On Theeory Of Geometric Continuity In Surface Reconstruction Of Reverse Engineering

This thesis studies the problems of geometric continuity about surfaces reconstruction in Reverse Engineering. Objects, even simple ones, often have complex surfaces, so it is extremely difficult to represent them by a global approach. On the other hand, if one subdivides the surface into a large or small numbers of pieces, and represents each piece by a NURBS surface patch, one can model a complex surface much more easily. The object' s surface often possesses some particular geometric continuities such as the continuity of position, of the tangent plane and even of curvature. When one models a surface with a piecewise representation, it is indispensable to be able to control the desired geometric continuity between adjacent surface patches for the resulting surface to be continuous.Geometric continuity between adjacent parametric patches has been playing a very important role in CAD/CAM, geometric modeling and reverse engineering, etc.. This is not only because geometric continuity provides free shape parameters which can be used to construct and modify those very complicated geometric objects, but also because this type of continuity is actually the essential continuity between surface patches, i.e., it avoids dependencies on the parametrizations of the involved patches. So it is applied in theoretical researches and engineer applications extensively. As in the design of aircraft fuselage, car bodies and ship hulls, surfaces with 2nd~order continuity are sometimes necessary because of the background in physics. The thesis investigates some effective modeling approaches for enhancing modeling ability of NURBS surfaces.In most of the modeling situations, the patches are defined over a topologically rectangular mesh. However, this kind of mesh does not allow the representation of more complex shapes such as the so-called suitcase corners and the branching of tree-like volume. It is necessary to research to construct surfaces with any topology meshes by rectangular NURBS patches , i. e. with any number of patches meeting at a node. To guarantee the geometriccontinuities of the resulting surfaces, one must solve the smooth connection problem between adjacent patches.It is interesting to note that despite the fact that B-spline surface isone of the most popular representation of toolsin CAD, geometric modeling, animation and reverse engineering industries, etc.. And in the past 20 years, the conditions of geometric continuity between two adjacent Bezier patches have been extensively studied in many literatures, but little attention has been paid to the conditions for geometric continuity of the B-spline surfaces.This thesis has investigated the conditions of G1 and G2 continuity about NURBS surfaces. The G1 and G2 continuity conditions between two adjacent NURBS patches are obtained. These conditions are directly represented by the control points of the NURBS patches. The intrinsic conditions for the control points of the common boundary of two adjacent G1 and G2smooth B-spline surfaces are obtained . These conditions imply that inserting single interior knots for the boundary curve does not increase the freedom of control points of the boundary curves. This thesis induces a local scheme to the geometric continuous constraints of the smooth connection of the surfaces around the corner. It is proved that the existence of a local scheme of constructing G1 continuous bicubic B-spline surface models only requires two pairs of interior double knots which can be placed anywhere of the knot vectors. In the case of G2 continuous bicubic B-spline surface models, it requires four pairs of interior triple double knots.