Parametric Equation

We present a finite element approximation of motion by minus the Laplacian of curvature and related flows. The proposed scheme covers both the closed curve case, and the case of curves that are connected via triple junctions. On introducing a parametric finite element approximation, we prove stability bounds and compare our scheme with existing approaches. It turns out that the new scheme has very good properties with respect to area conservation and the equidistribution of mesh points. We state also an extension of our scheme to Willmore flow of curves and discuss possible further generalizations.

Summary In this paper, we develop a new hybrid curve fitting model using fuzzy set and rough set theory. The developed method differs from classical curve fitting techniques and algorithms. For produced rule sets of model to create by using fuzzy logic technique and ...

Algorithms are presented for consoucting parametric equations of rational, implicitly defined quadric and cubic hypersurfaces for dimensions greater than and equal to three. • SlIfIponcd in part by ARO COlllIlltL DAAG29-SS-C00l8, NSF grant DMSS8-16286 and O:NR COIlLrllct NOOOl4-8S•K-04lJ2.

A general technique is presented for automatic generation of B-spline curves in a spatially constrained environment, subject to specified intrinsic shape properties. Spatial constraints are characterized by a distance metric relating points on the curve to polyhedral models of obstacles which the curve should avoid. The shape of the curve is governed by constraints based on intrinsic curve properties such as parametric variation and curvature. To simultaneously address the independent goals of global obstacle avoidance and local control of intrinsic shape properties, curve synthesis is formulated as a combinatorial optimization problem and solved via simulated annealing. Several example applications are presented which demonstrate the robustness of the technique. The synthesis of both uniform and nonuniform B-spline curves is also demonstrated. An extension of the technique to general sculptured surface model synthesis is briefly described, and a preliminary example of simple surface synthesis presented.

In this paper, the popular ®nite forward differencing algorithm for fast evaluation of points on a cubic parametric curve is modi®ed for its application to CNC systems so that the inherent accumulative error in the calculation of one piece of curve will not propagate to the whole composite curve. Using speci®ed tolerances, a computationally non-intensive method of analysis for the computation of step size has been developed. Tolerance, velocity, acceleration and jerk analyses are conducted based on computer simulation results. The interpolation scheme method developed in this paper has been analyzed and proven to be accurate, adaptive to the curvature of a curve, reduces jerk, and can be easily implemented in CNC systems. Finally, the paper explains why the algorithm developed here is natural for cubic parametric curve interpolations.

Parametric splines curves are typically constructed so that the first n parametric derivatives agree where the curve segments abut. This type of continuity condition has become known as C n or n th order parametric continuity. It has previously been shown that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves. We define nthorder geometric continuity (G'~), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity. G '~ continuity provides for the introduction of n quantities known as shape parameters which can be made available to a designer in a computer aided design environment to modify the shape 49 of curves without moving control vertices. Several applications of the theory are discussed, along with topics of future research.

The parametric or geometric continuity of a rational polynomial curve h a s often been obtained by requiring the homogeneous polynomial curve associated with the rational curve to possess parametric or geometric continuity, respectively. Recently this approach has been shown overly restrictive. We m a k e use of the necessary and su cient conditions of rational parametric continuity for de ning basis functions for the homogeneous representation of a rational curve. These functions are represented in terms of shape parameters of rational continuity, which a r e i n troduced due to these exact conditions. The shape parameters may b e v aried globally, a ecting the entire curve, or modi ed locally thereby a ecting only a few segments. Moreover, the local parameters can be represented as continuous or discrete functions. Based on these properties, we i n troduce three classes of basis functions which can be used for the homogeneous representation of rational parametric curves.

Background: Several recent studies have reported significant health effects of air pollution even at low levels of air pollutants, but in most of these studies linear non-threshold relations were assumed. Aims: To investigate the NO 2 mortality dose-response association in nine cities participating in the APHEA-2 project using two different methods: the meta-smooth and the cubic spline method. Methods: The meta-smooth method developed by Schwartz and Zanobetti is based on combining individual city non-parametric smooth curves; the cubic spline method developed within the APHEA-2 project combines individual city estimates of cubic spline shaped dose-response relations. The metasmooth method is easier and faster to implement, but the cubic spline method is more flexible for further investigation of possible heterogeneity in the dose-response curves among cities. Results: In the range of the pollutant common to all cities the two methods gave similar and comparable curves. Using the cubic spline method it was found that smoking prevalence acts as an effect modifier with larger NO 2 effects on mortality at lower smoking prevalence. Conclusions: The NO 2 -mortality association in the cities included in the present analysis, could be adequately estimated using the linear model. However, investigation of the city specific dose-response curves should precede the application of linear models.

