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.     

Canal surfaces with rational contour curves and blends bypassing the obstacles

In this paper, we will present an algebraic condition, see (20), which guarantees that a canal surface, given by its rational medial axis transform (MAT), possesses rational generalized contours (i.e., contour curves with respect to a given viewpoint). The remaining computational problem of this approach is how to find the right viewpoint. The canal surfaces fulfilling this distinguished property are suitable for being taken as modeling primitives when some rational approximations of canal surfaces are required. Mainly, we will focus on the low-degree cases such as quadratic and cubic MATs that are especially useful for applications. To document a practical usefulness of the presented approach, we designed and implemented two simple algorithms for computing rational offset blends between two canal surfaces based on the contour method which do not need any further advanced formalism (as e.g. interpolations with MPH curves). A main advantage of the designed blending technique is its simplicity and also an adaptivity to choose a suitable blend satisfying certain constrains (avoiding obstacles, bypassing other objects, etc.). Compared to other similar methods, our approach requires only one SOS decomposition for the whole family of rational canal surfaces sharing the same silhouette, which significantly simplifies the computational complexity.     

Blends of canal surfaces from polyhedral medial transform representations

We present a new method for constructing G¹ blending surfaces between an arbitrary number of canal surfaces. The topological relation of the canal surfaces is specified via a convex polyhedron and the design technique is based on a generalization of the medial surface transform. The resulting blend surface consists of trimmed envelopes of one- and two-parameter families of spheres. Blending the medial surface transform instead of the surface itself is shown to be a powerful and elegant approach for blend surface generation. The performance of our approach is demonstrated by several examples.     

Surfaces with Pythagorean normals along rational curves

A rational curve on a rational surface such that the unit normal vector field of the surface along this curve is rational will be called a curve providing Pythagorean surface normals (or shortly a PSN curve). These curves represent rational paths on the surface along which the surface possesses rational offset curves. Our aim is to study rational surfaces containing enough PSN curves. The relation with PN surfaces will be also investigated and thoroughly discussed. The algebraic and geometric properties of PSN curves will be described using the theory of double planes. The main motivation for this contribution is to bring the theory of rational offsets of rational surfaces closer to the practical problems appearing in numerical-control machining where the milling cutter does not follow continuously the whole offset surface but only certain chosen trajectories on it. A special attention will be devoted to rational surfaces with pencils of PSN curves.     

Rational adaptive blends among obstacles in 3D by contour method

In this paper we will continue in investigating ‘contour method’ and its using for the computation of rational parameterizations of canal surfaces without a need of sum of squares (SOS) decomposition. Further approaches for constructing flexible smooth transitions between canal surfaces will be presented. Mainly, we focus on one particular application of recently introduced rational envelope curves, newly constructed over an arbitrary planar rational curve in space. Using this type of curves significantly simplifies the previous methods discussed in Bizzarri (2015), and mainly new situations, which could not have been handled with the previous setup, are successfully solved, now. Especially a method for constructing rational adaptive blends which bypass a given obstacle (or more given obstacles when needed) is thoroughly discussed and its functionality is demonstrated on a number of examples. The designed approach works not only for simple obstacles represented by one-dimensional medial axis transforms but also for more general obstacles described by two-dimensional medial surface transforms.     

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。     

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.     

Conchoid surfaces of rational ruled surfaces

The conchoid surface G of a given surface F with respect to a point O is roughly speaking the surface obtained by increasing the radius function of F with respect to O by a constant d. This paper studies real rational ruled surfaces in this context and proves that their conchoid surfaces possess real rational parameterizations, independently of the position of O. Thus any rational ruled surface F admits a rational radius function r(u,v) with respect to any point in space. Besides the general skew ruled surfaces and examples of low algebraic degree we study ruled surfaces generated by rational motions.     

Quaternion rational surfaces: Rational surfaces generated from the quaternion product of two rational space curves

A quaternion rational surface is a surface generated from two rational space curves by quaternion multiplication. The goal of this paper is to demonstrate how to apply syzygies to analyze quaternion rational surfaces. We show that we can easily construct three special syzygies for a quaternion rational surface from a μ-basis for one of the generating rational space curves. The implicit equation of any quaternion rational surface can be computed from these three special syzygies and inversion formulas for the non-singular points on quaternion rational surfaces can be constructed. Quaternion rational ruled surfaces are generated from the quaternion product of a straight line and a rational space curve. We investigate special μ-bases for quaternion rational ruled surfaces and use these special μ-bases to provide implicitization and inversion formulas for quaternion rational ruled surfaces. Finally, we show how to determine if a real rational surface is also a quaternion rational surface.     

一种参数化可调的NURBS曲面G~2光滑拼接算法

提出一种用于NURBS曲面G2光滑拼接算法。在创建拼接曲面时,采用"参数可调"的思想,用拼接函数和指重参数先统一两基曲面的参数,计算出拼接曲面上的插值点,并以这些插值点为参考点根据G2连续的几何性质对拼接曲面的内部控制点进行修正。此算法适用于各类曲面的拼接,通过调整平衡因子和指重参数可以得到在满足G2连续的前提下各种曲率的拼接曲面,简化曲面拼接的计算过程。     

有理曲面的区间隐式化

利用一个低阶多项式区间隐式曲面来包围所给的参数式有理曲面,并构造了一些关于区间隐式曲面厚度和微分张量的目标函数.在最小化这些目标函数的条件下,该区间隐式曲面的中心曲面可以近似地逼近有理曲面,其逼近的误差可以利用区间隐式曲面的区间宽度进行估计.最后提供了具体的算法和一些实例.     

