On the convergence of hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we derive the bounds on the magnitude of l  th (l=2,3)$(l=2,3)$ order derivatives of rational Bézier curves, estimate the error, in the _L∞$L_{\infty }$ norm sense, for the hybrid polynomial approximation of the l  th (l=1,2,3)$(l=1,2,3)$ order derivatives of rational Bézier curves. We then prove that when the hybrid polynomial approximation converges to a given rational Bézier curve, the l  th (l=1,2,3)$(l=1,2,3)$ derivatives of the hybrid polynomial approximation curve also uniformly converge to the corresponding derivatives of the rational curve. These results are useful for designing simpler algorithms for computing tangent vector, curvature vector and torsion vector of rational Bézier curves.     

Approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces

An attractive method for approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces is introduced. The main result is that the arbitrary given order derived vectors of a polynomial triangular surface converge uniformly to those of the approximated rational triangular Bézier surface as the elevated degree tends to infinity. The polynomial triangular surface is constructed as follows. Firstly, we elevate the degree of the approximated rational triangular Bézier surface, then a polynomial triangular Bézier surface is produced, which has the same order and new control points of the degree-elevated rational surface. The approximation method has theoretical significance and application value: it solves two shortcomings-fussy expression and uninsured convergence of the approximation-of Hybrid algorithms for rational polynomial curves and surfaces approximation.     

Sample-based polynomial approximation of rational Bézier curves

We present an iteration method for the polynomial approximation of rational Bézier curves. Starting with an initial Bézier curve, we adjust its control points gradually by the scheme of weighted progressive iteration approximations. The L_p-error calculated by the trapezoidal rule using sampled points is used to guide the iteration approximation. We reduce the L_p-error by a predefined factor at every iteration so as to obtain the best approximation with a minimum error. Numerical examples demonstrate the fast convergence of our method and indicate that results obtained using the L₁-error criterion are better than those obtained using the L₂-error and L_∞-error criteria.     

Approximating conic sections by constrained Bézier curves of arbitrary degree

In this paper, an algorithm for approximating conic sections by constrained Bézier curves of arbitrary degree is proposed. First, using the eigenvalues of recurrence equations and the method of undetermined coefficients, some exact integral formulas for the product of two Bernstein basis functions and the denominator of rational quadratic form expressing conic section are given. Then, using the least squares method, a matrix-based representation of the control points of the optimal Bézier approximation curve is deduced. This algorithm yields an explicit, arbitrary-degree Bézier approximation of conic sections which has function value and derivatives at the endpoints that match the function value and the derivatives of the conic section up to second order and is optimal in the _L2$L_2$ norm. To reduce error, the method can be combined with a curve subdivision scheme. Computational examples are presented to validate the feasibility and effectiveness of the algorithm for a whole curve or its part generated by a subdivision.     

Approximate Bézier curves by cubic LN curves

In order to derive the offset curves by using cubic Bézier curves with a linear field of normal vectors (the so-called LN Bézier curves) more efficiently, three methods for approximating degree n Bézier curves by cubic LN Bézier curves are considered, which includes two traditional methods and one new method based on Hausdorff distance. The approximation based on shifting control points is equivalent to solving a quadratic equation, and the approximation based on L₂ norm is equivalent to solving a quartic equation. In addition, the sufficient and necessary condition of optimal approximation based on Hausdorff distance is presented, accordingly the algorithm for approximating the degree n Bézier curves based on Hausdorff distance is derived. Numerical examples show that the error of approximation based on Hausdorff distance is much smaller than that of approximation based on shifting control points and L₂ norm, furthermore, the algorithm based on Hausdorff distance is much simple and convenient.     

Dual bases for Wang-Bézier basis and their applications

This paper presents the dual bases for Wang-Bézier curves with a position parameter L, which include Bézier curve, Wang-Ball curve and some intermediate curves. The Marsden identity and the transformation formulas from Bézier curve to Wang-Bézier curve are also given. These results are useful for the application of Wang-Bézier curve and their popularization in Computer Aided Geometric Design.     

