均匀B样条曲线曲面的扩展

在多项式空间提出了一种带k个形状参数的k次均匀B样条,这类曲线与标准k次均匀B样条类似,每段曲线由k+1个控制顶点生成,它们不仅具有k次均匀B样条许多常见性质,而且利用形状参数的不同取值能够整体或局部调控曲线曲面形状。包含标准均匀B样条为其特例。     

Compactly-supported smooth interpolators for shape modeling with varying resolution

In applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape itself. Additionally, models for shape representation often need to provide local shape control and they need to be able to reproduce common shape primitives such as ellipsoids, spheres, cylinders, or tori. We present a general framework to construct families of compactly-supported interpolators that are piecewise-exponential polynomial. They can be designed to satisfy regularity constraints of any order and they enable one to build parametric deformable shape models by suitable linear combinations of interpolators. They allow to change the resolution of shapes based on the refinability of B-splines. We illustrate their use on examples to construct shape models that involve curves and surfaces with applications to interactive modeling and character design.     

Control Points for Multivariate B-Spline Surfaces over Arbitrary Triangulations

This paper describes first results of a test implementation that implements the new multivariate B-splines as recently developed by Dahmen et al. 10for quadratics and cubics. The surface scheme is based on blending functions and control points and allows the modelling of   C ^{k − 1}  -continuous piecewise polynomial surfaces of degree k over arbitrary triangulations of the parameter plane. The surface scheme exhibits both affine invariance and the convex hull property, and the control points can be used to manipulate the shape of the surface locally. Additional degrees of freedom in the underlying knot net allow for the modelling of discontinuities. Explicit formulas are given for the representation of polynomials and piecewise polynomials as linear combinations of B-splines.     

Triangular NURBS and their dynamic generalizations

Triangular B-splines are a new tool for modeling a broad class of objects defined over arbitrary, nonrectangular domains. They provide an elegant and unified representation scheme for all piecewise continuous polynomial surfaces over planar triangulations. To enhance the power of this model, we propose triangular NURBS, the rational generalization of triangular B-splines, with weights as additional degrees of freedom. Fixing the weights to unity reduces triangular NURBS to triangular B-splines. Conventional geometric design with triangular NURBS can be laborious, since the user must manually adjust the many control points and weights. To ameliorate the design process, we develop a new model based on the elegant triangular NURBS geometry and principles of physical dynamics. Our model combines the geometric features of triangular NURBS with the demonstrated conveniences of interaction within a physics-based framework. The dynamic behavior of the model results from the numerical integration of differential equations of motion that govern the temporal evolution of control points and weights in response to applied forces and constraints. This results in physically meaningful hence highly intuitive shape variation. We apply Lagrangian mechanics to formulate the equations of motion of dynamic triangular NURBS and finite element analysis to reduce these equations to efficient numerical algorithms. We demonstrate several applications, including direct manipulation and interactive sculpting through force-based tools, the fitting of unorganized data, and solid rounding with geometric and physical constraints.     

插值Doo-Sabin表面形状调节

修改了插值的Doo-Sabin细分表面的初始控制网格，在第一次细分的同时加入了表面调节参数。这个方案具有以下特征：1）满足插值所有顶点或某些顶点的同时可以由参数调节极限表面；增加了对极限表面的调节自由度。2）整个的计算复杂度为O(k),其中k是顶点的数量。在最后也对结果表面的形状处理进行了讨论。     

Controllable Locality in C2 Interpolating Curves by B2-splines / S-splines

This paper presents a systematic scheme for controlling the local behaviour of C² interpolating curves, based on the cubic B2-splines and the quartic S-splines. Both splines have an additional control point obtained by knot- insertion or degree-elevation in each span of the conventional uniform cubic interpolating B-splines. The shape designer can choose the desired range of locality for each span and get the corresponding additional control point as a barycentric combination of interpolation points within the range, without solving any variational problem and simultaneous equations. The scheme is consistent over the entire curve subject to some typical end conditions. The class of the curves derived includes the conventional cubic interpolating B-splines. Examples demonstrate the behaviour of the new interpolating curves and the capability of the scheme.     

