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A new method for approximation of conic section by quartic Bezier curve is presented, based on the quartic Bezier approximation of circular arcs. Here we give an upper bound of the Hausdorff distance between the conic section and the approximation curve, and show that the error bounds have the approximation order of eight. Furthermore, our method yields quartic G2 continuous spline approximation of conic section when using the subdivision scheme, and the effectiveness of this method is demonstrated by some numerical examples.

In this paper, we present a C 3 quartic B-spline approximation of circular arcs. The Hausdorff distance between the C 3 quartic B-spline curve and the circular arc is obtained in closed form. Using this error analysis, we show that the approximation order of our approximation method is six. For a given circular arc and error tolerance we find the C 3 quartic B-spline curve having the minimum number of control points within the tolerance. The algorithm yielding the C 3 quartic B-spline approximation of a circular arc is also presented.

Quartic C-Bezier curves possess similar properties with the traditional Bezier curves including terminal property, convex hull property, affine invariance, and approaching the shape of their control polygons as the shape parameter α decreases. In this paper, by adjusting the shape parameter α on the basis of the utilization of the least square approximation and nonlinear functional minimization together with fairing of a quartic C-Bezier curve with G1 continuity of quartic C-Bezier curve segments, we develop a fairing and G1 continuity algorithm for any given stitching coefficients λk(k = 1,2,…,n − 1). The shape parameters αi(i=1, 2, …, n) can be adjusted by the value of control points. The curvature of the resulting quartic C-Bezier curve segments after fairing is more uniform than before. Moreover, six examples are provided in the paper to demonstrate the efficacy of the algorithm and illustrate how to apply this algorithm to the computer-aided design/computer-aided manufacturing modeling systems. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we focus on the geometric shapes of the C-Bezier curves for the space span { 1 , t , ? , t n , sin t , cos t } . First, any C-Bezier curve is divided into a Bezier curve and a trigonometric part. So any C-Bezier curve describes the trajectory of a point orbiting around a center in an elliptical orbit while the orbital plane is moving as the ellipse center translating along a Bezier curve. Second, the geometric characters of the C-Bezier curve (the control points of the center Bezier curve, the trajectories of the vertices and the foci of the ellipse, etc.), can all be explicitly presented by the control points of the C-Bezier curve. Third, considering some special cases, we give the sufficient and necessary conditions of C-Bezier basis forming Bezier curve, ellipse, circle, common helix, and so on. Lastly, we show how to build some geometrically intuitive curves through the C-Bezier basis without rational forms. Transformation matrixes between any C-Bezier curve and its separated form.Geometric shape and the explicit expressions of the geometric characters of any C-Bezier curve.Sufficient and necessary conditions for the C-Bezier basis constructing some classical curves.Some classical curves can be constructed intuitively.

We describe the structure of the asymptotic lines near an inflection point of a Lagrangian surface, proving that in the generic situation it corresponds to two of the three possible cases when the discriminant curve has a cusp singularity. Besides being stable in general, inflection points are proved to exist on a compact Lagrangian surface whenever its Euler characteristic does not vanish.

A very general surface of degree at least four in projective space of dimension three contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces of degree at least five which contain some elliptic quartic curve. We also compute the degree of the locus of quartic surfaces containing an elliptic quartic curve, a case not covered by that formula., Minor typos corrected. To appear on Proceedings of the American Mathematical Society

An algorithm for representing planar curves in B-splines is presented in this paper. The representing problem is different from the approximation to data points; planar curve provided more information than data points. To make full use of the information, we propose a three-step representing approach: 1.Sample data points along with their tangent vectors from the planar curve according to the given accuracy. 2. Fit the sampled points by Bezier segments using local interpolation; compose these segments to an interpolation curve. 3. Approximate the interpolation curve using the best least approximation to get the final B-spline curve. Tangent information is used in the second step to construct the interpolation curve. In the third step, the system is always positive because of using the best least square approximation, so we can get more freedoms to approximate the interpolation curve. Finally, some examples of this algorithm demonstrate its usefulness and quality.

There is considerable interest in the properties of PH curves in geometric modeling and CAGD because PH curves can be computed at speeds comparable to polynomial curves and used to calculate curve arc lengths and isometric lines. The purpose of this paper is to develop a general approximation of the H-Bezier curve based on PH curves. We call the resulting approximations PHH-Bezier curves for convenience. First, a necessary and sufficient condition for a cubic plane H-Bezier curve to be a PH curve is obtained. Second, based on the H-Bezier curve, the control polygon of the cubic PHH-Bezier curve is constructed from an orderly triangle. The cubic PHH-Bezier curve is then introduced and a new algorithm for their construction is proposed. According to geometric modeling, the error analysis between H-Bezier curve and PHH-Bezier curve is estimated. Finally, the proposed algorithm is verified experimentally. It is demonstrated that cubic PHH-Bezier curves can accurately approximate H-Bezier curves but that the selection of the middle two control points of the PHH-Bezier curve has a profound impact on the quality of the approximation. Error analyses demonstrate that when the middle two control points of the PHH-Bezier curve are mixed with the corresponding original control points, a good approximation is achieved. The new algorithm may thus have considerable potential for use in geometry modeling.