On parametrization of interpolating curves

In parametric curve interpolation there is given a sequence of data points and corresponding parameter values (nodes), and we want to find a parametric curve that passes through data points at the associated parameter values. We consider those interpolating curves that are described by the combination of control points and blending functions. We study paths of control points and points of the interpolating curve obtained by the alteration of one node. We show geometric properties of quadratic Bézier interpolating curves with uniform and centripetal parameterizations. Finally, we propose geometric methods for the interactive modification and specification of nodes for interpolating Bézier curves.     

A planar cubic Bézier spiral

A planar cubic Bézier curve segment that is a spiral, i.e., its curvature varies monotonically with arc-length, is discussed. Since this curve segment does not have cusps, loops, and inflection points (except for a single inflection point at its beginning), it is suitable for applications such as highway design, in which the clothoid has been traditionally used. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS (nonuniform rational B-splines) and is thus suitable for general curve design applications in which fair curves are important.     

Circle approximation using LN Bézier curves of even degree and its application

We present an approximation method of circular arcs using linear-normal (LN) Bézier curves of even degree, four and higher. Our method achieves G^m$G^m$ continuity for endpoint interpolation of a circular arc by a LN Bézier curve of degree 2m  , for m=2,3$m=2,3$. We also present the exact Hausdorff distance between the circular arc and the approximating LN Bézier curve. We show that the LN curve has an approximation order of 2m+2$2m+2$, for m=2,3$m=2,3$. Our approximation method can be applied to offset approximation, so obtaining a rational Bézier curve as an offset approximant. We derive an algorithm for offset approximation based on the LN circle approximation and illustrate our method with some numerical examples.     

Necessary and sufficient conditions for rational quartic representation of conic sections

Conic section is one of the geometric elements most commonly used for shape expression and mechanical accessory cartography. A rational quadratic Bézier curve is just a conic section. It cannot represent an elliptic segment whose center angle is not less than π$\pi$. However, conics represented in rational quartic format when compared to rational quadratic format, enjoy better properties such as being able to represent conics up to 2π$2\pi$ (but not including 2π$2\pi$) without resorting to negative weights and possessing better parameterization. Therefore, it is actually worth studying the necessary and sufficient conditions for the rational quartic Bézier representation of conics. This paper attributes the rational quartic conic sections to two special kinds, that is, degree-reducible and improperly parameterized; on this basis, the necessary and sufficient conditions for the rational quartic Bézier representation of conics are derived. They are divided into two parts: Bézier control points and weights. These conditions can be used to judge whether a rational quartic Bézier curve is a conic section; or for a given conic section, present positions of the control points and values of the weights of the conic section in form of a rational quartic Bézier curve. Many examples are given to show the use of our results.     

On the conditions for the coincidence of two cubic Bézier curves

In a recent article, Wang et al. [2] derive a necessary and sufficient condition for the coincidence of two cubic Bézier curves with non-collinear control points. The condition reads that their control points must be either coincident or in reverse order. We point out that this uniqueness of the control points for polynomial cubics is a straightforward consequence of a previous and more general result of Barry and Patterson, namely the uniqueness of the control points for rational Bézier curves. Moreover, this uniqueness applies to properly parameterized polynomial curves of arbitrary degree.     

On the Convergence of Polynomial Approximation of Rational Functions

This paper investigates the convergence condition for the polynomial approximation of rational functions and rational curves. The main result, based on a hybrid expression of rational functions (or curves), is that two-point Hermite interpolation converges if all eigenvalue moduli of a certainr×rmatrix are less than 2, whereris the degree of the rational function (or curve), and where the elements of the matrix are expressions involving only the denominator polynomial coefficients (weights) of the rational function (or curve). As a corollary for the special case ofr=1, a necessary and sufficient condition for convergence is also obtained which only involves the roots of the denominator of the rational function and which is shown to be superior to the condition obtained by the traditional remainder theory for polynomial interpolation. For the low degree cases (r=1, 2, and 3), concrete conditions are derived. Application to rational Bernstein–Bézier curves is discussed.     

