Rational Quadratic Bezier Curves Research Articles

Asymptotic preserving schemes on conical unstructured 2D meshes

AbstractIn this article, we consider the hyperbolic heat equation. This system is linear hyperbolic with stiff source terms and satisfies a diffusion limit. Some finite volume numerical schemes have been proposed which reproduce this diffusion limit. Here, we extend such schemes, originally defined on polygonal meshes, to conical meshes (using rational quadratic Bezier curves). We obtain really new schemes that do not reduce to the polygonal version when the conical edges tend to straight lines. Moreover, these schemes can handle curved unstructured meshes so that geometric error on initial data representation is reduced and geometry of the domain is improved. Extra flux coming from conical edge (through his midedge point) has a deep impact on the stabilization when compared with the original polygonal scheme. Cross‐stencil phenomenon of polygonal scheme has disappeared, and issue of positivity for the diffusion problem (although unresolved on distorted mesh and/or with varying cross‐section) has been in some sense improved.

ON THE GEOMETRY OF RATIONAL BÉZIER CURVES

The purpose of this paper is to assign a movable frame to an arbitrary point of a rational Bezier curve on the 2-sphere S2 in Euclidean 3-space R3 to provide a better understanding of the geometry of the curve. Especially, we obtain the formula of geodesic curvature for a quadratic rational Bezier curve that allows a curve to be characterized on the surface. Moreover, we give some important results and relations for the Darboux frame and geodesic curvature of a such curve. Then, in specific case, given characterizations for the quadratic rational Bezier curve are illustrated on a unit 2-sphere.

Necessary and Sufficient Conditions for Expressing Quadratic Rational Bézier Curves

Quadratic rational Bezier curve transformation is widely used in the field of computer geometry. In this paper, we offer several important characteristics of quadratic rational Bezier curve. More precisely, on the basis of proving its monotonicity, the necessary and sufficient conditions for transforming quadratic rational Bezier curve into point, line segment, parabola, elliptic arc, circular arc and hyperbola are proved respectively.

Rational Quadratic Bézier Spirals

A quadratic Bezier representation withholds a curve segment with free from loops, cusps and inflection points. Furthermore, this rational form provides extra freedom to generate visually pleasing curves due to the existence of weights. In this paper, we propose sufficient conditions for rational quadratic Bezier curves to possess monotonic increasing/decreasing curvatures by means of monotone curvature tests which are based on the derivative of curvature functions. We have derived a simple interval of the middle weight that assures the construction of a family of rational quadratic Bezier curves to be planar spirals, which is characterized by the turning angle, end curvatures and the chords of control polygon. The proposed formulation can be used by CAD systems for aesthetic product design, highway/railway design and robot trajectory design avoiding unwanted curvature oscillations.

ISOGONAL AND ISOTOMIC CONJUGATES OF QUADRATIC RATIONAL Bézier CURVES

In this paper we characterize the isogonal and isotomic conjugates of conic. Every conic can be expressed by a quadratic rational Bezier curve having control polygon b0b1b2 with weight w > 0. We show that the isotomic conjugate of parabola and hyperbola with respect to △b0b1b2 is ellipse, and that the isotomic conjugate of ellipse with the weight w = 1-2 is identical. We also find all cases of the isogonal conjugate of conic with respect to △b0b1b2. Our characterizations are derived easily due to the expression of conic by the quadratic rational Bezier curve in standard form.

Improved derivative bounds of the rational quadratic Bézier curves

A new derivative bound of the rational quadratic Bézier curves is obtained. It is an improvement over the bound which was regarded as a sharp bound in a recent paper. Also the new bound is superior to the existing bounds given by some other authors.

Increasing bending strength in spur gears using shape optimisation of cutting tool profile

The geometrical shape of gear tooth fillet profile, usually cut out by the cutter tip, plays a significant role in the evaluation of the gear bending stress. In order to improve the teeth bending strength, a novel curve (quadratic rational Bezier curve) is introduced to describe the cutter tip. The gear tooth finite element models are founded by APDL in the ANSYS software. With the maximum bending stress (von Mises stress) as the objective function, SUB-PROBLEM and FIRST-ORDER optimisation methods in ANSYS are used to optimise the cutter tip. Single-tooth bending test is also conducted to prove the optimisation result. The study reveals that the relationship between the design variable and tooth root bending stress is non-linear, and the gear cut by the optimised cutter exhibits higher bending strength rather than the gear cut by standard cutting tool.

Isoparametric Curve of Quadratic F-Bézier Curve

AbstractIn this thesis, we consider isoparametric curves of quadratic F-Bezier curves. F-Bezier curves unify C-Bezier curves whose basis is aš•
’iŠ–š
’i
’iIae and H-Bezier curves whose basis is aš•
’iŠ–š
’i
’iIae . Thus F-Bezier curves are more useful in Geometric Modeling or CAGD(Computer Aided Geometric Design). We derive the relation between the quadratic F-Bezier curves and the quadratic rational Bezier curves. We also obtain the geometric properties of isoparametric curve of the quadraticF-Bezier curves at both end points and prove the continuity of the isoparametric curve. Key words : F-Bezier Curve, Quadratic Rational Bezier Curve, Isoparametric Curve, Q-Bezier Curve, H-Bezier Curve 1 Graduate School of Education, Chosun University, Gwangju 501-759, Korea 2 Department of Mathematics Education, Chosun University, Gwangju 501-759, Korea † Corresponding Author : ahn@chosun.ac.kr(Received : January 2, 2013, Revised : March 20, 2013, Accepted : March 25, 2013) 1. Introduction C-Bezier curve

