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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. more...

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. more...

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. more...

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 more...

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. more...

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 more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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

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 more...

1. The equation of the scroll R of tangents of the curve Γ of intersection of two quadrics in general position in [3] was found by Cayley in 1850 ((l), p. 50). Cayley also considered special positions of the quadrics but we shall not be concerned with these. was reproduced in Salmon's treatise ((8), p. 191) and Salmon, on this same page, provides an equation in covariant form and so applicable to pairs of quadrics whether in canonical form, or in general position, or not. more...

§1. Introduction . In this paper, as in [4], we are concerned with integer points ( m, n ) lying close to the curve in the sense that where ║ t ║ denotes the distance of the real number t from the nearest integer. We shall always suppose that and that F ( x ) is at least twice continuously differentiable. Let R be the number of solutions of (1.1) with m an integer, O ≤ m ≤ L . The obvious method of estimating R uses the row-of-teeth or rounding error function with more...

A new mathematical method is developed for interpolation from a given set of data points in a plane and for fitting a smooth curve to the points. This method is devised in such a way that the resultant curve will pass through the given points and will appear smooth and natural. It is based on a piecewise function composed of a set of polynomials, each of degree three, at most, and applicable to successive intervals of the given points. In this method, the slope of the curve is determined at each given point locally, and each polynomial representing a portion of the curve between a pair of given points is determined by the coordinates of and the slopes at the points. Comparison indicates that the curve obtained by this new method is closer to a manually drawn curve than those drawn by other mathematical methods. more...

1·1. The points, tangents, osculating planes, …, osculating primes of a curve may be said to form a system which is characterised by the number of these elements which are incident with a prime, …, line, point, respectively. For the normal rational quartic curve the system is (4, 6, 6, 4); projection of this system from a general point gives the system (4, 6, 6) in [3]; section by a general prime gives the system (6, 6, 4). These two systems in [3], which are the systems with which we are concerned in this paper, are duals of one another, and will be called systems of the first and second kinds respectively. more...

Summary Most of a typical log-normal curve lies very close to two straight lines, namely the tangents at its points of inflection. This holds good for nearly all the rising part of the curve and for most of the top half beyond the maximum. When loge (t – to) is normally distributed with standard deviation s, whether the whole curve can be observed or not, s, to, the mean, the mode and the area under the whole curve can be derived from the triangle formed by these two tangents and the base line. The triangle also provides a simple test of goodness-of-fit. The corresponding inflection triangles of two other skew distributions are investigated in outline namely a chi2 (or gamma) distribution and one arising in Brownian movement or random walk in one dimension with drift. These are compared with the log-normal one. The normal distribution and its inflection triangle are given as a limiting case of the log-normal one and it is shown that the curve on the cover of Statistica Neerlandica cannot be a normal one. The application is to the extrapolation problem in heart dye-dilution curves, which record the distribution of passage times of an indicator through the lungs and heart. The usual semi-log extrapolation for determining the area under the complete curve is discussed critically; it is shown how the inflection triangle provides a simpler procedure that is probably more accurate. The general formulae, an outline of the application and numerical tables for 0 < s 0.8 are given in the main text; the mathematical derivations of all the basic formulae are in an appendix (sections 11 to 16). The information needed for practical applications without studying the theory is contained in sections 3 to 6 and the associated tables and figures. more...

In this paper we obtain, for a class of plane curves, extensions of the well-known relation of inflection points, double points and bitangencies established by Fabricius-Bjerre for closed curves.

The object of this note is to correct an error in my paper “Extensions of a theorem of Segre's…,” the notation used being the same. The curve C4 dealt with is regarded as given by its canonical representationand at one point in the paper we sought the locus of the lines analogous to the line A2A4 of the figure of reference for each of the ∞2 representations of this type (p. 671, small print). In the space representation of the locus there is an additional principal curveand the order of the locus must be reduced by that of the form corresponding to the points of this conic. The locus sought is in fact none other than the cubic form, locus of chords of C4, the present system of lines being the directrix systemt†. This follows at once from the following results, which can be shown immediately using the above representation:(1) The space joining such a line g to any tangent cuts the curve again in coincident points, and thus contains a second tangent;(2) The line joining the points of contact of these tangents meets g, and the points give the involution more...

Let C denote a smooth, plane, projective curve of degree n>2, defined over 112 or any other algebraically closed field of characterization 0. For a line L let i(L, C) denote the number of points in which L intersects C (counted without multiplicity) and let t(L, C) denote the number of points of C having L as tangent (again, counted without multiplicity) finally, put 2(L, C)=i(L, C)(i(L, C ) l ) ( n 1 ) ( n 2 t ( L , C)). (1) more...