Transcendental Curves Research Articles

A 3D-printable machine for conics and oblique trajectories

We propose an original machine that traces conics and some transcendental curves (oblique trajectories of confocal conics) by the solution of inverse tangent problems. For such a machine, we also provide the 3D-printable model to be used as an intriguing supplement for geometry, calculus, or ordinary differential equations classes.

Overlapping Schwarz methods for isogeometric analysis based on generalized B-splines

Generalized B-splines (GB-splines) are a special class of Tchebycheff B-splines that are smooth piecewise function with sections in more general spaces. GB-splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation. In this paper, we present overlapping Schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on GB-splines. An h-analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes. Numerical results in two- and three-dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters, the domain deformation and the jumping coefficients.

Points of bounded height on oscillatory sets

— We show that transcendental curves in R (not necessarily compact) have few rational points of bounded height provided that the curves are well behaved with respect to algebraic sets and can be parametrized by functions belonging to a specified algebra of infinitely differentiable functions. The method is based on that of Bombieri and Pila, and extends to the non-o-minimal setting some earlier results by others on Pfaffian curves and special functions.

A polynomial Hermite interpolant for [formula omitted] quasi arc-length approximation

Transcendental curves, or in general those resulting from offsetting, do not admit an exact rational or polynomial representation and must hence be approximated in order to incorporate them into most commercial CAD systems. We present a simple, yet general geometric tool for polynomial approximation, based on piecewise Hermite interpolation with C2 quasi arc-length parameterization, a desirable property for robotics or CNC. We take the osculatory Hermite interpolation, prescribing position, tangent direction and curvature at the endpoints, and add quasi arc-length conditions, by imposing unit speed and vanishing tangential acceleration. These new conditions fit naturally into this scheme, yielding a quintic with Bézier points that turn out to display extremely simple geometry. In addition we consider a lower degree alternative to the quintic, namely a cubic B-spline. Finally, we include two examples of applications (the approximations of regular offsets and the clothoid) and compare our results with those from commercial systems or existing methods.

A Geometric Modeling Method Based on TH-Type Uniform B-Splines

A geometric modeling method based on TH-type uniform B-splines which are composed of trigonometric and hyperbolic polynomial with parameters is introduced in this paper. The new splines possess many important properties of quadratic and cubic B-splines. Taking different values of the parameters, one can not only locally adjust the shape of the curves, but also change the type of some segments of a curve between trigonometric and hyperbolic functions as well. The given curves can also interpolate directly control polygon locally by selecting special parameters. Moreover, the introduced splines can represent some quadratic curves and transcendental curves with selecting proper control points and parameters.

The rectification of quadratures as a central foundational problem for the early Leibnizian calculus

Transcendental curves posed a foundational challenge for the early calculus, as they demanded an extension of traditional notions of geometrical rigour and method. One of the main early responses to this challenge was to strive for the reduction of quadratures to rectifications. I analyse the arguments given to justify this enterprise and propose a hypothesis as to their underlying rationale. I then go on to argue that these foundational concerns provided the true motivation for much ostensibly applied work in this period, using Leibniz’s envelope paper of 1694 as a case study.

The return periods and risk assessment of severe dust storms in Inner Mongolia with consideration of the main contributing factors

This study presents a methodology for return period analysis and risk assessment of severe dust storm disaster. Meteorological observation data, soil moisture data, and remote sensing data from 30 meteorological stations in Inner Mongolia (western China) from 1985 to 2006 were used for the study. A composite index of severe dust storm disaster (Index I (SDS)) based on the influence mechanisms of the main contributing factors was developed by using the analytic hierarchy process and the weighted comprehensive method, and the hazard risk curves (i.e., the transcendental probability curves of I (SDS)) for the 30 stations were established using the parameter estimation method. We then analyzed the risk of the occurrence of severe dust storm under different scenarios of 5-, 10-, 20-, and 50-year return periods. The results show that the risk decreased from west to east across Inner Mongolia, and there are four severe dust storm occurrence peak value centers, including Guaizihu, Jilantai, Hailisu, and Zhurihe-Erenhot. The severity of dust storms in seven places will be intolerable in the 50-year return period scenario and in three places in the 20-year return period scenario. These results indicate that these locations should concentrate forces on disaster prevention, monitoring, and early warning. The I (SDS) was developed as an easily understandable tool useful for the assessment and comparison of the relative risk of severe dust storm disasters in different areas. The risk assessment was specifically intended to support local and national government agencies in their management of severe dust storm disasters in their efforts to (1) make resource allocation decisions, (2) make high-level planning decisions, and (3) raise public awareness of severe dust storm risk.

