形状可调插值曲线曲面的参数选择

目的 因大多数插值基函数中的参数都是全局参数,从而导致插值曲线曲面的形状无法进行局部调整。另外,当插值曲线曲面形状可调时,也存在如何选择参数才能获得形状较为理想的曲线曲面的问题,为此给出一种无需反求控制顶点、包含局部形状调整参数、具有显式表达式、能重构部分二次曲线曲面的插值曲线曲面构造方法,同时给出易于使用的形状参数确定方案。方法 基于经典3次Hermite插值曲线的Bernstein基函数表达形式,将其中的Bernstein基换成已证明具有全正性的一组三角基函数,根据三角基的端点性质调整曲线表达式以保证其插值性,然后设定插值数据点处的导向量,在其中引入参数,并保证相邻曲线段之间的连续性,得到了一种新的三角基插值曲线。结果 新曲线可以整理成以待插值数据点为控制顶点与一组插值基函数的线性组合形式,插值基表达式简单,插值曲线含一组局部形状调整参数,一个参数的改变只影响一条曲线段的形状,相邻曲线段之间G¹连续,曲线可以重构椭圆。根据不同目标给出了3种用于确定曲线中形状参数的准则,每种准则都提供了可以直接使用的公式。相应的插值曲面具有与插值曲线类似的性质。结论 形状参数选取准则的给出使含参数插值曲线曲面的设计由随意变为确定,这使得采用本文方法更易于得到满意的结果。本文所给插值基函数的构造方法具有一般性,可以采用相同的思路构造其他函数空间上性质类似的插值基。     

A surface modeling method based on the envelope template

In this paper, we present a novel surface modeling scheme based on an envelope template. A two-parameter family of interpolating surfaces is generated by repeated bicubic interpolation of the given data points, and then a solution to the envelope condition and the envelope of the family are constructed. The continuity conditions of two adjacent patches along the common boundary are derived by analyzing the geometric properties of the envelope patch. In order to facilitate surface modeling, an envelope template is constructed, which has many desirable advantages including simple structure, good local features and so on. G² or C² composite surfaces can be obtained utilizing the envelope template sweeping over the data points.     

Curvature continuous interpolation of curve meshes

This paper presents a method for local construction of a curvature continuous (GC²) piecewise polynomial surface which interpolates a given rectangular curvature continuous quintic curve mesh. First, we create a C² quintic basic curve mesh, which interpolates the same grid points, preserves the tangent slopes and curvatures but not derivative vectors at the grid points. After estimating twist and higher order mixed partial derivatives for each grid point, we generate locally the so-called C² biquintic basic patches, which serve to compute the first and second order cross-derivative functions of the final interpolation surface. In order to match the tangents and second order derivative vectors of the original boundary curves at the grid points, these basic patches are reparametrized by 5 × 3 and 3 × 5 functions, which lead to vector-valued first and second order cross-derivative functions of degrees 7 and 9 of the final surface patches, and eventually lead to a GC² piecewise polynomial surface of degree 9 × 9, which is then converted to a GC² Bézier composite surface. By arranging the surface patches in a chess-board fashion, the degrees of the final surface patches can be 9 × 5 and 5 × 9. An example for the construction of a GC² ship hull, together with its color-coded curvature maps, is given to illustrate the method. This method is attractive because the final surface has a much lower degree than other similar methods, it allows flexible local modification of the original curve mesh and local editing of the interpolation surface, and it is easily integrated into state-of-the-art geometric modeling systems.     

Flexible G1 interpolation of quad meshes

Transforming an arbitrary mesh into a smooth G¹ surface has been the subject of intensive research works. To get a visual pleasing shape without any imperfection even in the presence of extraordinary mesh vertices is still a challenging problem in particular when interpolation of the mesh vertices is required. We present a new local method, which produces visually smooth shapes while solving the interpolation problem. It consists of combining low degree biquartic Bézier patches with minimum number of pieces per mesh face, assembled together with G¹-continuity. All surface control points are given explicitly. The construction is local and free of zero-twists. We further show that within this economical class of surfaces it is however possible to derive a sufficient number of meaningful degrees of freedom so that standard optimization techniques result in high quality surfaces.     

GC 1 multisided Bézier surfaces

This paper presents a new method for generating a tangent-plane continuous (GC¹) multisided surface with an arbitrary number of sides. The method generates piecewise biquintic tensor product Bézier patches which join each other with G¹-continuity and which interpolate the given vector-valued first order cross-derivative functions along the boundary curves. The problem of the twist-compatibility of the surface patches at the center points is solved through the construction of normal-curvature continuous starlines and by the way the twists of surface patches are generated. This avoids the inter-relationship among the starlines and the twists of surface patches at the center points. The generation of the center points and the starlines has many degrees of freedom which can be used to modify and improve the quality of the resulting surface patches. The method can be used in various geometric modeling applications such as filling n-sided holes, smoothing vertices of polyhedral solids, blending multiple surfaces, and modeling surface over irregular polyhedral line and curve meshes.     

