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.     

Construction of Bézier surface patches with Bézier curves as geodesic boundaries

Given four polynomial or rational Bézier curves defining a curvilinear rectangle, we consider the problem of constructing polynomial or rational tensor-product Bézier patches bounded by these curves, such that they are geodesics of the constructed surface. The existence conditions and interpolation scheme, developed in a general context in earlier studies, are adapted herein to ensure that the geodesic-bounded surface patches are compatible with the usual polynomial/rational representation schemes of CAD systems. Precise conditions for four Bézier curves to constitute geodesic boundaries of a polynomial or rational surface patch are identified, and an interpolation scheme for the construction of such surfaces is presented when these conditions are satisfied. The method is illustrated with several computed examples.     

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.     

On the convexity of parametric Bézier triangular surfaces

In this paper, we discuss the convexity of parametric Bézier triangular patches, give some sufficient conditions of it to be convex, which only depend on the edge vectors and twist vectors. All the conditions we obtained can be served as the extension of the convexity preserving conditions of functional Bézier triangular patches.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Periodic Bézier curves

We construct closed trigonometric curves in a Bézier-like fashion. A closed control polygon defines the curves, and the control points exert a push-pull effect on the curve. The representation of circles and derived curves turns out to be surprisingly simple. Fourier and Bézier coefficients of a curve relate via Discrete Fourier Transform (DFT). As a consequence, DFT also applies to several operations, including parameter shift, successive differentiation and degree-elevation. This Bézier model is a particular instance of a general periodic scheme, where radial basis functions are generated as translates of a symmetric function. In addition to Bézier-like approximation, such a periodic scheme subsumes trigonometric Lagrange interpolation. The change of basis between Bézier and Lagrange proceeds via DFT too, which can be applied to sample the curve at regularly spaced parameter values. The Bézier curve defined by certain control points is a low-pass filtered version of the Lagrange curve interpolating the same set of points.     

The intersection of a bicubic Bézier patch and a plane

Let S be an arbitrarily given bicubic Bézier patch and P an arbitrarily given plane. In this paper, a necessary and sufficient condition for S ∩ P to be non-empty is given, and some properties of S ∩ P, if non-empty, are included. Furthermore an efficient and robust algorithm for finding S ∩ P is described, and many interesting examples are shown.     

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.     

Multi-degree reduction of triangular Bézier surfaces with boundary constraints

Given a triangular Bézier surface of degree n, the problem of multi-degree reduction by a triangular Bézier surface of degree m with boundary constraints is investigated. This paper considers the continuity of triangular Bézier surfaces at the three corners, so that the boundary curves preserve endpoints continuity of any order . The l₂- and L₂-norm combined with the constrained least-squares method are used to get the matrix representations for the control points of the degree reduced surfaces. Both methods can be applied to piecewise continuous triangular patches or to only a triangular patch with the combination of surface subdivision. And the resulting piecewise approximating patches are globally C⁰ continuous. Finally, error estimation is given and numerical examples demonstrate the effectiveness of our methods.     

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.