Mildness and the Density of Rational Points on Certain Transcendental Curves

We use a result due to Rolin, Speissegger, and Wilkie to show that definable sets in certain o-minimal structures admit definable parameterizations by mild maps. We then use this parameterization to prove a result on the density of rational points on curves defined by restricted Pfaffian functions.

A class of algebraic–trigonometric blended splines

This paper presents a new kind of algebraic–trigonometric blended spline curve, called x y B curves, generated over the space { 1 , t , sin t , cos t , sin 2 t , sin 3 t , cos 3 t } . The new curves not only inherit most properties of usual cubic B-spline curves in polynomial space, but also enjoy some other advantageous properties for modeling. For given control points, the shape of the new curves can be adjusted by using the parameters x and y . When the control points and the parameters are chosen appropriately, the new curves can represent some conics and transcendental curves. In addition, we present methods of constructing an interpolation x y B -spline curve and an x y B -spline curve which is tangent to the given control polygon. The generation of tensor product surfaces by these new spline curves is straightforward. Many properties of the curves can be easily extended to the surfaces. The new surfaces can exactly represent the rotation surfaces as well as the surfaces with elliptical or circular sections.

From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms

In this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension. In particular, throughout this paper we will focus on the derivation of 6-point interpolating schemes that turn out to be unique in combining vital ingredients like C 2 -continuity, simplicity of definition, ease of implementation, user independency, tension control and ability to reproduce salient trigonometric and transcendental curves.

On local behavior of holomorphic functions along complex submanifolds of ℂ N

In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of ℂ N . As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in ℂ2.

Convexity-preserving interpolation of trigonometric polynomial curves with a shape parameter

In computer aided geometric design (CAGD), it is often needed to produce a convexity-preserving interpolating curve according to the given planar data points. However, most existing pertinent methods cannot generate convexity-preserving interpolating transcendental curves; even constructing convexity-preserving interpolating polynomial curves, it is required to solve a system of equations or recur to a complicated iterative process. The method developed in this paper overcomes the above draw-backs. The basic idea is: first to construct a kind of trigonometric polynomial curves with a shape parameter, and interpolating trigonometric polynomial parametric curves with C2 (or G1) continuity can be automatically generated without having to solve any system of equations or do any iterative computation. Then, the convexity of the constructed curves can be guaranteed by the appropriate value of the shape parameter. Performing the method is easy and fast, and the curvature distribution of the resulting interpolating curves is always well-proportioned. Several numerical examples are shown to substantiate that our algorithm is not only correct but also usable.

AHT Bézier Curves and NUAHT B-Spline Curves

In this paper, we present two new unified mathematics models of conics and polynomial curves, called algebraic hyperbolic trigonometric (AHT) Bezier curves and non-uniform algebraic hyperbolic trigonometric (NUAHT) B-spline curves of order n, which are generated over the space span{sin t, cos t, sinh t, cosh t, 1, t,..., tn-5}, n ≥ 5. The two kinds of curves share most of the properties as those of the Bezier curves and B-spline curves in polynomial space. In particular, they can represent exactly some remarkable transcendental curves such as the helix, the cycloid and the catenary. The subdivision formulae of these new kinds of curves are also given. The generations of the tensor product surfaces are straightforward. Using the new mathematics models, we present the control mesh representations of two classes of minimal surfaces.