Shape aware normal interpolation for curved surface shading from polyhedral approximation

Independent interpolation of local surface patches and local normal patches is an efficient way for fast rendering of smooth curved surfaces from rough polyhedral meshes. However, the independently interpolating normals may deviate greatly from the analytical normals of local interpolating surfaces, and the normal deviation may cause severe rendering defects when the surface is shaded using the interpolating normals. In this paper we propose two novel normal interpolation schemes along with interpolation of cubic Bézier triangles for rendering curved surfaces from rough triangular meshes. Firstly, the interpolating normal is computed by a Gregory normal patch to each Bézier triangle by a new definition of quadratic normal functions along cubic space curves. Secondly, the interpolating normal is obtained by blending side-vertex normal functions along side-vertex parametric curves of the interpolating Bézier surface. The normal patches by these two methods can not only interpolate given normals at vertices or boundaries of a triangle but also match the shape of the local interpolating surface very well. As a result, more realistic shading results are obtained by either of the two new normal interpolation schemes than by the traditional quadratic normal interpolation method for rendering rough triangular meshes.     

A two-level approach to implicit surface modeling with compactly supported radial basis functions

We describe a two-level method for computing a function whose zero-level set is the surface reconstructed from given points scattered over the surface and associated with surface normal vectors. The function is defined as a linear combination of compactly supported radial basis functions (CSRBFs). The method preserves the simplicity and efficiency of implicit surface interpolation with CSRBFs and the reconstructed implicit surface owns the attributes, which are previously only associated with globally supported or globally regularized radial basis functions, such as exhibiting less extra zero-level sets, suitable for inside and outside tests. First, in the coarse scale approximation, we choose basis function centers on a grid that covers the enlarged bounding box of the given point set and compute their signed distances to the underlying surface using local quadratic approximations of the nearest surface points. Then a fitting to the residual errors on the surface points and additional off-surface points is performed with fine scale basis functions. The final function is the sum of the two intermediate functions and is a good approximation of the signed distance field to the surface in the bounding box. Examples of surface reconstruction and set operations between shapes are provided.     

Marching along surfacesurface intersection curves with an adaptive step length

A method for generating points on the intersection of two C² smooth parametric surfaces is presented. In each generated point the tangent and the curvature of the intersection curve are obtained from the surface positions, first and second derivatives. Initial approximation of the next point lies on a parabola approximating the intersection curve in a vicinity of the last point found. The length of the parabola between the two points is evaluated so that its maximal deviation from the chord joining the points is not greater than a given deviation tolerance. The new point is relaxed to the intersection curve.     

A Multi‐sided Bézier Patch with a Simple Control Structure

A new n‐sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding cross‐derivatives are specified as conventional Bézier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented. This Generalized Bézier (GB) patch is based on a new displacement scheme that builds up multi‐sided patches as a combination of a base patch, n displacement patches and an interior patch; this is considered to be an alternative to the Boolean sum concept. The input ribbons may have different degrees, but the final patch representation has a uniform degree. Interior control points—other than those specified by the user—are placed automatically by a special degree elevation algorithm. GB patches connect to adjacent Bézier surfaces with G¹continuity. The control structure is simple and intuitive; the number of control points is proportional to those of quadrilateral control grids. The scheme is introduced through simple examples; suggestions for future work are also discussed.     

Smooth multiple B-spline surface fitting with Catmull%ndash;Clark subdivision surfaces for extraordinary corner patches

This paper presents an algorithm for simultaneously fitting smoothly connected multiple surfaces from unorganized measured data. A hybrid mathematical model of B-spline surfaces and Catmull–Clark subdivision surfaces is introduced to represent objects with general quadrilateral topology. The interconnected multiple surfaces are G ² continuous across all surface boundaries except at a finite number of extraordinary corner points where G ¹ continuity is obtained. The algorithm is purely a linear least-squares fitting procedure without any constraint for maintaining the required geometric continuity. In case of general uniform knots for all surfaces, the final fitted multiple surfaces can also be exported as a set of Catmull–Clark subdivision surfaces with global C ² continuity and local C ¹ continuity at extraordinary corner points. Published online: 14 May 2002 Correspondence to: W. Ma     

