The Experts below are selected from a list of 663 Experts worldwide ranked by ideXlab platform
Ying Huifen - One of the best experts on this subject based on the ideXlab platform.
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Poisson basis functions and Poisson Curves
Journal of Zhejiang Normal University, 2007Co-Authors: Ying HuifenAbstract:Transcendental Curve is a class of Curves widely used in geometric modeling and industry design.However,they cannot be exactly represented by Bezier Curve and B-spline Curve.There were investigated the properties of Poisson basis functions and the corresponding Poisson Curves,which were similar to Bernstein basis functions and Bezier Curves respectively.Based on the excellent geometric and algebraic properties of Poisson Curves,some common Transcendental Curves were expressed as Poisson-type Curves.
Laura Capuano - One of the best experts on this subject based on the ideXlab platform.
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Some estimates about rational points of bounded height and applications to Diophantine Geometry
2013Co-Authors: Laura CapuanoAbstract:The problem of counting rational points has been studied since a long time. In 2006 Pila and Wilkie succeded in finding good estimates for a big class of sets (definable in an O-minimal structure) using some powerful tools due to logical theory. They estimated the number of rational points of bounded height T that lie in the set but not in its "algebraic part". In his later works Pila generalized the previous theorem showing that similar estimates can be obtained if you eliminate from the set a "smaller part" of the algebraic one, introducing the new notion of "blocks". These tools have lots of applications in Diophantine Geometry; we will see how this method can be used for counting rational points of bouded degree on a Transcendental Curve.
F. Ongay - One of the best experts on this subject based on the ideXlab platform.
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On approximating the distance trisector Curve
Boletín de la Sociedad Matemática Mexicana, 2018Co-Authors: J. Monterde, F. OngayAbstract:Recently we have shown that the distance trisector Curve is a Transcendental Curve. Since from the computational point of view this implies that no closed expression to describe the Curve in algebraic terms can be found, it is still of interest to know how to approximate it efficiently by means of polynomial or rational functions. We discuss here some of the remarkable properties of this Curve, that among other things lead to very good approximations.
J. Monterde - One of the best experts on this subject based on the ideXlab platform.
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On approximating the distance trisector Curve
Boletín de la Sociedad Matemática Mexicana, 2018Co-Authors: J. Monterde, F. OngayAbstract:Recently we have shown that the distance trisector Curve is a Transcendental Curve. Since from the computational point of view this implies that no closed expression to describe the Curve in algebraic terms can be found, it is still of interest to know how to approximate it efficiently by means of polynomial or rational functions. We discuss here some of the remarkable properties of this Curve, that among other things lead to very good approximations.
Marco Frego - One of the best experts on this subject based on the ideXlab platform.
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Path reconstruction using Bezier splines and clusterization
2018Co-Authors: Enrico Bertolazzi, Marco FregoAbstract:A problem often encountered in the inexact sciences is the smoothing of data which is affected by some kind of noise. It is not desirable to directly use the collected data in computer simulations and a treatment must be performed in order to minimize the effect of the inexact sampling phase. There are essentially two approaches to filter and approximate measured data that come from an experiment, the first is straight interpolation, which is feasible if the noise is low enough to be ignored, the second is the smoothing. The difference is that an interpolating spline (or surface) passes through each of the sampled points, while the fitted spline can in general not contain either point of the original data. The advantage of the second method is that a lower degree Curve can be used, whereas for interpolation, usually high degree polynomials are involved.\\In literature there is a huge amount of techniques for both interpolation and fitting, with many subfields for each method: the motivation depends often on the application and the dimension of the problem. Most of the literature is devoted to the approximation of data made of points in the real plane $\mathbb{R}^2$ or in the real space $\mathbb{R}^3$, and only a minority of articles treat the most general case of points in $\mathbb{R}^n$. This is due to the specific properties of Curves in the plane, where the study of the curvature is easier than in the three dimensional space where also the torsion in involved. The general case requires the abstract concept of multi dimensional curvature which does not permit to exploit many simplifications that are possible in the low dimensional cases. Nevertheless there are methods that can be naturally extended from the two dimensional case to higher dimension, whereas there are methods that is not possible to extend from the plane to the three dimensional space (at least without to pay the price of introducing new concepts or new definitions). A typical example of Curve that can be used in all dimensions without any essential change in the method is the \bez{} Curve, at the contrary, the extension to three dimension of a Transcendental Curve, like it could be the clothoid, is not possible without a heavy change of the definition and the introduction, for example, of hyperclothoids. On the other side, (cubic) polynomial splines are difficult to handle if conditions on the curvature are posed, while Transcendental Curves usually permit an easier treatment.The question of the curvature is so important in many applications, that a classification of the methods is performed on the basis of how the curvature and the tangents are considered. In general tangents and curvature are not considered along the whole Curve, but attention is focused only at the joints of the various spline segments that forms the Curve. If the Curve at the joints is only continuous, it is classified as $C^0$, if the first derivative is continuous, is called $C^1$, but if the tangents at the joints are only proportional (that is, it is possible to parametrize the Curve to made it $C^1$), it is called $G^1$, i.e. geometrically differentiable. The same concept of geometric continuity is extended to the case of the curvature ($G^2$ continuity), and in particular cases also the third derivative is considered, $G^3$ case.In this paper a quasi--$G^2$ fitting with cubic Hermite splines is proposed, in order to take advantage of the possibility to fit multidimensional sampled points while trying to approximate the benefit of a Curve with continuous curvature and low degree. The proposed algorithm follows the following steps: we cluster the pointsusing a single cubic and measuring the standard deviation of the fitted data via an almost orthogonal projection of the data on the cubic. To prevent non natural cusps and loops, we introduce a regularization parameter $\lambda$ that multipliesthe ingratl of the square of the second derivative of the cubic. The choice of $\lambda$ is decided minimising the Generalised Cross Validation. We gather theseinformation of the local cubics (one for each cluster of data) to build the globalmatrix of the optimisation problem, which takes into account the contribution of theleast squares distance and the integral of the second derivative. This problem isthen solved via a QR factorisation. As an application example we propose the reconstruction of a Formula 1 circuit track,which is acquired as a list of sampled planar points. The data are first smoothed with theproposed algorithm and then the refined points and tangents are interpolated withclothoid Curves, which are desirable Curves in the automotive field.