Cyclides in surface and solid modeling

It is shown that Dupin cyclides (C.P. Dupin, 1822), as surfaces in computer-aided geometric design (CAGD), have attractive properties such as low algebraic degree, rational parametric forms, and an easily comprehensible geometric representation using simple and intuitive geometric parameters. Their alternative representations permit the transition between forms when one or the other is more convenient for a specific purpose. Cyclides provide is useful extension of geometric coverage in solid modeling, primarily as blending surfaces for many commonly occurring situations. The geometry, properties, and uses of the Dupin cyclide in free-form surface modeling and blending are discussed     

Blending Parametric Patches with Subdivision Surfaces

In this paper the problem of blending parametric surfaces using subdivision patches is discussed. A new approach, named removing-boundary, is presented to generate piecewise-smooth subdivision surfaces through discarding the outmost quadrilaterals of the open meshes derived by each subdivision step. Then the approach is employed both to blend parametric bicubic B-spline surfaces and to fill n-sided holes. It is easy to produce piecewise-smooth subdivision surfaces with both convex and concave corners on the boundary, and limit surfaces are guaranteed to be C2 continuous on the boundaries except for a few singular points by the removing-boundary approach. Thus the blending method is very efficient and the blend-ing surface generated is of good effect.     

基于动态偏微分方程构造过渡面

为了实现交互式的偏微分方程曲面造型,针对传统静态偏微分方程构造过渡面存在的不足,提出了基于动态偏微分方程构造C1连续的过渡面,并引入迭代有限差分法求解偏微分方程的数值解,在此基础上构造了光滑过渡曲面.讨论了形状控制因子、密度、阻尼系数等物理参数的变化对曲面形状的影响,其中形状控制因子对生成曲面的形状影响最为明显.造型实例表明,利用动态偏微分方程构造过渡面具有更高的灵活性,大大提高了工业几何设计的交互性,在CAD/CAM中具有重要的应用价值.     

Controllable C1 continuous blending of time-dependent parametric surfaces

This paper proposes the concept of blending time-dependent varying surfaces, and develops a new method to create a controllable C1 continuous blending surface between primary parametric surfaces whose position and shape change with time. We treat it as a boundary-valued problem defined by the mathematical model of a vectored dynamic fourth-order partial differential equation subjected to time-dependent C1 continuous blending boundary constraints. High performance blending surface generation is achieved through the development of an approximate analytical solution of the mathematical model. We investigate the accuracy and efficiency of the solution, study the effective shape control of the blending surfaces, and apply the obtained solution to tackle surface blending problems. The applications demonstrate that our proposed approach is very effective and efficient in dealing with controllable C1 continuous surface blending between time-dependent varying parametric surfaces.     

Blending an implicit with a parametric surface

A method for blending an implicit surface with a parametric surface is introduced. The blending surface is defined as a collection of intersection curves of correlated pairs of surfaces. With a simple additional condition one can achieve G²-continuous transitions between the blending surface and the given surfaces.     

Computational methods for blending surface approximation

Blending surfaces form a smooth transition between two distinct, intersecting surfaces or smoothly join two or more disconnected surfaces and are normally procedural surfaces which are difficult to exchange and to interrogate in a reliable and efficient manner. In this paper, an approximation method for blending surfaces which are curvature continuous to the underlying surfaces with a non-uniform rational B-spline (NURBS) surface is presented. The use of NURBS is important since it facilitates the exchange of geometric information between various computer aided design and manufacturing systems. In the method, linkage curves on the underlying surfaces are approximated to within a specified tolerance and cross-link curves are created using the linkage curves, a directional curve and the parametric partial derivatives of the underlying surfaces. Cross-link curves are lofted to form the blending surface and an adaptive sampling procedure is used to test the blending surface against specified tolerances. Cross-link curves are added, where necessary, and the surface relofted until the continuity conditions are satisfied to within specified tolerances. Examples illustrate the applicability of the method.     