In 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

Ball curve； curve design； shape parameter Abstract. Ball curve is found similar to Bézier curve，also it has a good property of shape preserving，and in some respects，it has better properties than the Bézier curve. Therefore, In the shape design，Ball curve is paid more and more attention, it has a wide application. By introducing the concept of weights in NURBS curve into a blending technique, the paper extends the representation of the cubic Ball curve. The generalized cubic Ball curve is denoted as α extension cubic Ball curve, whose shape-control capability is shown to be much better than that of Ball curve. The representation and properties of the extension curve are studied. The curve is easy and intuitive to reshape by varying the parameters; so it is useful in some applications of CAD/CAM.

In this paper, we describe an algorithm for generating a C-shaped G^2 multispiral transition curve between two non-parallel straight lines. The G^2 multispiral is a curve that consists of two or more log-aesthetic curve segments connected with curvature continuity, and it has inflection endpoints. Compound-rhythm log-aesthetic curves are not directly applicable to the generation of transition curves between two straight lines, which is important in highway and railroad track design, because both endpoints are required to be inflection points. Thus, a new approach for generating transition curves is necessary. The two log-aesthetic curve segments with shape parameter @a

We introduce the notion of the principal (binormal)-direction curve and principal (binormal)-donor curve of a Frenet curve in E 3 and give the relationship of curvature and torsion of its mates. As application, we characterize general helices and slant helices in terms of their associated curves and give a canonical method to construct them. Also, we give a new characterization of Bertrand curve.

Concentration dependencies of components' activities in a binary non-ideal solution have at least one inflection point. It is proved that in the case of the regular solution, there is only one inflection point.

In CAGD, the Said-Ball representation for a polynomial curve has two advantages over the Bezier 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 polynomial curve runs twice as fast as the de Casteljau algorithm of Bezier curve. Another is that the operations of degree elevation and reduction for a polynomial curve in Said-Ball form are simpler and faster than in Bezier form. However, Said-Ball curve can not exactly represent conics which are usually used in aircraft and machine element design. To further extend the utilization of Said-Ball curve, this paper deduces the representation theory of rational cubic and quartic Said-Ball conics, according to the necessary and sufficient conditions for conic representation in rational low degree Bezier form and the transformation formula from Bernstein basis to Said-Ball basis. The results include the judging method for whether a rational quartic Said-Ball curve is a conic section and design method for presenting a given conic section in rational quartic Said-Ball form. Many experimental curves are given for confirming that our approaches are correct and effective.

The methods uniform curve, tolerance based curve and interpolate curve were used to fit curves with the data point groups in reconstruction of 3D models and the convolution was computed and error analyzed. The parameterization of the curve got with the method of uniform curve is uniform and the tolerance based curve is opposite. The interpolate curve connects every point of the data point clouds in order. The interpolate curve with many control points and great curvature is not smooth.

A new method for digital curve length calculation based on the approximation with a B-spline has been introduced where the control points of a B-spline curve are the pixel center points. An approximate length of the digital curve is determined by calculating the length of the continuous B-spline curve. In the paper several examples are presented and the calculated lengths are compared to other methods found in the literature. The advantage of the proposed method is in the approximation of the digital curve with the continuous curve rather than with piecewise linear sections used by most other methods.

In this paper, we study a necessary and sufficient condition for a planar quartic Bezier curve to possess a Pythagorean-hodograph (PH). Based on the definition of PH curve and complex representation of planar curve, we deduce geometric conditions in terms of the legs of the control polygon which guarantee the PH property. We also discuss the problem of G^1 Hermite interpolation by planar PH quartics.

The quartic curve of Han [X. Han, Piecewise quartic polynomial curves with shape parameter, Journal of Computational and Applied Mathematics 195 (2006) 34–45] can be considered as the generalization of the cubic B-spline curve incorporating shape parameters into the polynomial basis functions. We show that this curve can be considered as the linear blending of the original cubic B-spline curve and a fixed quartic curve. Moreover, we present the Bézier form of the curve, which is useful in terms of incorporating the curve into existing CAD systems. Geometric effects of the alteration of shape parameters is also discussed, including design oriented computational methods for constrained shape control of the curve.

