Gregory-type patches bounded by low degree intergral curves for G continuity

G² continuity of free-form surfaces is sometimes very important in engineering applications. The conditions for G² continuity to connect two Bézier patches were studied and methods have been developed to ensure it. However, they have some restrictions on the shapes of patches of the Bézierpatch formulation. Gregory patch is a kind of free-form surface patch which is extended from Bézier patch so that four first derivatives on its boundary curves can be specified without restrictions of the compatibility condition. Several types of Gregory patches have been developed for intergral, rational, and NURBS boundary curves. In this paper, we propose some intergral boundary Gregorytype patches bounded by cubic and quartic curves for G² continuity.     

Local surface interpolation with Bézier patches

A surface interpolation method for meshes of cubic curves is described. A mesh of cubic curve is constructed between the given vertices. This mesh is filled with Bézier patches, so that the surface is represented as a union of geometrically continuous bicubic quadrilateral and/or quartic triangular Bézier patches. The method is local and uses Farin's [Farin '83] conditions of G¹ continuity between patches. The procedure for finding the needed control points of the Bézier patches is simple and efficient.     

G interpolation of free form curve networks by biquintic Gregory patches

The problem of interpolating a free form curve network with irregular topology is investigated in order to create a curvature continuous surface. The spanning curve segments are parametric quintic polynomials, the interpolating surface elements are biquintic Gregory patches. A necessary compatibility condition is formulated and proved which need to be satisfied at each node of the curve network. Constraints are derived for assuring G² continuity between biquintic Gregory patches, which share a common side or a common corner point. The above conditions still leave certain geometric freedom for defining the entire G² surface, so following some analysis a particular construction is presented, by which after computing the principle curvatures at each node the free parameters are locally set for each interpolating Gregory patch.     

Approximate G continuous interpolation of a rectangular network of rational cubic curves

The problem of spanning a rectangular network of rational cubic curves with a smooth surface is discussed in this paper. Provided the network is compatible with a smooth surface, then algorithms for patch construction, optimization and subdivision are developed to construct an ‘approximately smooth’ surface, that is, G¹ continuous to within some tolerance, composed of rational bicubic patches. The algorithms have been applied in the die and mould industry. The toolmaker constructs a wireframe model of an EDM (electro-discharge machining) electrode and the algorithms automatically construct the surface model. For toolmaking companies, this simplifies the surface modelling process making a highly-specialized and time-consuming task virtually automatic.     

PDE blending surfaces with C continuity

In this paper, we propose to use a general sixth-order partial differential equation (PDE) to solve the problem of C² continuous surface blending. Good accuracy and high efficiency are obtained by constructing a compound solution function, which is able to both satisfy the boundary conditions exactly and minimise the error of the PDE. This method can cope with much more complex surface-blending problems than other published analytical PDE methods. Comparison with the existing methods indicates that our method is capable of generating blending surfaces almost as fast and accurately as the closed-form method and it is more efficient and accurate than other extant PDE-based methods.     

双五次B-样条曲面的G2连续条件

引言 Coons曲面,B6zier曲面和B一样条(包括NURBs)曲面等参数曲面的构造是几何造型和CAD/CAM领域的基础问题.B一样条曲面由于它的局部支集和全局逼近性质已成为这些领域最为常用的几何表示方法.特别是近年来反向工程的发展,使得B一样条的研究越发受到重视.反向工程的关键问题之一是复杂曲面的重建,但是目     

A modification to the rational boundary Gregory patch

This paper presents a method for modifying the boundary derivatives of rational Bézier patches, preserving their directions at any parameter so as not to affect the G¹ continuity with adjacent patches. This method is applicable to reduce the complexity of rational boundary Gregory patches.     

自由曲面光滑拼接的数值优化方法

研究了B样条曲面的光滑拼接问题，介绍了参数曲面几何连续性的定义，提出了2种在B样条曲面间实现 光滑拼接的数值优化算法，即调整邻接曲面的公共边界曲线和调整其中一个曲面的邻近边界的控制顶点。实验结果表明，灵活使用文中的方法修改曲面外形可有效地解决曲面造型中的相关问题。     

Filling n-sided regions with NURBS patches

n -sided region with G ^ɛ continuous NURBS patches that interpolate boundary curves and approximate given cross-boundary derivatives. The NURBS surfaces joining along inner or boundary curves have normal vectors that do not deviate more than the user-specified angular tolerance ɛ. The method is general in that there are no restrictions on the number of boundary curves, and the cross-boundary derivatives can be specified independently. To satisfy all conditions, only one degree elevation is needed.     

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.     

A modified C Coons'' patch

A modified C² Coons' patch, free of any compatibility constraints, has recently been developed in [Barnhill '83]. In this short note we show how it can be modified and simplified to produce a computationally more efficient C² transfinite interpolation scheme over rectangles.     

