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.     

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.     

Improved bounds on partial derivatives of rational triangular Bézier surfaces

This paper applies inequality skill, degree elevation of triangular Bézier surfaces and difference operators to deduce the bounds on first and second partial derivatives of rational triangular Bézier surfaces. Further more, we prove that the new bounds are tighter and more effective than the known ones. All the results are obviously helpful for further optimization of geometric design systems.     

On rationally supported surfaces

We analyze the class of surfaces which are equipped with rational support functions. Any rational support function can be decomposed into a symmetric (even) and an antisymmetric (odd) part. We analyze certain geometric properties of surfaces with odd and even rational support functions. In particular it is shown that odd rational support functions correspond to those rational surfaces which can be equipped with a linear field of normal vectors, which were discussed by Sampoli et al. (Sampoli, M.L., Peternell, M., Jüttler, B., 2006. Rational surfaces with linear normals and their convolutions with rational surfaces. Comput. Aided Geom. Design 23, 179–192). As shown recently, this class of surfaces includes non-developable quadratic triangular Bézier surface patches (Lávička, M., Bastl, B., 2007. Rational hypersurfaces with rational convolutions. Comput. Aided Geom. Design 24, 410–426; Peternell, M., Odehnal, B., 2008. Convolution surfaces of quadratic triangular Bézier surfaces. Comput. Aided Geom. Design 25, 116–129).     

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.     

Link between Bézier and Lagrange curve and surface schemes

A class of curves that vary continuously between polynomial Lagrange interpolants and polynomial Bézier curves is discussed. An element in this class is specified by a real number which could be used as a shape parameter for Bézier curves. A geometric derivation of this scheme is given, and the connection to Pólya curves is pointed out. A generalization to the case of tensor product and triangular surface patches is also described.     

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.     

Dual Bézier curves and convexity preserving interpolation

A convexity preserving interpolation problem is analyzed from a geometrical point of view. A dualization of the usual Bézier techniques allows us to define a subdivision algorithm which generates certain conic sections. This algorithm can be used to define a rational convexity preserving interpolant. We also describe some particular dual Bézier curves which are particularly suitable for the design of convex functions.     

A simple algorithm for designing developable Bézier surfaces

An algorithm is presented that generates developable Bézier surfaces through a Bézier curve of arbitrary degree and shape. The algorithm has two important advantages. No (nonlinear) characterizing equations have to be solved and the control of singular points is guaranteed. Further interpolation conditions can be met.     

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.     

The rational cubic Bézier representation of conics

The rational cubic Bézier curve is a very useful tool in CAGD. It incorporates both conic sections and parametric cubic curves as special cases, so its advantage is that one can deal with curves of these two kinds in one computer procedure. In this paper, the necessary and sufficient conditions for representing conics by the rational cubic Bézier form in proper parametrization are investigated; these conditions can be divided into two parts: one for weights and the other for Bézier vertices.     

Matrix formulations of Bézier technique

At the start of this paper, a recurrence formula for calculating the Bézier functions of any order is proved. Based on this formula, the Bézier functions are written in matrix form which enables us to develop the Bézier technique in a concise way. There are some new identities of the Bézier functions which are potentially useful in CAGD. Surprisingly a matrix involving the Bézier functions has been shown to be a doubly-stochastic matrix and the convergence behaviour of its powers is determined. Based upon these facts, the Kelisky — Rivlin theorem for the Bernstein polynomials has been extended to the Bézier curves.     

Three-dimensional Bézier interpolation in solid modeling and grid generation

Bézier and B-spline patches are popular tools in surface modeling. With these methods, a surface is represented by the product of univariate approximations. The extension of this concept to three-dimensions is straightforward and can be applied to the problem of grid generation. This report will demonstrate how rational Bézier basis functions can be used to generate three-dimensional solids that can be used in solid modeling and in the generation of grids. Examples will be given demonstrating the ability to generated three-dimensional grids directly from a set of data points without having to first set up the boundary surfaces. Many geometric grid properties can be imposed by the proper choice of the control net, the weights, and the twist models.     

An adaptive method for smooth surface approximation to scattered 3D points

The construction of a surface from arbitrarily scattered data is an important problem in many applications. When there are a large number of data points, the surface representations generated by interpolation methods may be inefficient in both storage and computational requirements. This paper describes an adaptive method for smooth surface approximation from scattered 3D points. The approximating surface is represented by a piecewise cubic triangular Bézier surface possessing C¹ continuity. The method begins with a rough surface interpolating only boundary points and, in the successive steps, refines it by adding the maximum error point at a time among the remaining internal points until the desired approximation accuracy is reached. Our method is simple in concept and efficient in computational time, yet realizes efficient data reduction. Some experimental results are given to show that surface representations constructed by our method are compact and faithful to the original data points.     

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.     

An approximation approach of the clothoid curve defined in the interval [0, π/2] and its offset by free-form curves

We propose an algorithm to approximate the clothoid curve defined in the interval [0, π/2] and its offset curves with Bézier curves and the approximation errors converge to zero as the degree of the Bézier curves is increased. Secondly, we discuss how to approximate the clothoid curve by B-spline curves of low degrees. By employing our method, the clothoid curve and its offset can be efficiently incorporated into CAD/CAM systems, which are important for the development of 3D civil engineering CAD systems, especially for 3D highway road design systems. The proposed method has been implemented on AutoCAD R14.     

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.     

Computer-aided geometric design and panel generation for hull forms based on rational cubic Bézier curves

For generation of hull forms, a method using rational cubic Bézier curves is chosen because of their superior segmentwise local-weighted behavior. A hull form is defined by two sets of grid lines—transverse grid lines arranged in length direction and longitudinal grid lines arranged in depth direction. Transverse lines are first defined, the points on the transverse lines with the same curve parameter values are then fitted to define longitudinal lines. Thereby, each curve is described by a rational cubic Bézier curve in space. The bilge, flat side and flat bottom can be defined precisely and more flexibilities are provided for defining bow and stern regions. By the way, a hull surface can be generated which is useful to produce desired data for hydrostatic or panel generations.     

The bound on derivatives of rational Bézier curves

The estimation of bounds on derivatives of rational Bézier curves has important application in CAGD. This paper derives some new bounds according to the properties of derivation and recursion of Bernstein basis, and shows that the new bounds are better than existing ones and more effective.     

The octant of a sphere as a non-degenerate triangular Bézier patch

We construct a symmetric rational quartic map from the standard triangle onto an octant of a sphere. The surface is non-degenerate: all Bézier points are distinct and their associated weights are positive.