and surfaces, are defined as locus of the points which are at a distance d along the normal vector direction from the base curves and surfaces. Offset operation, as a significant geometric operation, is an important studying content in Computer Aided Geometric Design, which is widely used in NC, auto body design, calculation of robot’s trajectory and other fields. Meantime, offset operation is a kind of complex geometric operation, so the approximation method is usually adopted to generate offset curves and surfaces in actual application.At present, approximation methods for offset curves and surfaces are hard to solve the following problems well at the same time:the result is not the form of NURBS, so it is not applicable to common modeling system; it is hard to achieve a better accuracy control; the amount of data is too large; the efficiency of the algorithm is unsatisfactory. The research of this thesis is based on the Progressive Iterative Approximation algorithm, as well as the features and geometric properties of offset curves and surfaces, and gets some results.Firstly, a new algorithm for generating the approximation offset of planar curve based on the progressive iteration approximation is proposed. The idea of this algorithm is adaptive sampling to the base curve by using the tangent vector corner of that firstly and, then using the progressive iteration approximation to generate a B-spline curve to approximate the offset curves. The algorithm has the following performances:it can be apply to arbitrary planar parametric curves and function curves; the global error can be control well; the offset curve obtained is the form of B-spline; the number of control points is relatively less; there is no need to solve linear equations in the iterative process, so the algorithm is efficient.Then, an improved algorithm for offset curves which is based on the progressive iteration approximation is presented. Aiming at the problem of the overestimation in the process of sampling and calculating the approximation error, and the lack in the utilization of the geometric properties of offset curves in the approximation process for the algorithm of planar offset curves based on the progressive iteration approximation, this improved algorithm takes advantage of the height of curve segment controlling the sampling error and approximation error, which control the global error more exactly for the resulted B-spline curve approximating offset curve, and utilizes the normal vector and the consistency of parameterization to control the shape of offset approximation curve further.At last, combining the features of offset surfaces, and generalizing the algorithm of offset curves to offset surfaces by appropriate deformation, an algorithm for offset surfaces based on the progressive iteration approximation is proposed. This algorithm adopts the scheme of sampling several curves synchronously, and also considers the normal vector and the consistency of parameterization in the approximation process. This algorithm achieves a satisfactory effect in sampling of data points, approximation of offset surfaces, controlling error and other aspects.For the above three algorithms, some examples are presented to illustrate the validity of the algorithms.