We present an iterative method to define a two-parameter family of continuous functions fa,θ: I → ℂ such that f1/3,π/3 is the Koch curve. We consider the two-cases θ = π/3 and θ = π/4 of these generalized Koch curves fa,θ(I). In each case we determine the pivotal value of a, the largest value of a for which the corresponding curve is not simple. We give characterizations of the double points of the curve (points on the curve that have two inverse images). In the case where θ = π/3 double points are vertices of equilateral triangles. When θ = π/4 the double points form Cantor sets in the plane. We conclude with a more general result that proves that if the fixed set (attractor) of an iterated function system is connected, then it is a curve.

The log-aesthetic curves include the logarithmic (equiangular) spiral, clothoid, and involute curves. Although most of them are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. However, their input method proposed so far by use of three so-called control points can generate only an aesthetic curve segment with monotonic curvature variation and can not create a curve with the curvature-extremal point or the inflection point. Hence at first we propose a technique to input a log-aesthetic curve segment with an inflection point at its end. Then we propose a method to generate an log-aesthetic curve with G2 continuity from a sequence of 2D points input with, for example, a liquid crystal pen tablet.

In this article, we will study plane curves of a certain degree d with three or four total inflection points. In particular, we will study their image in the moduli spaces. Also a result on curves with five total inflection points is included.

Let L be an invertible sheaf on a smooth curve C. A generalized inflection point of L is an inflection point of $$L^{\otimes n}$$ for some integer n > 0. A generalized inflection point P of L is called strongly normal if there is a unique integer n > 0 such that P is an inflection point of $$L^{\otimes n}$$ and moreover its inflection weight is equal to 1. In case L is a very general invertible sheaf of degree x on C then all generalized inflection points of L are strongly normal.

The concept of pure gaps of a Weierstrass semigroup at several points of an algebraic curve has been used lately to obtain codes that have a lower bound for the minimum distance which is greater than the Goppa bound. In this work, we show that the existence of total inflection points on a smooth plane curve determines the existence of pure gaps in certain Weierstrass semigroups. We then apply our results to the Hermitian curve and construct codes supported on several points that compare better to one-point codes from that same curve.

this paper we study plane curves of degree d with e total inflection points, for nonzero natural numbers d and e.

We classify all semigroups each of which arises as a Weierstrass semigroup at a pair of inflection points of multiplicities d or d-1 on a smooth plane curve of degree d.

We show that all so-called C-curves are affine images of trochoids or sine curves and use this relation to investigate the occurrence of inflection points, cusps, and loops. The results are summarized in a shape diagram of C-Bezier curves, which is useful when using C-Bezier curves for curve and surface modeling.

We investigate low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve. We compute all quadratic points on these curves and use Coleman’s effective Chabauty method to obtain bounds for the number of cubic points on each of the former two curves.

Shape modeling using planar cubic algebraic curves calls for computing the real inflection points of these curves since inflection points represents important shape feature. A real inflection point is also required for transforming projectively a planar cubic algebraic curve to the normal form, in order to facilitate further analysis of the curve. However, the naive method for computing the inflection points of a planar cubic algebraic curve f = 0 by directly intersecting f = 0 and its Hessian curve H(f) = 0 requires solving a degree nine univariate polynomial equation, and thus is relatively inefficient. In this paper we present an algorithm for computing the real inflection points of a real planar cubic algebraic curve. The algorithm follows Hilbert's solution for computing the inflection points of a cubic algebraic curve in the complex projective plane. Hilbert's solution is based on invariant theory and requires solving only a quartic polynomial equation and several cubic polynomial equations. Through a detailed study with emphasis on the distinction between real and imaginary inflection points, we adapt Hilbert's solution to efficiently compute only the real inflection points of a cubic algebraic curve f = 0, without exhaustive but unnecessary search and root testing. To compute the real inflection points of f = 0, only two cubic polynomial equations need to be solved in our algorithm and it is unnecessary to solve numerically the quartic equation prescribed in Hilbert's solution. In addition, the invariants of f = 0 are used to analyze the singularity of a singular curve, since the number of the real inflection points of f = 0 depends on its singularity type.

A planar algebraic curve C has an implicit equation and a tangential equation. The tangential equation defines a dual curve to C. Starting with a parametrization of C, we find a parametrization of the dual curve, and the tangential equation and implicit equation of C in a novel way. We also find equations whose roots are the parameter values of the cusps and inflection points of C. Methods include polar reciprocation and the theory of envelopes.