A Blend of Subdivision Surfaces and NURBS

In present paper, the contour deletion method is developed both to blend surfaces and to fill N-sided holes, which is used for subdividing the NURBS surface. First, according to the non-uniform Catmull-Clark subdivision principle, surfaces are blended. The non-uniform Catmull-Clark subdivision method is constructed, which build the surface through interpolating corner vertices and boundary curves. Then the contour deletion method is adapted to remove the controlling mesh boundary contour in the process of segmentation iteration. Last, N sided-hole is filled to generate a integral smooth continuous surface. This method not only guarantee that the blending surface and base surface patches have C2 continuity at the boundary, but also greatly improve the smoothness of the N-side hole filling surface. The results show that, this method simplifies the specific computer-implemented process, broads the scope of application of subdivision surfaces, and solves the incompatible problem between the subdivision surface and classical spline. The resulting surface has both advantages of the subdivision surface and classical spline, and also has better filling effect.     

基于结式方法的代数曲面拼接

以同伦连续映射理论为基础,构造代数曲面拼接应该满足的代数方程组。然后,利用结式方法消去相关变元得到拼接曲面方程。两代数曲面拼接时,方程组是两个关于单位区间变元的方程。利用Sylvester结式消去该变元即可得到曲面拼接方程。对于多代数曲面,拼接过程可以考虑为不同种的连续映射。由此得到三种不同的曲面拼接方法,即串接法、过渡法和提升法。串接法可得到较低次的拼接曲面,但适用于代数曲面两两拼接且过渡曲面不相交的情况;过渡法适用于所有情况,但得到拼接曲面比较复杂;提升法是一种较好的算法,拼接时逐个将代数曲面并入拼接曲面中。该算法既可得到最低次拼接方程又适用于一般情况。上述方法的优点是无需考虑代数曲面方程中的变元,仅考虑对新增单位区间变元的处理。因此,算法的计算量小,并且能够预先得到拼接曲面时的计算量。     

Parametric G n blending of curves and surfaces

We introduce a simple blending method for parametric curves and surfaces that produces families of parametrically defined, G ⁿ –continuous blending curves and surfaces. The method depends essentially on the parameterizations of the curves/surfaces to be blended. Hence, the flexibility of the method relies on the existence of suitable parameter transformations of the given curves/surfaces. The feasibility of the blending method is shown by several examples. The shape of the blend curve/surface can be changed in a predictable way with the aid of two design parameters (thumb weight and balance).     

多项式混合曲线曲面方法构造

针对混合曲线表示及其求导和求积困难的问题,通过计算构造出一种多项式混合曲线曲面形式.当待混合曲线是多项式时,混合曲线也为多项式形式.该多项式混合公式可以推广得到任意参数连续C(n)和几何连续G(n)的混合曲线曲面.另外,在得到的混合曲线曲面族中构造出了新的更优能量光顺方程,通过设置参数可得到合适的混合曲线曲面.实验结果表明,文中提出的混合曲线曲面造型方法稳定、有效.     

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.     

三角域上三次DP曲面的扩展

给出了三角域上带双参数的四次DP混合函数,它是三角域上三次DP 基函数的扩展。分析了该组混合函数的性质并定义了三角域上带双形状参数的四次DP参数曲面。该组混合函数及其参数曲面分别具有与三次DP基函数及三次DP参数曲面类似的性质。当两参数为0时,可分别退化到三次DP基函数及三次DP参数曲面。研究表明,通过改变两个形状参数的取值,既可整体又可局部调整曲面的形状。     

Generating Blending Surfaces with a Pseudo-Lévy Series Solution to Fourth Order Partial Differential Equations

In our previous work, a more general fourth order partial differential equation (PDE) with three vector-valued parameters was introduced. This equation is able to generate a superset of the blending surfaces of those produced by other existing fourth order PDEs found in the literature. Since it is usually more difficult to solve PDEs analytically than numerically, many references are only concerned with numerical solutions, which unfortunately are often inefficient. In this paper, we have developed a fast and accurate resolution method, the pseudo-Lévy series method. Due to its analytical nature, the comparison with other existing methods indicates that the developed method can generate blending surfaces almost as quickly and accurately as the closed form resolution method, and has higher computational accuracy and efficiency than existing Fourier series and pseudo-spectral methods as well as other numerical methods. In addition, it can be used to solve complex surface blending problems which cannot be tackled by the closed form resolution method. To demonstrate the potential of this new method we have applied it to various surface blending problems, including the generation of the blending surface between parametric primary surfaces, general second and higher degree surfaces, and surfaces defined by explicit equations.     