高阶连续的形状可调三角多项式曲线曲面

目的目前使用的B样条曲线曲面存在着高连续阶与高局部调整性两者无法兼而有之的不足,且B样条曲线曲面的形状被控制顶点和节点向量唯一确定,这些因素影响着B样条方法的几何设计效果与方便性。本文旨在克服这种局限,以期构造具有高次B样条方法的高连续阶,低次B样条方法的高局部调整性,以及有理B样条方法权因子决定的形状调整性的曲线曲面。方法在三角函数空间上构造了一组含参数的调配函数,进而定义具有与3次B样条曲线曲面相同结构的新曲线与张量积曲面。结果新曲线曲面继承了B样条方法的凸包性、对称性、几何不变性等诸多性质。不同的是,同样是基于4点分段,3次均匀B样条曲线C2连续,而对于等距节点,在一般情况下,新曲线C5连续,当参数取特殊值时可达C7连续。新曲线在C5连续的情况下存在1个形状参数,能较好地调整曲线的形状同时又无须改变控制顶点。另外,将形状参数设为特定值,新曲线可以自动插值给定点列。新曲面具有与新曲线相应的优点。结论在强局部性下实现高阶连续性的形状可调分段组合曲线曲面,为高阶光滑曲线曲面的设计提供了可能,并且新曲线实现了逼近与插值的统一表示,能较好地应用于工程实际。调配函数的构造方法具有一般性,可用相同方式构造其他具有类似性质的调配函数。     

Triangular bubble spline surfaces

We present a new method for generating a Gⁿ-surface from a triangular network of compatible surface strips. The compatible surface strips are given by a network of polynomial curves with an associated implicitly defined surface, which fulfill certain compatibility conditions. Our construction is based on a new concept, called bubble patches, to represent the single surface patches. The compatible surface strips provide a simple Gⁿ-condition between two neighboring bubble patches, which are used to construct surface patches, connected with Gⁿ-continuity. For n≤2, we describe the obtained Gⁿ-condition in detail. It can be generalized to any n≥3. The construction of a single surface patch is based on Gordon–Coons interpolation for triangles.Our method is a simple local construction scheme, which works uniformly for vertices of arbitrary valency. The resulting surface is a piecewise rational surface, which interpolates the given network of polynomial curves. Several examples of G⁰, G¹ and G²-surfaces are presented, which have been generated by using our method. The obtained surfaces are visualized with reflection lines to demonstrate the order of smoothness.     

GenLab: A MATLAB®-based program for structural analysis of folds mapped by GPS or seismic methods

GenLab is a MATLAB^® code for structural analysis of digitally and seismically mapped horizons based on the geometrical model of the developable surface. This model is chosen because surfaces of this type result from the bending of a non-stretching stratum and because deviation from developable geometry implies that bed stretching during folding has occurred. The model is therefore linked with straining of the horizon and can be used for highlighting regions of deformation expressed by brittle fracturing or ductile deformation. Mathematically, a developable surface is one upon which straight lines (generators) can be drawn, and along these generators the orientations of tangent planes of the surface remain constant. Our method involves treating real geological surfaces as approximations to developable surfaces and determines the generator lines at different points on the surface. At these points we consider strip samples of the surface radiating in different directions and examine the variability of strike and dip within each sample strip. The radial scan-line length parameter is user-defined, and the optimal value can be easily identified by geostatistical analysis of the depth data. The local plunge lines (the approximation to the generatrix of a developable fold) are therefore identified as the direction associated with the least variation of surface attitude, i.e., the most constant strike and dip. The map pattern obtained by analyzing the plunges and trends of plunge lines across an area allows identification of domains where folding accords closely with a developable geometry. Such domains are recognized from straight plunge lines, defining parallel or convergent patterns. According to the model, deviations from these patterns correspond to regions of structural complexity associated with ductile or brittle straining of the surface being analyzed. The plunge-line analysis therefore offers a useful technique with potential for predicting strained zones and automatic fault recognition. GenLab allows the rapid implementation of this analysis and uses the matrix capabilities and contouring functions of MATLAB.     

Optimization methods for scattered data approximation with subdivision surfaces

We present a method for scattered data approximation with subdivision surfaces which actually uses the true representation of the limit surface as a linear combination of smooth basis functions associated with the control vertices. A robust and fast algorithm for exact closest point search on Loop surfaces which combines Newton iteration and non-linear minimization is used for parameterizing the samples. Based on this we perform unconditionally convergent parameter correction to optimize the approximation with respect to the L² metric, and thus we make a well-established scattered data fitting technique which has been available before only for B-spline surfaces, applicable to subdivision surfaces. We also adapt the recently discovered local second order squared distance function approximant to the parameter correction setup. Further we exploit the fact that the control mesh of a subdivision surface can have arbitrary connectivity to reduce the L^∞ error up to a certain user-defined tolerance by adaptively restructuring the control mesh. Combining the presented algorithms we describe a complete procedure which is able to produce high-quality approximations of complex, detailed models.     

