Construction of Endpoint Constrained Cubic Rational Curve with Chord-Length Parameterization

This paper discusses the problem that constructing a curve to satisfy the given endpoint constraints and chord-length parameters. Based on the research of Lu, the curve construction method for the entire tangent angles region （α0, α1）∈（-r, r）×（-r, r） is given. Firstly, to ensure the weights are always positive, the three characteristics of cubic rational Bezier curve is proved, then the segment construction idea for the other tangent angles are presented in view of the three characteristics. The curve constructed with the new method satisfies the endpoint constraint and chord-length parameters, it＇s G1 continuous in every segment curve, and the shapes of the curve are well.

APPROXIMATE COMMON DIVISORS OF POLYNOMIALS AND DEGREE REDUCTION FOR RATIONAL CURVES

Abstract This paper deals with how to perturb a given set of polynomials so as to include a common linear factor. An algorithm is derived for determining such a set of perturbation polynomials which are subject to certain constrains at the endpoints of a prescribed parametric interval and minimized in a certain sense. This result can be combined with subdivision technique to obtain a continuous piecewise approximation to a rational curve.

Judging or setting weight steady-state of rational Bézier curves and surfaces

Many works have investigated the problem of reparameterizing rational B^zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.

Direction monotonicity for a rational Bézier curve

The monotonicity of a rational Bezier curve, usually related to an explicit function, is determined by the used coordinate system. However, the shape of the curve is independent of the coordinate system. To meet the affine invariant property, a kind of generalized mono- tonicity, called direction monotonicity, is introduced for rational Bezier curves. The direction monotonicity is applied to both planar and space curves and to both Cartesian and affine co- ordinate systems, and it includes the traditional monotonicity as a subcase. By means of it, proper affine coordinate systems may be chosen to make some rational Bezier curves monotonic. Direction monotonic interpolation may be realized for some of the traditionally nonmonotonic data as well.

A New Fractional Rational Bézier Curve

Based on rational Bézier curves given by Ron Goldman, a new fractional rational Bézier curve was first defined in terms of fractional Bernstein bases. Moreover, some basic properties were dicussed and a theorem connected to Poisson curves was obtained. Some examples in this paper were given by the visual results.

The Relationship Between Weights and Control Vertices of Two Rational NURBS Curves Representing the Same Curve Parametrically and Geometrically

The paper discusses the relationship between weights and control vertices of two rational NURBS curves of degree two or three with all weights larger than zero when they represent the same curve parametrically and geometrically, and gives sufficient and necessary conditions for coincidence of two rational NURBS curves in non-degeneracy case.

An optimization method for rational Bézier curve design

Adjusting weights as a shape control tool in rational B6zier curve design is not easy because the weights have a global in- fluence. The curve could not approximate control polygon satisfactorily by an interactive manner. In order to produce a curve close enough to control polygon at every control vertex, an optimization model is established to minimize the distance between rational B6zier curve and its control points. This optimization problem is converted to a quadratic programming problem by separating and recombining the objective function. The new combined multi-objective optimization problem is reasonable and easy to solve. With an optimal parameter, the computing process is discussed. Comparative examples show that the designed curve is closer to control polygon and preserves the shape of the control polygon well.

THE RELATIONSHIP BETWEEN PROJECTIVE GEOMETRIC AND RATIONAL QUADRATIC B-SPLINE CURVES

Abstract For two rational quadratic B spline curves with same control vertexes, the cross ratio of four collinear points are represented: which are any one of the vertexes, and the two points that the ray initialing from the vertex intersects with the corresponding segments of the two curves, and the point the ray intersecting with the connecting line between the two neighboring vertexes. Different from rational quadratic Bézier curves, the value is generally related with the location of the ray, and the necessary and sufficient condition of the ratio being independent of the ray's location is showed. Also another cross ratio of the following four collinear points are suggested, i.e. one vertex, the points that the ray from the initial vertex intersects respectively with the curve segment, the line connecting the segments end points, and the line connecting the two neighboring vertexes. This cross ratio is concerned only with the ray's location, but not with the weights of the curve. Furthermore, the cross ratio is projective invariant under the projective transformation between the two segments.

Rational cubic/quartic Said-Ball conics

In CAGD, the Said-Ball representation for a polynomial curve has two advantagesover the B′ezier representation, since the degrees of Said-Ball basis are distributed in a step type.One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomialcurve runs twice as fast as the de Casteljau algorithm of B′ezier curve. Another is that theoperations of degree elevation and reduction for a polynomial curve in Said-Ball form are simplerand faster than in B′ezier form. However, Said-Ball curve can not exactly represent conics whichare usually used in aircraft and machine element design. To further extend the utilizationof Said-Ball curve, this paper deduces the representation theory of rational cubic and quarticSaid-Ball conics, according to the necessary and su？cient conditions for conic representation inrational low degree B′ezier form and the transformation formula from Bernstein basis to Said-Ballbasis. The results include the judging method for whether a rational quartic Said-Ball curve is aconic section and design method for presenting a given conic section in rational quartic Said-Ballform. Many experimental curves are given for confirming that our approaches are correct ande？ective.

