Rational hodographs

An equation for the hodograph control points for a rational degree n Bézier curve is derived. If, for a rational Bézier curve with positive weights, all weights are set to one, then the hodograph for this integral Bézier curve will bound the tangent directions of the rational curve.     

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.     

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.     

On the smooth convergence of subdivision and degree elevation for Bézier curves

Bézier subdivision and degree elevation algorithms generate piecewise linear approximations of Bézier curves that converge to the original Bézier curve. Discrete derivatives of arbitrary order can be associated with these piecewise linear functions via divided differences. Here we establish the convergence of these discrete derivatives to the corresponding continuous derivatives of the initial Bézier curve. Thus, we show that the control polygons generated by subdivision and degree elevation provide not only an approximation to a Bézier curve, but also approximations of its derivatives of arbitrary order.     

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.     

Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials

In a recent article, Ge et al. (1997) identify a special class of rational curves (Harmonic Rational Bézier (HRB) curves) that can be reparameterized in sinusoidal form. Here we show how this family of curves strongly relates to the class of p-Bézier curves, curves easily expressible as single-valued in polar coordinates. Although both subsets do not coincide, the reparameterization needed in both cases is exactly the same, and the weights of a HRB curve are those corresponding to the representation of a circular arc as a p-Bézier curve. We also prove that a HRB curve can be written as a combination of its control points and certain Bernstein-like trigonometric basis functions. These functions form a normalized totally positive B-basis (that is, the basis with optimal shape preserving properties) of the space of trigonometric polynomials {1, sint, cost, …. sinmt, cosmt} defined on an interval of length < π.     

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.     

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 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.     

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.     

Optimized refinable enclosures of multivariate polynomial pieces

An enclosure is a two-sided approximation of a uni- or multivariate function by a pair of typically simpler functions such that b⁻bb⁺ over the domain U of interest. Enclosures are optimized by minimizing the width max_Ub⁺−b⁻ and refined by enlarging the space . This paper develops a framework for efficiently computing enclosures for multivariate polynomials and, in particular, derives piecewise bilinear enclosures for bivariate polynomials in tensor-product Bézier form. Runtime computation of enclosures consists of looking up pre-optimized enclosures and linearly combining them with the second differences of b. The width of these enclosures scales by a factor 1/4 under midpoint subdivision.     

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.     

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 uniqueness of Bézier control points

This paper studies the uniqueness of the control points and weights of a Bézier curve in any dimensional space. If the curve cannot be degree reduced and is not improperly parametrized then the control points are uniquely defined and the weights can only be varied by a projective transformation of the parameter space.     

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.     

Cubic Bézier approximation of a digitized curve

In this paper we present an efficient technique for piecewise cubic Bézier approximation of digitized curve. An adaptive breakpoint detection method divides a digital curve into a number of segments and each segment is approximated by a cubic Bézier curve so that the approximation error is minimized. Initial approximated Bézier control points for each of the segments are obtained by interpolation technique i.e. by the reverse recursion of De Castaljau's algorithm. Two methods, two-dimensional logarithmic search algorithm (TDLSA) and an evolutionary search algorithm (ESA), are introduced to find the best-fit Bézier control points from the approximate interpolated control points. ESA based refinement is proved to be better experimentally. Experimental results show that Bézier approximation of a digitized curve is much more accurate and uses less number of points compared to other approximation techniques.     

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.     

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 the Bézier points of B-spline curves and surfaces

The well-known algorithm by de Boor for calculating a point of a B-spline curve can also be used to produce the Bézier points of a B-spline curve or surface.