Shape preserving interpolation by curvature continuous parametric curves

An interpolation scheme for planar curves is described, obtained by patching together parametric cubic segments and straight lines. The scheme has, in general, geometric continuity of order 2 (G² continuity) and is similar in approach to that of [Goodman & Unsworth ′86], but whereas this earlier scheme, when applied to cubics, produces curves with zero curvature at the interpolation points, the corresponding curvature values in this scheme are in general non-zero. The choice of a tangent vector at each interpolation point guarantees that the interpolating curve is local convexity preserving, and in the case of functional data it is single-valued and local monotonicity preserving. The algorithm for generating the cubic curve segments usually requires the solution of two non-linear equations in two unknowns, and lower bounds are obtained on the magnitude of the curvature at the relevant interpolation points in order that this system of equations has a unique solution. Particular attention is given to cubic segments which are adjacent to straight line segments. Two methods for calculating these segments are described, one which preserves G² continuity, and one which only gives G¹ continuity. A number of examples of the application of the scheme are presented.     

The singular cases for γ-spline interpolation

We derive a natural extension of Boehm's free-form γ-spline, the G² interpolating γ-spline. Primarily, the conditions under which singularities in the spline formulation occur are investigated. Also, the effect that these singularities have on the interpolant are studied. Comparisons are made to the behavior of the interpolating ν-spline.     

Generalized Coons'' patches

Surfaces designed in a computer graphics environment have many applications, including the design of cars, airplans, shipbodies and modeling robots. The choice of the surface form depends upon the application. A very useful and popular tool is the c²-biquintic Gordon-Coons' patch.

A major problem using this technique is to input the twist vectors. To avoid twist constraints a new method is presented based upon the replacement of the cross partial derivatives by the product of a convex combination of curvature functions and the normal vectors.     

Theories of contact specified by connection matrices

We begin by characterizing notions of geometric continuity represented by connection matrices. Next we present a set of geometric properties that must be satisfied by all reasonable notions of geometric continuity. These geometric requirements are then reinterpreted as an equivalent collection of algebraic constraints on corresponding sets of connection matrices. We provide a general technique for constructing sets of connection matrices satisfying these criteria and apply this technique to generate many examples of novel notions of geometric continuity. Using these constraints and construction techniques, we show that there is no notion of geometric continuity between reparametrization continuity of order 3, (G³), and Frenet frame continuity of order 3, (F³); that there are several notions of geometric continuity between G⁴ and F⁴; and that the number of different notions of geometric continuity between Gⁿ and Fⁿ grows at least exponentially with n.     

Non-four-sided patch expressions with control points

This paper introduces new mathematical expressions which describe surface patches with three, five or six sides. Each side of a patch is given as a space curve expressed in Bézier type of second or third degree. Patch shape is modified predictably by moving its control points in the same way as with a four-sided patch. Distribution of cross tangent vectors along a patch boundary is also expressed in the same form with a few control vectors belonging to the boundary concerned. Therefore it can be connected with its surrounding four-sided patches smoothly. A general method of derivation of these equations is explained for further applications.     

Biquartic C-surface splines over irregular meshes

C¹-surface splines define tangent continuous surfaces from control points in the manner of tensor-product (B-)splines, but allow a wider class of control meshes capable of outlining arbitrary free-form surfaces with or without boundary. In particular, irregular meshes with non-quadrilateral cells and more or fewer than four cells meeting at a point can be input and are treated in the same conceptual frame work as tensor-product B-splines; that is, the mesh points serve as control points of a smooth piecewise polynomial surface representation that is local and evaluates by averaging. Biquartic surface splines extend and complement the definition of C¹-surface splines in a previous paper (Peters, J SLAM J. Numer. Anal. Vol 32 No 2 (1993) 645–666) improving continuity and shape properties in the case where the user chooses to model entirely with four-sided patches. While tangent continuity is guaranteed, it is shown that no polynomial, symmetry-preserving construction with adjustable blends can guarantee its surfaces to lie in the local convex hull of the control mesh for very sharp blends where three patches join. Biquartic C¹-surface splines do as well as possible by guaranteeing the property whenever more than three patches join and whenever the blend exceeds a certain small threshold.     

Improvement of free-form surfaces for product styling applications

There are a number of ways of describing free-form surfaces within geometric modelling systems. For product styling and related activities, there is a need to ensure that surface quality is good and that patches join together smoothly. Additional parameters can be introduced to allow surfaces to be modified. This raises the question of whether these can be chosen automatically, and this in turn requires measures of what is a ‘fair’ surface. Measures based on surface curvature are discussed and applied to adjust surface patches presented in terms of point meshes.     

Smooth closed surfaces with discrete triangular interpolants

Discrete interpolants which involve cross boundary derivatives in an attempt to form C¹ surfaces have the following major problem: Requiring C¹ joins between patches makes sense only if the patch domains are adjacent in the domain space. This makes it impossible to form C¹ closed surfaces, or indeed any surface which contains more connections than can be achieved in the domain.

This paper develops a method of forming smooth closed (or otherwise complexly connected) surfaces from a discrete triangular interpolant by relaxing the C¹ property of an interpolant to ‘Visually C¹”.

The only constraint on the scheme is that the data to be interpolated define a unique tangent plane at each vertex where several triangles meet. Then each patch can be calculated independently of its neighbors, using only data defined at its vertices, and the domain for each triangular patch can be chosen without regarding the connectivity of the patch with others. This last feature could be of great interest to a designer of a surface since one can choose the domain of each patch to be an equilateral triangle, and give it no further thought.     

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．