基于曲率优化的机器人砂带磨削轨迹规划

针对复杂曲面零件砂带磨削编程效率低、精度差的问题,基于B样条曲线曲面重构和机器人离线编程技术,提出了一种根据关键接触点曲率值生成工业机器人磨削轨迹的方法.首先,利用零件表面上需要进行砂带磨削的关键接触点和积累弦长参数化法构造节点矢量,从而计算出磨削轨迹的B样条基函数;其次,根据控制顶点反求矩阵得到全部未知控制点和3次B样条加工曲线;然后,分析关键接触点之间的曲率变化率和弧长,对关键点细化生成符合磨削工艺要求的目标点;最后,通过求解双3次B样条插值曲面方程获得目标点的加工姿态.以水龙头磨削为例进行试验,结果表明曲率优化算法磨削的零件表面轮廓形状明显优于截面法,且其粗糙度值能稳定在0.082 μm左右,可以有效提高工件表面加工质量.     

Interpolating G Bézier surfaces over irregular curve networks for ship hull design

We propose a local method of constructing piecewise G¹ Bézier patches to span an irregular curve network, without modifying the given curves at odd- and 4-valent node points. Topologically irregular regions of the network are approximated by implicit surfaces, which are used to generate split curves, which subdivide the regions into triangular and/or rectangular sub-regions. The subdivided regions are then interpolated with Bézier patches. We analyze various singular cases of the G¹ condition that is to be met by the interpolation and propose a new G¹ continuity condition using linear and quartic scalar weight functions. Using this condition, a curve network can be interpolated without modification at 4-valent nodes with two collinear tangent vectors, even in the presence of singularities. We demonstrate our approach in a ship hull.     

Modifying CAD/CAM surfaces according to displacements prescribed at a finite set of points

This article presents a method for modifying CAD/CAM surfaces automatically in accordance with displacements prescribed at a finite set of points in R³, such as node displacements predicted by finite-element analysis. The method is based on the ‘morphing’ approach introduced by Sederberg and Parry in 1986. The input to the process consists of (a) a CAD/CAM model containing trimmed polynomial B-spline surfaces and (b) a set of points and associated displacement vectors in R³. These points are assumed to be close to, but not necessarily on, the objects of the CAD/CAM model. A rectangular volume, enclosing the CAD/CAM model and the input points in R³, is represented as a volume spline, i.e. a trivariate tensor-product spline. A modified volume spline is computed using (a) a least-squares fit based on the given point displacements, and (b) a smoothing functional. The modified CAD/CAM objects are defined as compositions of the original parametric functions and the modified volume spline (i.e. a morphing). In order to ensure compatibility with standard commercial CAD/CAM systems, the modified surfaces are fitted with appropriate splines using any standard, reasonably shape-preserving, fitting procedure applied in the parameter domains of the original surfaces.     

The conformal alpha shape filtration

Conformal alpha shapes are a new filtration of the Delaunay triangulation of a finite set of points in ℝ^d. In contrast to (ordinary) alpha shapes the new filtration is parameterized by a local scale parameter instead of the global scale parameter in alpha shapes. The local scale parameter conforms to the local geometry and is motivated from applications and previous algorithms in surface reconstruction. We show how conformal alpha shapes can be used for surface reconstruction of non-uniformly sampled surfaces, which is not possible with alpha shapes.     

Towards flattenable mesh surfaces

In many industries, products are constructed by assembled surface patches in ℜ³, where each patch is expected to have an isometric map to a corresponding region in ℜ². The widely investigated developable surfaces in differential geometry show this property. However, the method to model a piecewise-linear surface with this characteristic is still under research. To distinguish from the continuous developable surface, we name them as flattenable mesh surfaces since a polygonal mesh has the isometric mapping property if it can be flattened into a two-dimensional sheet without stretching. In this paper, a novel flattenable mesh surface (Flattenable Laplacian mesh) is introduced and the relevant modelling tool is formulated. Moreover, for a given triangular mesh which is almost flattenable, a local perturbation approach is developed to improve its flattenability. The interference between the meshes under process and their nearby objects has been prevented in this local flattenable perturbation. Both the computations of Flattenable Laplacian meshes and the flattenable perturbation are based on the constrained optimization technology.     

Hermite variational implicit surface reconstruction

We propose a new technique for reconstructing surfaces from a large set of unorganized 3D data points and their associated normal vectors. The surface is represented as the zero level set of an implicit vol-ume model which fits the data points and normal constraints. Compared with variational implicit sur-faces,we make use of surface normal vectors at data points directly in the implicit model and avoid of introducing manufactured off-surface points. Given n surface point/normal pairs,the proposed method only needs to solve an n×n positive definite linear system. It allows fitting large datasets effectively and robustly. We demonstrate the performance of the proposed method with both globally supported and compactly supported radial basis functions on several datasets.