The Problem Of Geometric Continuity In NURBS Surfaces Reconstruction

This thesis studies the problems of geometric continuity in NURBS surfaces reconstruction. In the fields of Reserve Engineering, CAD/CAM, Computer Aided Geometric Design, Computer Graphics, etc., the key problems are the reconstruction of complicated surfaces and the standard tools are NURBS curves and surfaces. Objects often have complex surfaces, so that it is extremely difficult to represent them using a single patch. On the other hand, if one subdivide the surface into a large or small number of pieces, and represents each piece by a NURBS surface patch, one can model a complex surface more easily. The object's surface often possesses some particular continuities, so it is indispensable to be able to control the desired continuities between adjacent NURBS patches. Because geometric continuity is not relative to special parametrizations, it is applied in theoretical researches and engineer applications extensively. The aims of this thesis are researching some typical NURBS surfaces' geometric continuous conditions and pointing out the effects of knot vector and the degree of surface on geometric continuities. Moreover, this thesis offers the mothod of generate smooth NURBS surfaces over arbitrary quadragulation. This mothod overcomes the defects of adopting so called simple collinear continuous condition to deal with the problem of geometric continuity.NURBS surfaces with single interior knots are simple and used widely. The thesis main aims at biuquartic, biquintic, k x fc-degree B-spline surfaces and bicubic NURBS with single interior knots. One of the contributions is to apply knot refinement and the geometric continuous conditions of Bezier surfaces to deduce the geometric continuous constraints of NURBS patches and obtain the intrinsic equations of common boundary curves, where the intrinsic equations are the special phenomena of NURBS surfaces. These results are more useful to improve the smoothness of NURBS models.The main difficulty along with approximating to object's surface with multi-NURBS patches is that one must deal with more control points, especially the control points around the JV-patch corner because of the intertwining of the geometric continuous constraints along the boundaries converging on the corner. In general, it is not wise to solve the global complex system composed with the geometric continuous constraints around the corner in that it will lose the abilities to local control the object's shapes. This thesis induces a local scheme to the geometric continuous constraints around the corner, i.e., isolating the involved control points around the corner, so it will only effect the local shape by adjusting these points. Generally, there are two approaches for constructing a Gn(n > 1) smooth and local scheme. The one is to localize the propagation of continuity constraints by refining surfaces in order to obtain supplementary degrees of freedom, the other is to increase the degrees of surfaces as appropriate according to the propagation of continuity constraints. Another approach of this thesis is to demonstrate that the lowest degree is 5 in order to make the B-spline patches hold the property of local adjustment if no specific111Abstractrestriction to the partition. Most existed surfaces reconstruction methods adopt Bezier tool and demand particular partition. The local scheme addressed in this thesis doesn't restrict the partitions of fitted surfaces, and the fitting tools is biquartic B-spline surfaces with double interior knots and biquintic B-spline surface with single interior knots. This thesis obtains a good local scheme which could overcome the defects arose by using simple collinear continuous condition and preserve fine geometric properties of adjacent patches. Using biquartic B-spline surfaces with double interior knots guarantees the patches are internally parametrically C2. This could fulfill the basic requirement in engineer fields.