Offset approximation based on reparameterizing the path of a moving point along the base circle

This paper presents a novel algorithm for planar curve offsetting. The basic idea is to regard the locus relative to initial base circle, which is formed by moving the unit normal vectors of the base curve, as a unit circular arc first, then accurately to represent it as a rational curve, and finally to reparameterize it in a particular way to approximate the offset. Examples illustrated that the algorithm yields fewer curve segments and control points as well as C^1 continuity, and so has much significance in terms of saving computing time, reducing the data storage and smoothing curves entirely.

RESEARCH ON A MORE RATIONAL CONFIGURATION OF THE INVERT CURVE OF A SPILLWAY

This paper is concerned with the research on the rational curve for thebucket of a spillway. A new type of rational curve against cavitation damage to spillway has beenproposed; and the numerical results show that the proposed curve (gradually and continuously variedcurvature curve, abbreviated as GCVC curve) can greatly raise the minimum cavitation number andcause the distribution of water pressure on the curved surface more reasonable. The proposed curve(GCVC curve) is simple, and can be conveniently used in practical hydraulic engineering.

PIECEWISE RATIONAL APPROXIMATIONS OF REAL ALGEBRAIC CURVES

We use a combination of both algebraic and numerical techniques to construct a C-1-continuous, piecewise (m, n) rational epsilon-approximation of a real algebraic plane curve of degree d. At singular points we use the classical Weierstrass Preparation Theorem and Newton power series factorizations, based on the technique of Hensel lifting. These, together with modified rational Pade approximations, are used to efficiently construct locally approximate, rational parametric representations for all real branches of an algebraic plane curve. Besides singular points we obtain an adaptive selection of simple points about which the curve approximations yield a small number of pieces yet achieve C-1 continuity between pieces. The simpler cases of C-1 and C-0 continuity are also handled in a similar manner. The computation of singularity, the approximation error bounds and details of the implementation of these algorithms are also provided.

VERIFYING THE IMPLICITIZATION FOR MULAE FORDEGREE n RATIONAL BEZIER CURVES

This is a continuation of short communication([1]). In [1] a verification of the implicitization equation for degree two rational Bezier curves is presented which does not require the use of resultants. This paper presents these verifications in the general cases, i.e., for degree n rational Bezier curves. Thus some interesting interplay between the structure of the n x n implicitization matrix and the de Casteljau algorithm is revealed.

Spline-based automatic path generation of welding robot

This paper presents a flexible method for the representation of welded seam based on spline interpolation. In this method, the tool path of welding robot can be generated automatically from a 3D CAD model. This technique has been implemented and demonstrated in the FANUC Arc Welding Robot Workstation. According to the method, a software system is developed using VBA of SolidWorks 2006. It offers an interface between SolidWorks and ROBOGUIDE, the off-line programming software of FANUC robot. It combines the strong modeling function of the former and the simulating function of the latter. It also has the capability of with on-line robot. The result data have shown its high accuracy and strong reliability in experiments. This method will improve the intelligence and the flexibility of the welding robot workstation.

Representing conics by low degree rational DP curves

A DP curve is a new kind of parametric curve defined by Delgado and Pefla(2003);it has very good properties when used in both geometry and algebra,i.e.,it is shape preserving and has a linear time complexity for evaluation.It overcomes the disadvantage of some generalized Ball curves that are fast for evaluation but cannot preserve shape,and the disadvantage of the Bézier curve that is shape preserving but slow for evaluation.It also has potential applications in computer-aided design and manufacturing(CAD/CAM) systems.As conic section is often used in shape design,this paper deduces the necessary and sufficient conditions for rational cubic or quartic DP representation of conics to expand the application area of DP curves.The main idea is based on the transformation relationship between low degree DP basis and Bernstein basis,and the representation theory of conics in rational low degree Bézier form.The results can identify whether a rational low degree DP curve is a conic section and also express a given conic section in rational low degree DP form,i.e.,give positions of the control points and values of the weights of rational cubic or quartic DP conics.Finally,several numerical examples are presented to validate the effectiveness of the method.

A CLASS OF INEQUALITIES ABOUT BERNSTEIN POLYNOMIAL AND THEIR APPLICATIONS

In this paper a class of new inequalities about Bernstein polynomial is established. With these inequalities, the estimation of heights, the derivative bounds of Bézier curves and rational Bézier curves can be improved greatly.

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

This paper considers the class of autonomous algebraic ordinary differential equations(AODEs)of order one,and studies their Liouvillian general solutions.In particular,let F(y,w)=0 be a rational algebraic curve over C.The authors give necessary and sufficient conditions for the autonomous first-order AODE F(y,y′)=0 to have a Liouvillian solution over C.Moreover,the authors show that a Liouvillian solutionαof this equation is either an algebraic function over C(x)or an algebraic function over C(exp(ax)).As a byproduct,these results lead to an algorithm for determining a Liouvillian general solution of an autonomous AODE of order one of genus zero.Rational parametrizations of rational algebraic curves play an important role on this method.

On the Space of Conics on Complete Intersections

We get sharp degree bound for generic smoothness and connectedness of the space of lines and conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on hypersurfaces.As a consequence,we prove that for a Fano complete intersection X with index≥2,the 1-Griffiths group generated by algebraic 1-cycles homologous to 0 modulo algebraic equivalence is trivial,which is a conjecture for general rationally connected varieties.