用有理双三次Bezier曲面片混合二次曲面

提出一种二次曲面混合方法，混合曲面由2张有理双三次B6zier曲面片构成，它们之间保持G^2连续，混合曲面与二次曲面间保持G^1连续．给出了混合曲面片控制顶点的显式表示，通过修改2类混合参数可以直观地调节混合方向及混合曲面的形状．另外，混合5个圆锥曲面的例子表明，该方法为多个二次曲面的混合问题提供了有效途径．     

Geometric texturing using level sets

We present techniques for warping and blending (or subtracting) geometric textures onto surfaces represented by high resolution level sets. The geometric texture itself can be represented either explicitly as a polygonal mesh or implicitly as a level set. Unlike previous approaches, we can produce topologically connected surfaces with smooth blending and low distortion. Specifically, we offer two different solutions to the problem of adding fine-scale geometric detail to surfaces. Both solutions assume a level set representation of the base surface which is easily achieved by means of a mesh-to-level-set scan conversion. To facilitate our mapping, we parameterize the embedding space of the base level set surface using fast particle advection. We can then warp explicit texture meshes onto this surface at nearly interactive speeds or blend level set representations of the texture to produce high-quality surfaces with smooth transitions.     

Blending multiple parametric normal ringed surfaces using implicit functional splines and auxiliary spheres

First presented by Hartmann, closings (implicit surfaces sealing the inlets or outlets of pipes) can bridge the gap between parametric pipe surfaces and implicit functional splines (a powerful tool for blending several implicit surfaces). This paper proposes auxiliary spheres instead of the initial pipe surfaces as the base surfaces in constructing closings, so that the closing based algorithm of two steps (constructing a closing for each pipe and blending the closings) can G¹-continuously connect multiple parametric normal ringed surfaces with freeform directrices and variable radii. The basic theory of an auxiliary sphere tangent to the normal ringed surface is addressed. Either one or two (yielding more design parameters) auxiliary spheres can be added. How the parameters influence the closing configuration is discussed. In addition, the blending shape can be optimized by genetic algorithm after assigning some fiducial points on the blend. The enhanced algorithm is illustrated with four practical examples.     

矩形域上有理Bezier曲面的广义离散算法及其应用

本文推广了有理Ｂｅｚｉｅｒ曲面的离散算法，得到了沿非等参数离散矩形域上有理Ｂｅｚｉｅｒ曲面的割角算法，并给出了从有理Ｂｅｚｉｅｒ矩形片到有理Ｂｅｚｉｅｒ三角片的割角转换。     

A note on two-sided holes in implicit quadric spline surfaces

Some implicit spline surface schemes require filling two-sided holes [Dahmen 1989; Guo 1993] and a blending technique of Warren (1987) can be used to fill two-sided holes in quadric spline surfaces. This note shows that there are important cases where the blending technique does not apply. These cases are identified and an applicatble technique is proposed.     

Splat representation of parametric surfaces

Point and splat-based representations have become a suitable technique both for modeling and rendering complex 3D shapes. Converting other kinds of models as parametric surfaces to splat-based representations will allow to mix surface and splat-based models and to take advantage of the existing point-based rendering methods. In this work, we present an approach to convert a parametric surface into a splat-based representation. It works in parametric space, performs an adaptive sampling based on the surface curvature and a given error tolerance and uses power Voronoi diagrams. The goal is to approximate the surface with an optimized set of elliptical splats.     

High-order implicit blending surfaces of low degree

The applicability of the so-called potential method for blending implicitly defined surfaces is extended. The extended method is able to produce blendings which are C^k-continuous, where k may be chosen arbitrarily large. The blendings consist of piecewise algebraic surfaces of low degree. Specifically, the degree is k + 1 for so-called convex corners, and 2k for an important class of non-convex corners. In general the degree is linear in k, the constant of proportionality depending on the geometry of the corner. The method is illustrated by a concrete example, where a C² blending surface of degree 4 is constructed for a certain non-convex corner.