Rational two-parameter families of spheres and rational offset surfaces

The present paper investigates two-parameter families of spheres in R³${\mathbb{R}}^3$ and their corresponding two-dimensional surfaces Φ$\Phi$ in R⁴${\mathbb{R}}^4$. Considering a rational surface Φ$\Phi$ in R⁴${\mathbb{R}}^4$, the envelope surface Ψ$\Psi$ of the corresponding family of spheres in R³${\mathbb{R}}^3$ is typically non-rational. Using a classical sphere-geometric approach, we prove that the envelope surface Ψ$\Psi$ and its offset surfaces admit rational parameterizations if and only if Φ$\Phi$ is a rational sub-variety of a rational isotropic hyper-surface in R⁴${\mathbb{R}}^4$. The close relation between the envelope surfaces Ψ$\Psi$ and rational offset surfaces in R³${\mathbb{R}}^3$ is elaborated in detail. This connection leads to explicit rational parameterizations for all rational surfaces Φ$\Phi$ in R⁴${\mathbb{R}}^4$ whose corresponding two-parameter families of spheres possess envelope surfaces admitting rational parameterizations. Finally we discuss several classes of surfaces sharing this property.     

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).     

A vertex-first parametric algorithm for polyhedron blending

This paper enhances the conventional parametric algorithms for polyhedron blending, by strategically inverting the edges-first approach to vertex-first, so that matching the vertex blending surface (using a triangular or tensor product Bézier surface, or an S-patch) with the edge blending surfaces (generated by Hartmann method) becomes essentially easier. Based on a study of cross boundary derivatives (those of S-patches are deduced herein), G^g-continuity between all the above surfaces and the primary planar faces is achieved by a novel trick as a first step: assigning the vertex, some edge points and some face points to be the proper control points. This still leaves enough free parameters usable for changing the blending configuration. The new algorithm is illustrated with two practical examples involving miscellaneous vertices up to 6-edge convex–concave.     

Approximate T-spline surface skinning

This paper considers the problem of constructing a smooth surface to fit rows of data points. A special class of $T$-spline surfaces is examined, which is characterized to have a global knot vector in one parameter direction and individual knot vectors from row to row in the other parameter direction. These $T$-spline surfaces are suitable for lofted surface interpolation or approximation. A skinning algorithm using these $T$-spline surfaces is proposed, which does not require the knot compatibility of sectional curves. The algorithm consists of three main steps: generating sectional curves by interpolating data points of each row by a $B$-spline curve; computing the control curves of a skinning surface that interpolates the sectional curves; and approximating each control curve by a $B$-spline curve with fewer knots, which results in a $T$-spline surface. Compared with conventional $B$-spline surface skinning, the proposed $T$-spline surface skinning has two advantages. First, the sectional curves and the control curves of a $T$-spline surface can be constructed independently. Second, the generated $T$-spline skinning surface usually has much fewer control points than a lofted $B$-spline surface that fits the data points with the same error bound. Experimental examples have demonstrated the effectiveness of the proposed algorithm.     

Blending Canal Surfaces Based on PH Curves

In this paper, a new method for blending two canal surfaces is proposed. The blending surface is itself a generalized canal surface, the spine curve of which is a PH (Pythagorean-Hodograph) curve. The blending surface possesses an attractive property - its representation is rational. The method is extensible to blend general surfaces as long as the blending boundaries are well-defined.     

Ribbon-based transfinite surfaces

One major issue in CAGD is to model complex objects using free-form surfaces of general topology. A natural approach is curvenet-based design, where designers directly create and modify feature curves. These are interpolated by smoothly connected, multi-sided patches, which can be represented by transfinite surfaces, defined as a combination of side interpolants or ribbons. A ribbon embeds Hermite data, i.e., prescribed positional and cross-derivative functions along boundary curves.The paper focuses on two transfinite schemes: the first is an enhanced and extended variant of a multi-sided generalization of the classical Coons patch (Várady et al., 2011); the second one is based on a new concept of combining doubly curved composite ribbons, each one interpolating three adjacent sides. Main contributions include various ribbon parameterizations that surpass former methods in quality and computational efficiency. It is proven that these surfaces smoothly interpolate the prescribed ribbon data. Both formulations are based on non-regular convex polygonal domains and distance-based, rational blending functions. A few examples illustrate the results.     

Computing exact rational offsets of quadratic triangular Bézier surface patches

The offset surfaces to non-developable quadratic triangular Bézier patches are rational surfaces. In this paper we give a direct proof of this result and formulate an algorithm for computing the parameterization of the offsets. Based on the observation that quadratic triangular patches are capable of producing C¹ smooth surfaces, we use this algorithm to generate rational approximations to offset surfaces of general free-form surfaces.     

Minimum distance between two sphere-swept surfaces

We present an efficient and robust approach for computing the minimum distance between two sphere-swept surfaces. As examples of sphere-swept surfaces, we consider canal surfaces and bivariate sphere-swept surfaces. For computing the minimum distance between two parametric surfaces, a simple technique is to find the two closest points from the given surfaces using the normal vector information. We suggest a novel approach that efficiently computes the minimum distance between two sphere-swept surfaces by treating each surface as a family of spheres. Rather than computing the complicated normal vectors for given surfaces, our method solves the problem by computing the minimum distance between two moving spheres. We prove that the minimum distance between two sphere-swept surfaces is identical to that between two moving spheres. Experimental results of minimum distance computation are given. We also reproduce the result of Kim [Kim K-J. Minimum distance between a canal surface and a simple surface. Computer-Aided Design 2003;35:871-9] based on the suggested approach.