The evaluation of points and the computations of inflection points or cusps on a curve are often necessary in CAGD applications. When a curve is represented in a B-spline form, such computations can be made easier once it is transformed into a set of piecewise polynomial curves in power form. The usual practice of the transformation of a B-spline curve into a set of piecewise polynomial curves in power form is done either by a knot refinement followed by basis conversions, or by applying a Taylor expansion on each knot span of a B-spline curve. Presented in this paper is a new algorithm to convert a B-spline curve into a set of piecewise polynomial curves in power form. Experiment shows that the proposed algorithm significantly outperforms the conventional approach when one or more control points of a B-spline curve are continuously moving.

A local shape preserving curve interpolation scheme in R3 which allows local modification of the curve is derived. A piecewise rational quartic Bezier curve which is F3 continuous is used to pass through the given sequence of data points. A formula for the approximation of torsion based on the given data points is also presented.

We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations., 31pages, TeX

We study the relationship between rational points and Galois points for a plane curve over a finite field. It is known that the set of Galois points coincides with that of rational points of the projective plane if the curve is the Hermitian, Klein quartic or Ballico-Hefez curves. We propose a problem: Does the converse hold true? When the curve of genus at most one has a rational point, we will have an affirmative answer., 7 pages

Let q be a power of an odd prime. For arbitrary positive integers h, n, m with n dividing m and arbitrary with ? ? 0 we determine the number of -rational points of the curve in many cases.

We discussed the transition curve of G 2 Hermite interpolation of boundary data that consist of two points, two unit tangent vectors, and two signed curvatures at those points, by using Bezier quartic spline of degree 4. It is shown that the solution of G 2 Hermite interpolation problem can be found by combining S-shape or C-shape transition curve with one or more quartic arcs. In this paper, an S-shaped interpolating curve made of one spiral and a C-shaped interpolating curve made of a curve with only one extreme curvature is constructed. As a result, we obtain a new construction of curvature continuous Bezier spline curves where those transition curves are considered as fair because it comprises with a small number of spirals.

We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincare-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.

(2000). On the equivalence of plane curve singularities. Communications in Algebra: Vol. 28, No. 9, pp. 4389-4401.

We obtain the distribution of inflection points and singularities on a parametric rational cubic curve segment with aid of Mathematica (A System of for Doing Mathematics by Computer). The reciprocal numbers of the magnitudes of the end slopes determine the occurrence of inflection points and singularities on the segment. Its use enables us to check whether the segment has inflection points or a singularity (a loop or a cusp) and to get an idea how to place control vertices and how to choose weights for the rational Bezier cubic curve segment to preserve the fair shape.

In this paper, a new transition curve to be used in the design of the horizontal geometry of routes has been examined. This curve joins two straight lines with a single element. It is an alternative curve to the well-known classical transition curve, which is formed by spiral curve-circular are spiral curve. In order to compare the suggested transition curve with the classical transition curve in terms of the vehicle-road dynamics, the curvature and superelevation functions of both curves are derived, and funcions of the lateral change of acceleration of both curves are obtained and illustrated graphically for the seven selected motion models. The comparisons that take into account the discontinuities in the form of jumps in the graphics of the lateral change of acceleration have shown that the new transition curve is superior to the classical transition curve for all seven motion models. Language: en

Using homogeneous coordinates, a rational curve can be represented in a nonrational form. Based on such a nonrational representation of a curve, a simple method to identify inflection points and cusps on 2-D and 3-D rational curves is proposed.

This paper proposes an algorithm of constructing planar quartic Bezier curve with four shape parameters in discretionary tangent polygon, and curve segments are joined together with G1-continuity (first order geometrical continuity). Bezier curve segments are all shape-preserving to their tangent polygons. In a certain range, every curve segment can be adjusted by controlling parameters of the shape and parameters of control vertex. Finally, examples illustrate that the method given in this paper is effective.

A method to construct transition curves using a family of the quartic Bezier spiral is described. The transition curves discussed are S-shape and C-shape of G2 contact, between two separated circles. A spiral is a curve of monotone increasing or monotone decreasing curvature of one sign. Thus, a spiral cannot have an inflection point or curvature extreme. The family of quartic Bezier spiral form which is introduced has more degrees of freedom and will give a better approximation. It is proved that the methods of constructing transition curves can be simplified by the transformation process and the ratio of two radii has no restriction, which extends the application area, and it gives a family of transition curves that allow more flexible curve designs.

A recent result shows that a general smooth plane quartic can be recovered from its 24 inflection lines and a single inflection point. Nevertheless, the question of whether or not a smooth plane curve of degree at least 4 is determined from its inflection lines is still open. Over a field of characteristic 0, we show that it is possible to reconstruct any smooth plane quartic with at least 8 hyperinflection points by its inflection lines. Our methods apply also in positive characteristic, where we show a similar result, with two exceptions in characteristic 13.