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.     

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.     

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.     

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.     

Curvature continuous curves and surfaces

A simple methods is given for constructing the Bézier points of curvature continuous cubic spline curves and surfaces from their B-spline control points. The method is similar to the well-known construction of Bézier points of C² splines from their B-spline control points. The new construction allows the use of all results of the powerful Bernstein-Bézier technique in the realm of geometric splines.

A simple introduction to nu- and beta-splines is also derived, as well as some simple geometric properties of beta-splines.     

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.     

Interpolating G-splines with rational offsets

In (Pottmann, 1995), a geometric characterization of rational PH-curves is presented. Using this result we developed an explicit Bézier representation for interpolating G¹-Hermite PH-splines referring to local coordinate systems. Furthermore a simple geometric criterion for avoiding singularities is proposed.     

Approximate conversion of a rational boundary gregory patch to a nonuniform B-spline surface

A rational boundary Gregory patch is characterized by the facts that anyn-sided loop can be smoothly interpolated and that it can be smoothly connected to an adjacent patch. Thus, it is well-suited to interpolate complicated wire frames in shape modeling. Although a rational boundary Gregory patch can be exactly converted to a rational Bézier patch to enable the exchange of data, problems of high degree and singularity tend to arise as a result of conversion. This paper presents an algorithm that can approximately convert a rational boundary Gregory patch to a bicubic nonuniform B-spline surface. The approximating surface hasC ¹ continuity between its inner patches.     

Control point surfaces over non-four-sided areas

This paper constructs control point surfaces of arbitrary degree over 3-, 5- and 6-sided areas. These surface patches behave like rectangular Bézier surface patches along their boundaries and can be connected smoothly with surrounding rectangular Bézier patches.     

Change of basis algorithms for surfaces in CAGD

The computational complexity of general change of basis algorithms from one bivariate polynomial basis of degree n to another bivariate polynomial basis of degree n using matrix multiplication is O(n⁴). Applying blossoming and duality, we derive change of basis algorithms with computational complexity O(n³) between two important classes of polynomial bases used for representing surfaces in CAGD: B-bases and L-bases. Change of basis algorithms for B-bases follow from their blossoming property; change of basis algorithms for L-bases follow from the duality between L-bases and B-bases. The Bézier and multinomial bases are special cases of both B-bases and L-bases, so these algorithms can be used to convert between the Bézier and multinomial forms. We also show that the bivariate Horner evaluation algorithm for the multinomial basis is dual to the bivariate de Boor evaluation algorithm for B-patches.     

Bézier representation for cubic surface patches

The paper describes a new method for creating rectangular Bézier surface patches on an implicit cubic surface. Traditional techniques for representing surfaces have relied on parametric representations of surfaces, which, in general, generate surfaces of implicit degree 8 in the case of rectangular Bézier surfaces with rational biquadratic parameterization. The method constructs low-degree algebraic surface patches by reducing the implicit degree from 8 to 3. The construction uses a rectangular biquadratic Bézier control polyhedron that is embedded within a tetrahedron and satisfies a projective constraint. The control polyhedron and the resulting cubic surface patch satisfy all of the standard properties of parametric Bézier surfaces, including interpolation of the corners of the control polyhedron and the convex-hull property.     

Tool path generation for free form surfaces using Bézier curves/surfaces

Presented in this paper is a tool path generation method for multi-axis machining of free-form surfaces using Bézier curves and surfaces. The tool path generation includes two core steps. First is the forward-step function that determines the maximum distance, called forward step, between two cutter contact (CC) points with a given tolerance. The second component is the side step function which determines the maximum distance, called side step, between two adjacent tool paths with a given scallop height. Using the Bézier curves and surfaces, we generate cutter contact (CC) points for free-form surfaces and cutter location (CL) data files for post processing. Several parts are machined using a multi-axis milling machine. As part of the validation process, the tool paths generated from Bézier curves and surfaces are analyzed to compare the machined part and the desired part.     

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.     

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.     

An analogue approach to surface definition

This note describes a method for producing cubic curves made up from Bézier functions on an oscilloscope screen using simple analogue techniques. Ruled surface patches are illustrated in a simple extension of the method. It also indicates how generalized Bézier patches could be produced.     

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 Bézier points for curves and surfaces from boundary information

This paper presents efficient methods for directly generating Bézier points of curves and surfaces explicitly from the given compatible arbitrary order boundary information of Hermite curves, Coons-Hermite Cartesian sum patches and Coons-Boolean sum patches. The explicit expressions for the generalized Hermite functions are also developed. Furthermore, a method for determining the twist control points and higher level sets of interior control points from their boundary and lower level sets of control points by using the Coons-Boolean sum schema presented. Many interesting and useful examples are also given in this paper.     

Adaptive patch-based mesh fitting for reverse engineering

In this paper,  we propose a novel adaptive mesh fitting algorithm that fits a triangular model with G¹ smoothly stitching bi-quintic Bézier patches. Our algorithm first segments the input mesh into a set of quadrilateral patches, whose boundaries form a quadrangle mesh. For each boundary of each quadrilateral patch, we construct a normal curve and a boundary-fitting curve, which fit the normal and position of its boundary vertices respectively. By interpolating the normal and boundary-fitting curves of each quadrilateral patch with a Bézier patch, an initial G¹ smoothly stitching Bézier patches is generated. We perform this patch-based fitting scheme in an adaptive fashion by recursively subdividing the underlying quadrilateral into four sub-patches. The experimental results show that our algorithm achieves precision-ensured Bézier patches with G¹ continuity and meets the requirements of reverse engineering.     

The convexity of parametric Bézier triangular patches of degree 2

This paper discusses the convexity of parametric Bézier patches of degree 2 over triangles. A necessary and sufficient condition for the convexity of the Bézier patches is presented.     

稀疏三角网格模型的Gregory曲面光顺插值

稀疏网格模型精细光顺重建时,网格顶点的法曲率不一致问题仍没有解决,导致渲染阴影。通过推导获得四次三角域Gregory顶点拼接处法曲率变化一致的约束条件,并基于该约束条件对稀疏三角网格模型进行精细重建。重建后的模型不但保证所有相邻三角Gregory曲面片G¹光顺连续,而且拼接顶点处的法曲率变化最小,从而可获得高质量的视觉效果。实验结果验证了在只有原始模型1%~2%网格数目的情况下可获得光顺的视觉渲染效果,结果模型亦具有高精细特征。