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.     

High accuracy geometric Hermite interpolation

We describe a parametric cubic spline interpolation scheme for planar curves which is based on an idea of Sabin for the construction of C¹ bicubic parametric spline surfaces. The method is a natural generalization of [standard] Hermite interpolation. In addition to position and tangent, the curvature is prescribed at each knot. This ensures that the resulting interpolating piecewise cubic curve is twice continuously differentiable with respect to arclength and can be constructed locally. Moreover, under appropriate assumptions, the interpolant preserves convexity and is 6-th order accurate.     

Hermite interpolation with a pair of spirals

The Hermite interpolation problem in the plane considered here is to join two points and to match given unit tangent vectors and signed curvatures at the two points with various G² curves consisting of a pair of spirals. The rotation of the tangent vector of the interpolating curve from one point to the other is restricted to being less than π. The necessary and sufficient conditions for the existence of each of the various curves are given.     

Shape preserving interpolation by space curves

Shape preserving interpolation for planar data has been well studied while little has been done for shape preserving curve interpolation in space. We consider some criteria for shape preserving interpolation by space curves: convexity and inflections of the projections of the curve onto certain planes, the sign of the torsion, coplanarity and collinearity. Based upon these criteria we then derive an algorithm for interpolating given points in space with a shape preserving piecewise rational cubic curve. The scheme is local and produces curves which are unit tangent continuous and also continuous in curvature magnitude apart from some exceptional cases where the curve contains linear segments. We illustrate the scheme with some graphical examples.     

Local control of interval tension using weighted splines

Cubic spline interpolation and B-spline sums are useful and powerful tools in computer aided design. These are extended by weighted cubic splines which have tension controls that allow the user to tighten or loosen the curve on intervals between interpolation points. The weighted spline is a C¹ piecewise cubic that minimizes a variational problem similar to one that a C² cubic spline minimizes. A B-spline like basis is constructed for weighted splines where each basis function is nonnegative and nonzero only on four intervals. The basis functions sum up identically to one, thus curves generated by summing control points multiplied by the basis functions have the convex hull property. Different weights are built into the basis functions so that the control point curves are piecewise cubics with local control of interval tension. If all weights are equal, then the weighted spline is the C² cubic spline and the basis functions are B-splines.     

Planar G transition between two circles with a fair cubic Bézier curve

Consumer products such as ping-pong paddles, can be designed by blending circles. To be visually pleasing it is desirable that the blend be curvature continuous without extraneous curvature extrema. Transition curves of gradually increasing or decreasing curvature between circles also play an important role in the design of highways and railways. Recently planar cubic and Pythagorean hodograph quintic spiral segments were developed and it was demonstrated how these segments can be composed pairwise to form transition curves that are suitable for G² blending. It is now shown that a single cubic curve can be used for blending or as a transition curve with the guarantee of curvature continuity and fairness. Use of a single curve rather than two segments has the benefit that designers and implementers have fewer entities to be concerned with.     

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.     

Generating the Bézier points of a β-spline curve

We present an efficient algorithm for computing the Bézier points of a generalized cubic β-spline curve and show the connection with multiple knot insertion. We also consider the inverse problem of determining the β-spline vertices of a composite G² Bézier curve. Finally, we briefly discuss how to construct the Bézier net of a tensor product β-spline surface.     

Algorithms of discretization of algebraic spatial curves on homogeneous cubical grids

In this paper new methods of discretization (integer approximation) of algebraic spatial curves in the form of intersecting surfaces P(x, y, z) = 0 and Q(x, y, z) = 0 are analyzed.

The use of homogeneous cubical grids G(h³) to discretize a curve is the essence of the method. Two new algorithms of discretization (on 6-connected grid G_6c(h³) and 26-connected grid G₂₆(h³)) are presented based on the method above. Implementation of the algorithms for algebraic spatial curves is suggested. The elaborated algorithms are adjusted for application in computer graphics and numerical control of machine tools.     

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.     

A pythagorean hodograph quintic spiral

A polynomial curve with a Pythagorean hodograph has the properties that its arc-length is a polynomial of its parameter, and its offset is a rational algebraic expression. A quintic is the lowest degree Pythagorean hodograph curve that may have an inflection point and that inflection point allows a segment of it to be joined to a straight line segment while preserving continuity of curvature, continuity of position, and continuity of tangential direction. The curvature of a spiral varies monotonically with arc-length. Spiral segments are useful in the design of fair curves. A Pythagorean hodograph quintic spiral is presented which allows the design of fair curves in a based system. It is also suitable for applications such as highway design in which the clothoid has traditionally been used.     

曲率连续的有理二次样条插值的一种优化方法

人们通常用有理三次曲线样条来构造整体曲率连续的曲线.提出利用有理二次样条曲线插值整体曲率连续的曲线的一种方法.首先导出了两相邻二次曲线段间曲率连续的拼接条件,然后提出了求解平面上一个闭的点列中每一点处的切线的最优算法.最后给出了闭曲线插值的一些实例以检验方法的有效性.     

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.     

基于能量优化G2连续插值三次样条曲线

给出了一种在能量优化意义下构造G2连续保形插值三次参数样条曲线的方法.具体步骤如下:(1)以曲线应变能最小为目标构造目标函数,通过解线性方程组,求出优化意义下的每个插值点处的最优切矢方向;(2)用文中给出的简易公式求出各插值点的曲率,进而计算出插值点处的切矢模长,使曲线满足G2连续、保形插值的条件;(3)用Hermite插值方法求出相邻两插值点间的曲线.实验结果显示了方法的有效性.     

基于能量优化G2连续插值三次样条曲线

给出了一种在能量优化意义下构造G²连续保形插值三次参数样条曲线的方法。具体步骤如下：（1）以曲线应变能最小为目标构造目标函数，通过解线性方程组，求出优化意义下的每个插值点处的最优切矢方向；（2）用文中给出的简易公式求出各插值点的曲率，进而计算出插值点处的切矢模长，使曲线满足G²连续、保形插值的条件；（3）用Hermite插值方法求出相邻两插值点间的曲线。实验结果显示了方法的有效性。     

Ensuring compatibility of G-continuous surface patches around a nodepoint

The problem of ensuring compatibility of mixed partial derivative vectors of surface patches joining G²-continuously around a common nodepoint is essential in modelling G²-continuous n-sided surfaces. Although the compatibility constraints can be removed by using C² Gregory patches, these patches have singularities at their corner points. This paper presents conditions for ensuring the compatibility of the mixed partial derivative vectors of surface patches joining G²-continuously around a common nodepoint. After investigating the solvability of these compatibility conditions, a new solution method exploiting G³-continuity of surface patches at a common nodepoint is given. Example surfaces based on this solution method are also provided.     

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.