B样条曲线曲面的节点消去算法研究

在分析了Tiller给出的B样条曲线节点消去算法的基础上,提出了改进算法。改进算法充分地利用了B样条曲线的局部性质,无需考虑节点消去的顺序,一次消去多个节点。实验表明,与Tiller的算法相比较,改进后的算法效率有较大提高。     

B-spline scale-space of spline curves and surfaces

We present iterative algorithms for $B$-spline scale-space smoothing of geometric data and recovery of high frequency information in the smoothing process. The scale-space representation is based on a directional smoothing process using $B$-splines. If the geometric data are approximated or modelled by uniform $B$-splines or box-splines then the scale-space smoothing produces $B$-spline curves or box-spline surfaces. The method is applicable to geometric data processing and geometric modelling of free-form curves and surfaces from quadrilateral polyhedra with extraordinary vertices.     

Alternative representation for parametric cubic curves and surfaces

A representation for parametric cubic curves and surfaces is presented which incorporates the polygonal approach popularized by the Bézier and B-spline schemes. Since the curves more closely mimic the polygon than their counterparts employing the Bézier or B-spline schemes, the method is potentially useful for creating and manipulating geometric models. In connection with the logical extension of the approach from curves to surfaces, a discussion of the relationship between continuity and redundant data storage is included.     

Offsets of curves on rational B-spline surfaces

The construction of offset curves is an important problem encountered often in design processes and in interrogation of geometric models. In this paper the problem of construction of offsets of curves lying on the same parametric surface is addressed. A novel algorithm is introduced, whose main feature is the use of geodesic paths to determine points of the offset. The offset is then approximated in the underlying surface parameter space by B-splines interpolating data points obtained by traveling a known distance along the geodesics departing from corresponding points of the progenitor in a direction perpendicular to the latter. A comprehensive error checking scheme has been devised allowing adaptive improvement of the approximation of the offset. The applicability of the algorithm is demonstrated by number of numerical examples.     

Exact and approximate computation of B-spline curves on surfaces

Curves on surfaces are important elements in computer aided geometric design. After presenting a method to explicitly compute these curves in three-dimensions, practical algorithmic issues are discussed concerning the efficiency of the implementation. Good approximations are important because of the quite high degree of exact curves on surfaces. We present two approximate solutions to the problem. The first is derived from the exact representation, while the second extends conventional least-squares approximation by incorporating the geometry of the surface as well. The efficiency and behaviour of the algorithms are evaluated by means of examples.     

Parametrization of Bézier-type B-spline curves and surfaces

Sections of parametric surfaces defined by equally spaced parameter values can be very unevenly spaced physically. This can cause practical problems when the surface is to be drawn or machined automatically. This paper describes a method for imposing a good parametrization on a curve constructed by Bézier's method and based on B-splines. The extension of the method to the parametrization of surfaces is considered briefly.     

Recursive algorithms for the representation of parametric curves and surfaces

Recursive algorithms for the representation of parametric curves and surfaces are presented which are based upon a geometric property of the de Casteljau algorithm. The algorithms work with triangular and pyramidal arrays that provide an easy handling of the curve and the surface ‘in a large’ design.     

Reconstruction of B-spline curves and surfaces by adaptive group testing

Point clouds as measurements of 3D sensors have many applications in various fields such as object modeling, environment mapping and surface representation. Storage and processing of raw point clouds is time consuming and computationally expensive. In addition, their high dimensionality shall be considered, which results in the well known curse of dimensionality. Conventional methods either apply reduction or approximation to the captured point clouds in order to make the data processing tractable. B-spline curves and surfaces can effectively represent 2D data points and 3D point clouds for most applications. Since processing all available data for B-spline curve or surface fitting is not efficient, based on the Group Testing theory an algorithm is developed that finds salient points sequentially. The B-spline curve or surface models are updated by adding a new salient point to the fitting process iteratively until the Akaike Information Criterion (AIC) is met. Also, it has been proved that the proposed method finds a unique solution so as what is defined in the group testing theory. From the experimental results the applicability and performance improvement of the proposed method in relation to some state-of-the-art B-spline curve and surface fitting methods, may be concluded.     

Cubic B-spline curves and surfaces in computer aided geometric design

Representations of cubic and bicubic splines are given, combining the advantages of B-splines with the handiness of Bézier technique. The Bézier points of spline curves and surfaces are found by forming convex combinations of nodes. The given algorithms are suited especially for computer aided geometric design.     