A family of quasi-cubic blended splines and applications

A class of quasi-cubic B-spline base functions by trigonometric polynomials are established which inherit properties similar to those of cubic B-spline bases. The corresponding curves with a shape parameter a, defined by the introduced base functions, include the B-spline curves and can approximate the B-spline curves from both sides. The curves can be adjusted easily by using the shape parameter a, where dpi(a,t) is linear with respect to da for the fixed t. With the shape parameter chosen properly, the defined curves can be used to precisely represent straight line segments, parabola segments, circular arcs and some transcendental curves, and the corresponding tensor product surfaces can also represent spherical surfaces, cylindrical surfaces and some transcendental surfaces exactly. By abandoning positive property, this paper proposes a new C2 continuous blended interpolation spline based on piecewise trigonometric polynomials associated with a sequence of local parameters. Illustration showed that the curves and surfaces constructed by the blended spline can be adjusted easily and freely. The blended interpolation spline curves can be shape-preserving with proper local parameters since these local parameters can be considered to be the magnification ratio to the length of tangent vectors at the interpolating points. The idea is extended to produce blended spline surfaces.

Jackson-type theorems on some transcendental curves in [formula omitted

“Efficient curves” in the sense of the best rate of multivariate polynomial approximation to contractive functions on these curves, were first introduced by D.J. Newman and L. Raymon in 1969. They proved that algebraic curves are efficient, but claimed that the exponential curve γ : = { ( t , e t ) : 0 ⩽ t ⩽ 1 } is not. We prove to the contrary that this exponential curve and its generalization to higher dimensions are indeed efficient. We also investigate helical curves in R d and show that they too are efficient. Transcendental curves of the form { ( t , t λ ) : δ ⩽ t ⩽ 1 } are shown to be efficient for δ > 0 , contradicting another claim of Newman and Raymon.

Tangential Markov Inequalities on Transcendental Curves

Abstract. We show that on the curves γ:=(x,et(x)) , x∈ [a,b] , where t(x) is a fixed polynomial, there holds a tangential Markov inequality of exponent four for algebraic polynomials PN(x,y) of degree at most N in each variable x,y: ||(PN(x,et(x)))'||[a,b]≤ CN4||PN||γ , and the exponent four is sharp. On the other hand, the corresponding tangential Markov factors on curves y=xα with irrational α grow exponentially in the degree of the polynomials.

A New Notion of Defect for Holomorphic Curves and Their Sequences

In this paper, we study transcendental curves in ℙn and also infinite sequences of rational curves using methods of the value distribution theory of holomorphic mappings and, in particular, the notions of Nevanlinna and Valiron defects. In contrast to the classical case, we study the defects of points in ℙn rather than of divisors. It is shown how the main notions of the theory should be changed to make them meaningful in this context. Analogs of the main theorems are proved. It is established that the sets of defective points are sparse in the sense of an adequately introduced Hausdorff h-measure.

ALMOST INTEGRABLE MAPPINGS

We analyze birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. We concentrate on 4 × 4 matrices and elementary transpositions of two entries. This analysis brings out six classes of birational transformations. Three classes correspond to integrable mappings, their iteration yielding elliptic curves. Generically, the iterations corresponding to the three other classes are included in higher dimensional non-trivial algebraic varieties. Nevertheless some orbits of the parameter space lie on (transcendental) curves. These transformations act on fifteen (or q2 − 1) variables, however one can associate to them remarkably simple non-linear recurrences bearing on a single variable. The study of these last recurrences gives a complementary understanding of these amazingly regular non-integrable mappings, which could provide interesting tools to analyze weak chaos.

Curvatura integra, handle‐number, and genus of transcendental curves

Communications on Pure and Applied MathematicsVolume 44, Issue 8-9 p. 1057-1066 Article Curvatura integra, handle-number, and genus of transcendental curves† H. P. McKean, H. P. McKean Courant InstituteSearch for more papers by this author H. P. McKean, H. P. McKean Courant InstituteSearch for more papers by this author First published: October ‐ December 1991 https://doi.org/10.1002/cpa.3160440816Citations: 1 † Dedicated to Natascha Brunswick. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume44, Issue8-9October ‐ December 1991Pages 1057-1066 RelatedInformation

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