The current cubic spline curve interpolation scheme is derived based on the implicit assumption that the magnitude of the first derivative of the curve is close to a constant. However, the assumption is not realized by the interpolation scheme. A knot choosing technique for parametric cubic spline interpolating curve construction that accommodates this assumption is presented. A comparison of the new method with several existing methods is performed and test results are included.

New algorithms for the classical problem of fairing cubic spline curves ans bicubic spline surfaces are presented. To fair a cubic spline curve or a bicubic spline surface with abonormal portions, the algorithms (automatically or interactively) identify the 'bad' data points and replace them with new pointsproduced by minimizing the strain energy of the new cureve or suface. The proposed algorithms are more general than the existing algorithms in that the new algorithms can adjust more than one'bad' data point in each modification step and they include the existing algorithms in that the new algorithms can adjust more than one 'bad' data point in each modification step and they include the existing algorithms [Computer-Aided Design 15 (5) (1983) 288-293; 28 (1996) 59-66] as special cases. Test results ofthe new algorithms are included.

A new method to scale a trimmed NURBS surface while holding the shape and size of specific features (trimming curves) unchanged is presented. The new method is fix-and-stretch based: the new surface is formed by fixing selected regions of the given trimmed NURBS surface that contain the trimming curves and stretching the remaining part of the surface to reach certain boundary conditions. The stretching process is performed using an optimization process to ensure that the resulting surface reflects the shape and curvature distribution of the scaled version of the given surface. The resulting surface maintains a NURBS representation and, hence, is compatible with most of the current data-exchange standards. The new approach is more robust than a previous, attach-and-deform based approach (Zhang P, Zhang C, Cheng F. Constrained shape scaling of trimmed NURBS surfaces. In: Proceedings of the 1999 ASME Design Theory and Methodology Conference. Las Vegas, Nevada, 1999) in that it can tolerate scaling factors of bigger values (up to 2 in some cases). The new approach also guarantees that the features remain exactly the same after scaling. Test results on several car parts with trimming curves and comparison with the previous approach are included. The quality of the resulting surfaces is examined using the highlight line model. The presented technique is important for integrating standard parts into a sculptured product.

讨论了构造C2连续的四次样条插值曲面问题。把四次样条函数降为C2连续可提供额外的自由度，用于提高曲面的插值精度和控制曲面的形状。给出了一个确定自由度的方法和C2连续的四次样条曲线需满足的连续性方程，提出了构造C2连续的四次样条插值曲面的新方法。新方法的特点是曲面需满足的连续性方程是三对角占优势的，曲面的不连续点在给定的数据点处。所构造的曲面具有四次多项式插值精度。最后以实例对新方法和现有三、四次样条函数方法的插值精度做了比较。

The construction of a GC1 cubicinterpolatiing curve that lies on the same side of a given straight line as the data points is studied. The main task is to choose appropriate approaches to modify tangent vectors at the data points for the desired curve. Three types of approches for changing the magnitudes of the tangent vectors are presented.The first-type approach modifies the tanggent vectors by applying a constraint to the curve segment. The second one does thework by optimization techniques. The thired one is a modification of the existing method. Three critera are prsented to compare the three types of approaches with the existing method. the experiments that test the effectiveess of the approaches are included.