Extension of ALE methodology to unstructured conical meshes

We propose a bi-dimensional finite volume extension of a continuous ALE method on unstructured cells whose edges are parameterized by rational quadratic Bezier curves. For each edge, the control point possess a weight that permits to represent any conic (see for example [LIGACH]) and thanks to [WAGUSEDE,WAGU], we are able to compute the exact area of our cells. We then give an extension of scheme for remapping step based on volume fluxing [MARSHA] and self-intersection flux [ALE2DHAL]. For the rezoning phase, we propose a three step process based on moving nodes, followed by control point and weight re-adjustment. Finally, for the hydrodynamic step, we present the GLACE scheme [GLACE] extension (at first-order) on conic cell using the same formalism. We only propose some preliminary first-order simulations for each steps: Remap, Pure Lagrangian and finally ALE (rezoning and remapping).

ARC-LENGTH ESTIMATIONS FOR QUADRATIC RATIONAL B´ ezier CURVES COINCIDING WITH ARC-LENGTH OF SPECIAL SHAPES

In this paper, we present arc-length estimations for quadratic rational Bezier curves using the length of polygon and distance between both end points. Our arc-length es-timations coincide with the arc-length of the quadratic rational Bezier curve exactly when the weight w is 0, 1 and ∞. We show that for all w ＞ 0 our estimations are strictly increasing with respect to w. Moreover, we find the parameter μ * which makes our estimation coincide with the arc-length of the quadratic rational Bezier curve when it is a circular arc too. We also show that μ * has a special limit, which is used for optimal estimation. We present some numerical examples, and the numerical results illustrates that the estimation with the limit value of μ * is an optimal estimation.

Parametric curve interpolation by combination of two conic sections

The problem of constructing a parametric cubic rational curve to interpolate a set of distinct data points is discussed. Unlike the existing methods which include the determination of knots, the new method constructs parametric curve without the process of determining knots. For each point, a quadratic rational Bezier curve is constructed by the five convex data points. For the data point, if there is no five convex data points, the quadratic rational Bezier curve is constructed with four or three points under a constraint. Between each pair of the two adjacent points, a parametric cubic rational curve is constructed by the combination of the two quadratic rational Bezier curves. The constructed cubic rational curve reproduces a conic section exactly if the given data points are taken from the conic section. The comparison of the new method with other ones is included.

Solving Kepler’s Equation using Bézier curves

This paper presents a non-iterative approach to solve Kepler’s Equation, M = E − e sin E, based on non-rational cubic and rational quadratic Bezier curves. Optimal control point coordinates are first shown to be linear with respect to orbit eccentricity for any eccentric anomaly range. This property yields the development of a piecewise (e.g., 3, 4) solving technique providing accuracies better than 10−13 degree for orbit eccentricity e ≤ 0.99. The proposed method does not require large pre-computed discretization data, but instead solves a cubic/quadratic algebraic equation and uses a single final Halley iteration in only a few lines of code. The method still provides accuracies better than 10−5 degree for the near parabolic worst case (e = 0.9999) with very small mean anomalies (M < 0.0517 deg). The complexity of the proposed algorithm is constant, independent of the parameters e and M. This makes the method suitable for extensive orbit propagations.

Quadratic approximation to plane parametric curves and its application in approximate implicitization

Expressing complex curves with simple parametric curve segments is widely used in computer graphics, CAD and so on. This paper applies rational quadratic B-spline curves to give a global C1 continuous approximation to a large class of plane parametric curves including rational parametric curves. Its application in approximate implicitization is also explored. The approximated parametric curve is first divided into intrinsic triangle convex segments which can be efficiently approximated with rational quadratic Bezier curves. With this approximation, we keep the convexity and the cusp (sharp) points of the approximated curve with simple computations. High accuracy approximation is achieved with a small number of quadratic segments. Experimental results are given to demonstrate the operation and efficiency of the algorithm.

A Bidirectional Generating Algorithm for Rational Parametric Curves

A generating algorithm for rational quadratic Bezier curves is provided. In the algorithm, we find the recurrence relation to get every pixel along the curve, using the bidirectional strategy for rendering and the forward and backward difference method for computing. The algorithm has a fast generating speed, and the curves keep a reasonable rendering accuracy. This algorithm can easily be generalized to render higher-degree rational Bezier curves and other rational parametric curves; it has broad applications. Source code is available online at the website listed at the end of this paper.

Geometric meanings of the parameters on rational conic segments

Using algebraic and geometric methods, functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bezier curve. That is, the inverse mappings of the mappings represented by the expressions of rational conic segments are given. These formulae relate some triangular areas or some angles, determined by the selected point on the curve and the control points of the curve, as well as by the weights of the rational Bezier curve. Also, the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment, as well as by the weights of the rational Bezier curve. These results are greatly useful for optimal parametrization, reparametrization, etc., of rational Bezier curves and surfaces.

The relationship between projective geometric and rational quadratic b-spline curves

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 Bezier 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.

２次有理Ｂｅｚｉｅｒ曲線および２次ＮＵＲＢＳの同次表現について

２次有理Ｂｅｚｉｅｒ曲線および２次ＮＵＲＢＳの同次表現について

CONSTRAINING A QUADRATIC RATIONAL BEZIER TO BE AN ELLIPTICAL ARC

Analytic constraint solvers have a limited set of available geometries, typically lines and circles. This paper examines a method for constraining the control net of a rational quadratic Bezier curve so that it is always an elliptical arc.