Smoothing rational B-spline curves using the weights in an optimization procedure

Freeform shape design is typically accomplished in an interactive manner and shapes generated by a computer are rarely immediately acceptable. The available techniques for any subsequent modifications depend on the chosen representation for the geometry. In many computer aided styling and design systems which use nonuniform rational B-splines (NURBS) for representation of geometry, the use of the weights as a shape control tool is very inadequately supported. In fact they are often hidden from the user and therefore remain unused. This paper investigates the possibilities of entering the weights in an automatic fairing process. In order to produce a curve with a more gradual change in curvature and the smallest deviation from its initial shape the perturbation of the weights is stated as an optimization problem. Examples of applications to automotive shape design are presented and discussed.     

D-NURBS: a physics-based framework for geometric design

Presents dynamic non-uniform rational B-splines (D-NURBS), a physics-based generalization of NURBS. NURBS have become a de facto standard in commercial modeling systems. Traditionally, however, NURBS have been viewed as purely geometric primitives, which require the designer to interactively adjust many degrees of freedom-control points and associated weights-to achieve the desired shapes. The conventional shape modification process can often be clumsy and laborious. D-NURBS are physics-based models that incorporate physical quantities into the NURBS geometric substrate. Their dynamic behavior, resulting from the numerical integration of a set of nonlinear differential equations, produces physically meaningful, and hence intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and setting weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. We use Lagrangian mechanics to formulate the equations of motion for D-NURBS curves, tensor-product D-NURBS surfaces, swung D-NURBS surfaces and triangular D-NURBS surfaces. We apply finite element analysis to reduce these equations to efficient numerical algorithms computable at interactive rates on common graphics workstations. We implement a prototype modeling environment based on D-NURBS and demonstrate that D-NURBS can be effective tools in a wide range of computer-aided geometric design (CAGD) applications     

An adaptive B-spline representation of piecewise polynomial functions for multilevel approximation

Let I be some real interval endowed with an arbitrary partition τ. The aim of this work is to establish a new B-spline representation of the piecewise polynomial functions (p.p.f.) defined on I which can be used for a multilevel approximation. First, we show that any p.p.f. S can be written in terms of B-splines having τ as sequence of simple knots and the same smoothness order as S. This new family of B-splines has interesting properties similar to those of the well known B-splines. In order to illustrate the interest of this representation, we establish two methods which allow to approximate or project a given p.p.f. S in spaces with high smoothness order. At the end of this paper, we use this representation for constructing quasi-interpolants based on these methods.     

B样条曲线同时插入多个节点的快速算法

基于离散B样条的一个新的递推公式，提出B样条曲线同时插入多个节点的新算法。不同于Cohen等插入节点的Oslo算法，本算法用新的方法离算离散B样条，求每个离散B样条的值只需O(1)的运算量，从而使本算法高效，其时间复杂性为O(sk n)，其中k为B样条曲线的阶，n k 1为原节点数，s为新插入节点的个数,本算法的通用性强，适用于端点插值的和非端点插值的B样条曲线，可同时在曲线定义域内外的任意位置上插入任意个节点。     

一种n次均匀B样条曲线细分算法

利用 次均匀B样条细分的掩模与Pascal三角形关系,并借助控制多边形在每次加细过程中新旧控制顶点对应的几何位置关系,给出一种新的 次均匀B样条曲线细分算法,基于该算法构造出带有形状参数的局部插值约束的奇次均匀B样条细分曲线。通过理论和算例说明,该算法几何直观性强、新旧点对应明确、应用灵活且能保持良好的参数连续性。     

Shape Modification and Deformation of Parametric Surfaces

A new method for shape modification or deformation of all kinds of parametric surfaces is presented. Instead of using any control mesh and designing any auxiliary tools for modification or deformation, this method only needs to multiply the equation of the original surface by several shape operator matrixes so that deformation effects are achieved. Due to introducing interactive control parameters with different properties those parameters can be changed to get more ideal effects of deformation or shape modification. The core of the method is that so-called cosine extension function is proposed by which the shape operator matrix can be created. Experiment shows that the method is simple, intuitive and easy to control, and diversified local deformations or shape modifications can easily be made via small and random variations of interactive control parameters.     