Knot Insertion Algorithms for ECT B-spline Curves

Knot insertion algorithm is one of the most important technologies of B-spline method. By inserting a knot the local prop- erties of B-spline curve and the control flexibility of its shape can be fiu＇ther improved, also the segmentation of the curve can be rea- lized. ECT spline curve is drew by the multi-knots spline curve with associated matrix in ECT spline space; Muehlbach G and Tang Y and many others have deduced the existence and uniqueness of the ECT spline function and developed many of its important properties .This paper mainly focuses on the knot insertion algorithm of ECT B-spline curve.It is the widest popularization of B-spline Behm algorithm and theory. Inspired by the Behm algorithm, in the ECT spline space, structure of generalized P61ya poly- nomials and generalized de Boor Fix dual functional, expressing new control points which are inserted after the knot by linear com- bination of original control vertex the single knot, and there are two cases, one is the single knot, the other is the double knot. Then finally comes the insertion algorithm of ECT spline curve knot. By application of the knot insertion algorithm, this paper also gives out the knot insertion algorithm of four order geometric continuous piecewise polynomial B-spline and algebraic trigonometric spline B-spline, which is consistent with previous results.     

Fast variational design of multiresolution curves and surfaces with B-spline wavelets

Multiresolution modeling provides a powerful tool for complex shape editing. To achieve a better control of deformations and a more intuitive interface, variational principles have been used in such multiresolution models. However, when handling multiresolution constraints, the existing methods often result in solving large optimization systems. Hence, the computational time may become too excessive to satisfy the requirements for interactive design in CAD. In this paper, we present a fast approach for interactive variational design of multiresolution models. By converting all constraints at different levels to a target level, the optimization problem is formulated and solved at the lower level. Thus, the unknown coefficients of the optimization system are significantly reduced. This improves the efficiency of variational design. Meanwhile, to avoid smoothing out the details of the shape in variational modeling, we optimize the change in the deformation energy instead of the total energy of the deformed shape. Several examples and the experimental results are given to demonstrate the effectiveness and efficiency of this approach.     

Construction of flexible blending parametric surfaces via curves

The main purpose of this paper is to provide a method that allows to solve the blending problem of two parametric surfaces. The blending surface is constructed with a collection of space curves defined by point pairs on the blending boundaries of given primary surfaces. Bézier and C-cubic curves are used to interpolate the blending boundaries. The blending surface is Gⁿ continuously connected to the primary surfaces.     

Generating the Bézier points of B-spline curves and surfaces

The well-known algorithm by de Boor for calculating a point of a B-spline curve can also be used to produce the Bézier points of a B-spline curve or surface.     

Systolic architecture for B-spline surfaces

B-spline surfaces are among the most commonly used types of surfaces for modeling objects in computer graphics and CAD applications. One of the time consuming operations in B-spline surface generation is that of inversion. An efficient algorithm is proposed for solving this problem. This algorithm is implemented on a systolic architecture in order to facilitate fast interactive surface design.     

Method for fairing B-spline surfaces

A method of fairing tensor product B-spline surfaces is described. The technique is based on automatic repositioning of the surface control points by a constrained minimization algorithm. The objective function is based on a measure of the surface curvature, and the constraint is a measure of the distance between the original and the modified surfaces. Changes in surface shapedare analysed from illustrations of plane sections, contour plots of surface slope and Gaussian curvature, and colour plots of Gaussian curvature. These illustrations indicate that the present form of the technique is effective in some circumstances and suggest that improved implementations will produce more generally useful results.     

Convex B-spline surfaces

This paper gives a definition for the convexity of B-spline surfaces and points out the conditions, on which the convexity depends.A back shift smoothing method is introduced. This method is built on the basis of the convexity conditions. Application of this smoothing method gives a strictly convex curve.     

Parametrization and shape of B-spline curves for CAD

It is found that Bézier-type B-spline curves cannot, in general, be given an arc length parametrization. In view of this, two ways of choosing knots are discussed: an iterative method and a simple formula. The formula, already published in the context of ab initio design, is found to be useful when applied to interpolating B-apline curves; when the B-spline nodes are used as parameter values, good shape and good parametrization are usually achieved.