NURBS曲面重构中的几何连续性问题
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摘要
本文研究NURBS曲面重构中的几何连续性问题。在反向工程、CAD／CAM、计算机辅助几何设计(CAGD)和计算机图形学等领域，一个关键的问题是复杂曲面的重建，所使用的标准工具是NURBS曲面。由于世间物体表面复杂多样，用单片曲面很难精确描述其曲面形状，如果我们将其表面分解成相对较小的曲面片，并用NURBS曲面来拟合这些曲面片，就能比较容易的构造复杂物体的表面模型。物体的表面一般都具有一定的光滑性，因此拟合小曲面片的这些NURBS曲面之间就必须达到一定程度的光滑连接。由于几何连续与具体的参数选取无关，因此在理论研究和工程应用中被广泛采用。本文的目的是研究一些典型的NURBS曲面的几何连续条件，指出节点向量和曲面片的次数对几何连续性的影响。并针对任意拓扑的四边形剖分，给出了构造光滑NURBS曲面的方法，克服了采用所谓的简单共线法处理几何光滑连接的缺陷。
节点向量是内部单节点的NURBS曲面是最简单也是最常用的，本文重点针对单节点的双四次、双五次和双k次B样条曲面及双三次NURBS曲面，运用节点插入技术和Bézier曲面的几何连续理论成果，导出了它们之间的几何光滑拼接条件，同时得到了公共边界曲线所必须满足的本征方程，其中本征方程是NURBS曲面所独有的现象。这些结果对我们改进工程中所使用的NURBS曲面模型有着重要的指导意义。
用NURBS曲面分片逼近物体表面带来的问题就是要处理大量的控制顶点，特别是N面角附近的控制顶点更需要特别处理，因为曲面片之间的几何连续约束在角点处互相缠结。一般不采用整体求解所有角点附近的几何连续约束组成的方程组来调整角点附近的控制顶点，因为这样就丧失局部调整曲面形状的功能。通常都采用“局部”方法来处理角点附近的几何连续约束，即将约束涉及到的控制顶点从相邻曲面片中“剥离”出来，这样调整这些顶点只会影响角点附近的曲面形状。利用“局部格式”构造G~1光滑的曲面模型，一般有两种方式可供选择：一是加细原始剖分将连续性约束局部化以提供足够的自由度；二是根据连续性约束的传播性质适当提高曲面片的次数。本文论证了对剖分不作特殊要求时，内部单节点的B样条曲面要具备这种“局部”调整的特征，双五次是最低的要求。已有的曲面重构方法大多采用Bézier曲面工具，并且采取了加细原始剖分的方式。本文论述的“局部格式”调整方法不对剖分区域施加任何限制，主要采用内部具有二重节点的双四次B样条曲面和单节点的双五次B样条曲面做为拟合工具，给出了“局部”调整的方法，该方法能很好的保持拼接曲面的几何特征，克服了许多已有的重构方法仅采用简单共线法处理几何光滑性的弊端。采用内部具有二重结点的双四次B样条曲面做为拟合工具保证了单片曲面内部是二阶参数连续的，并且适当降低了曲面片的次数，基本上能够满足工程的需要。
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 specific
111
Abstract
restriction 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.
节点向量是内部单节点的NURBS曲面是最简单也是最常用的，本文重点针对单节点的双四次、双五次和双k次B样条曲面及双三次NURBS曲面，运用节点插入技术和Bézier曲面的几何连续理论成果，导出了它们之间的几何光滑拼接条件，同时得到了公共边界曲线所必须满足的本征方程，其中本征方程是NURBS曲面所独有的现象。这些结果对我们改进工程中所使用的NURBS曲面模型有着重要的指导意义。
用NURBS曲面分片逼近物体表面带来的问题就是要处理大量的控制顶点，特别是N面角附近的控制顶点更需要特别处理，因为曲面片之间的几何连续约束在角点处互相缠结。一般不采用整体求解所有角点附近的几何连续约束组成的方程组来调整角点附近的控制顶点，因为这样就丧失局部调整曲面形状的功能。通常都采用“局部”方法来处理角点附近的几何连续约束，即将约束涉及到的控制顶点从相邻曲面片中“剥离”出来，这样调整这些顶点只会影响角点附近的曲面形状。利用“局部格式”构造G~1光滑的曲面模型，一般有两种方式可供选择：一是加细原始剖分将连续性约束局部化以提供足够的自由度；二是根据连续性约束的传播性质适当提高曲面片的次数。本文论证了对剖分不作特殊要求时，内部单节点的B样条曲面要具备这种“局部”调整的特征，双五次是最低的要求。已有的曲面重构方法大多采用Bézier曲面工具，并且采取了加细原始剖分的方式。本文论述的“局部格式”调整方法不对剖分区域施加任何限制，主要采用内部具有二重节点的双四次B样条曲面和单节点的双五次B样条曲面做为拟合工具，给出了“局部”调整的方法，该方法能很好的保持拼接曲面的几何特征，克服了许多已有的重构方法仅采用简单共线法处理几何光滑性的弊端。采用内部具有二重结点的双四次B样条曲面做为拟合工具保证了单片曲面内部是二阶参数连续的，并且适当降低了曲面片的次数，基本上能够满足工程的需要。
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 specific
111
Abstract
restriction 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.
引文
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