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.     

A Subdivision Scheme for Continuous-Scale B-Splines and Affine-Invariant Progressive Smoothing

Multiscale representations and progressive smoothing constitutean important topic in different fields as computer vision, CAGD,and image processing. In this work, a multiscale representationof planar shapes is first described. The approach is based oncomputing classical B-splines of increasing orders, andtherefore is automatically affine invariant. The resultingrepresentation satisfies basic scale-space properties at least ina qualitative form, and is simple to implement.The representation obtained in this way is discrete in scale,since classical B-splines are functions in , where k isan integer bigger or equal than two. We present a subdivisionscheme for the computation of B-splines of finite support atcontinuous scales. With this scheme, B-splines representationsin are obtained for any real r in [0, ), andthe multiscale representation is extended to continuous scale.The proposed progressive smoothing receives a discrete set ofpoints as initial shape, while the smoothed curves arerepresented by continuous (analytical) functions, allowing astraightforward computation of geometric characteristics of theshape.     

Automatic Shape Control of Triangular B-Splines of Arbitrary Topology

Triangular B-splines are powerful and flexible in modeling a broader class of geometric objects defined over arbitrary, non-rectangular domains. Despite their great potential and advantages in theory, practical techniques and computational tools with triangular B-splines are less-developed. This is mainly because users have to handle a large number of irregularly distributed control points over arbitrary triangulation. In this paper, an automatic and efficient method is proposed to generate visually pleasing, high-quality triangular B-splines of arbitrary topology. The experimental results on several real datasets show that triangular B-splines are powerful and effective in both theory and practice.     

Artifact analysis on triangular box-splines and subdivision surfaces defined by triangular polyhedra

Surface artifacts are features in a surface which cannot be avoided by movement of control points. They are present in B-splines, box splines and subdivision surfaces. We showed how the subdivision process can be used as a tool to analyse artifacts in surfaces defined by quadrilateral polyhedra ( [Sabin et al., 2005] and [Augsd?rfer et al., 2011]).In this paper we are utilising the subdivision process to develop a generic expression which can be employed to determine the magnitude of artifacts in surfaces defined by any regular triangular polyhedra. We demonstrate the method by analysing box-splines and regular regions of subdivision surfaces based on triangular meshes: Loop subdivision, Butterfly subdivision and a novel interpolating scheme with two smoothing stages. We compare our results for surfaces defined by triangular polyhedra to those for surfaces defined by quadrilateral polyhedra.     

A hyper elasticity method for interactive virtual design of hearing aids

We present a computational efficient method for isotropic hyper elasticity based on functional analysis. By selecting a class of shape functions, we arrive at a computational scheme which yields very sparse tensors. This enables fast computations of the hyper elastic energy potential and its derivatives. We achieve efficiency and performance through the use of shape functions that are linear in their parameters and through rotation into the eigenspace of the right Cauchy–Green strain tensor. This makes near real time evaluation of hyper elasticity of complex meshes on CPU relatively easy to implement. The approach does not rely on a specific shape function or material model but offers a general framework for isotropic hyper elasticity. The method is aimed at interactive and accurate non-linear hyper elastic modeling for a wide range of industrial virtual design applications, which we exemplify by insertion of hearing aid domes into the ear canal. We validate the method for tetrahedral meshes with linear shape functions with an Ogden material model by comparing simulations to deformations of real material. We illustrate the use of other shape functions and models using uniform cubic B-splines in combination with Riemannian elasticity.     

非均匀B样条曲线升阶的新算法

实践证明，传统的Ｂ样条曲线升阶算法只能解决端点插值Ｂ样条曲线的升阶问题，当用于其它非均匀Ｂ样条曲线以及均匀Ｂ样条曲线的升阶进均会出现严重错误，本文基于一个新的Ｂ样条恒等式，提出了一个Ｂ样条曲线升阶的新算法，该算法可用于任何均匀和非均匀的Ｂ样条曲线的升阶，当用于一段均匀Ｂ样条曲线的升阶时，不需要的节点矢量中间插入任何节点，升阶后仍为一条均匀Ｂ样条曲线，其计算简便、速度快。本文最后还得到两个新结论：（