Recursive polynomial curve schemes and computer-aided geometric design

A class of polynomial curve schemes is introduced that may have widespread application to CAGD (computer-aided geometric design), and which contains many well-known curve schemes, including Bézier curves, Lagrange polynomials, B-spline curve (segments), and Catmull-Rom spline (segments). The curves in this class can be characterized by a simple recursion formula. They are also shown to have many properties desirable for CAGD; in particular they are affine invariant, have the convex hull property, and possess a recursive evaluation algorithm. Further, these curves have shape parameters which may be used as a design tool for introducing such geometric effects as tautness, bias, or interpolation. The link between probability theory and this class of curves is also discussed.Communicated by Klaus Höllig.     

Design and G connection of developable surfaces through Bézier geodesics

For potential application in shoemaking and garment manufacture industries, the G¹ connection of (1, k) developable surfaces with abutting geodesic is important. In this paper, we discuss the developable surface which contains a given 3D Bézier curve as geodesic and prove the corresponding conclusions in detail. Primarily we study G¹ connection of developable surfaces through abutting cubic Bézier geodesics and give some examples.     

Conditions for coincidence of two cubic Bézier curves

This paper presents a necessary and sufficient condition for judging whether two cubic Bézier curves are coincident: two cubic Bézier curves whose control points are not collinear are coincident if and only if their corresponding control points are coincident or one curve is the reversal of the other curve. However, this is not true for degree higher than 3. This paper provides a set of counterexamples of degree 4.     

Pseudo-geodesics on three-dimensional surfaces and pseudo-geodesic meshes

We present two numerical methods to approximate the shortest path or a geodesic between two points on a three-dimensional parametric surface. The first one consists of minimizing the path length, working in the parameter domain, where the approximation class is composed of Bézier curves. In the second approach, we consider Bézier surfaces and their control net. The numerical implementation is based on finding the shortest path on the successive control net subdivisions. The convergence property of the Bézier net to the surface gives an approximation of the required shortest path. These approximations, also called pseudo-geodesics, are then applied to the creation of pseudo-geodesic meshes. Experimental results are also provided.     

Curve design with rational Pythagorean-hodograph curves

The dual Bézier representation offers a simple and efficient constructive approach to rational curves with rational offsets (rational PH curves). Based on the dual form, we develop geometric algorithms for approximating a given curve with aG ² piecewise rational PH curve. The basic components of the algorithms are an appropriate geometric segmentation andG ² Hermite interpolation. The solution involves rational PH curves of algebraic class 4; these curves and important special cases are studied in detail.     

Smooth polynomial approximation of spiral arcs

Constructing fair curve segments using parametric polynomials is difficult due to the oscillatory nature of polynomials. Even NURBS curves can exhibit unsatisfactory curvature profiles. Curve segments with monotonic curvature profiles, for example spiral arcs, exist but are intrinsically non-polynomial in nature and thus difficult to integrate into existing CAD systems. A method of constructing an approximation to a generalised Cornu spiral (GCS) arc using non-rational quintic Bézier curves matching end points, end slopes and end curvatures is presented. By defining an objective function based on the relative error between the curvature profiles of the GCS and its Bézier approximation, a curve segment is constructed that has a monotonic curvature profile within a specified tolerance.     

An extension of the Bézier model

In this paper, we first construct a new kind of basis functions by a recursive approach. Based on these basis functions, we define the Bézier-like curve and rectangular Bézier-like surface. Then we extend the new basis functions to the triangular domain, and define the Bernstein-Bézier-like surface over the triangular domain. The new curve and surfaces have most properties of the corresponding classical Bézier curve and surfaces. Moreover, the shape parameter can adjust the shape of the new curve and surfaces without changing the control points. Along with the increase of the shape parameter, the new curve and surfaces approach the control polygon or control net. In addition, the evaluation algorithm for the new curve and triangular surface are provided.