In this paper, we present an algorithm for reparametrizing algebraic plane curves from a numerical point of view. That is, we deal with mathematical objects that are assumed to be given approximately. More precisely, given a tolerance ǫ > 0 and a rational parametrization P with perturbed float coefficients of a plane curve C, we present an algorithm that computes a parametrization Q of a new plane curve D such that Q is an ǫ-proper reparametrization of D. In addition, the error bound is carefully discussed and we present a formula that measures the "closeness" between the input curve C and the output curve D.

We present and analyse the Hough transform (HT) to recognise and approximate space curves in digital models, a problem that is not currently addressed by the standard HT. Our method works on meshes and point clouds and applies to models even incomplete or affected by noise, thus being suitable for the analysis of digital models deriving from 3D scans. In our approach we take advantage of a recent HT formulation for algebraic curves to define both parametric and implicit space curve representations. We also provide a comparative analysis of the HT-based method when dealing with both curve representations, discussing the computational performance and the approximation accuracy of both strategies.

We present a technique to fitC2continuous parametric surfaces to scattered geometric data points forming frontiers delimiting physiologic structures in segmented images. Such mathematical representation is interesting because it facilitates a large number of operations in modeling. While the fitting ofC2continuous parametric curves to scattered geometric data points is quite trivial, the fitting ofC2continuous parametric surfaces is not. The difficulty comes from the fact that each scattered data point should be assigned a unique parametric coordinate, and the fit is quite sensitive to their distribution on the parametric plane. We present a new approach where a polygonal (quadrilateral or triangular) surface is extracted from the segmented image. This surface is subsequently projected onto a parametric plane in a manner to ensure a one-to-one mapping. The resulting polygonal mesh is then regularized for area and edge length. Finally, from this point, surface fitting is relatively t...

Based on earlier work by the authors, in this paper we introduce novel fully discrete, fully practical parametric finite element approximations for geometric evolution equations of curves in the plane. The fully implicit approximations are unconditionally stable and intrinsically equidistribute the vertices at each time level. We present iterative solution methods for the systems of nonlinear equations arising at each time level and present several numerical results. The ideas easily generalize to the evolution of curve networks and to anisotropic surface energies.

We construct multi-parametric rational solutions to the KdV equation. For this, we use solutions in terms of exponentials depending on several parameters and take a limit when one of these parameters goes to 0. Here we present degenerate rational solutions and give a result without the presence of a limit as a quotient of polynomials depending on 3N parameters. We give the explicit expressions of some of these rational solutions.

Stabilization by a parametric distributed control function plays a very important role in aeronautics, aerospace and physics. Choosing the right parameters is necessary for handling the distributed control. In the current paper, we introduce stabilization criteria for an n-order functional-differential equation with a parametric distributed control function in n-term integrals and 2n parameter sets. In our article, we use properties of unimodal and log-concave polynomials.

A Deus, criador dos céus e da terra, por mais essa graça alcançada. Obrigado, Senhor! Ao Professor Doutor Franck Bellemain, pelo trabalho de orientação desenvolvido com dedicação, empenho e amizade. Às Professoras Doutoras Verônica Gitirana e Paula Bautar, pelas sugestões, comentários e críticas que tanto contribuíram para a elaboração e evolução desta tese. Aos Professores: Doutora Jana Trgalová, Doutor Vincenzo Bongiovanni e Doutor Vladimir Andrade, que muito nos honraram em aceitar o convite para a participação da banca de defesa. Aos professores do Programa de Pós-Graduação em Educação Matemática e Tecnológica da UFPE e do Departamento de Expressão Gráfica da UFPE, por todo o incentivo dado durante o curso. Aos estudantes José Barbosa, Jorge Alexandre e Igor Melo, que contribuíram no processo de prototipação do Conics Studium 3D. Aos colegas professores que participaram da fase experimental. Aos amigos do Doutorado, pelo companheirismo e sugestões e, em especial, aos integrantes do LEMATEC (Laboratório de Ensino da Matemática e Tecnologia), pelas contribuições fornecidas durante as discussões nos seminários. Aos meus pais, Mariluce Beserra e José de Almeida (In memorian), irmãos, e demais familiares, os quais sempre forneceram apoio e estímulo em todas as fases acadêmicas. Em especial a minha querida esposa, Nívia Mendes, que em todos os momentos me incentivou, demonstrando compreensão, carinho e respeito pelo trabalho. Ao querido filho José Guilherme, um anjo que Deus colocou em nossas vidas. Ele nos ensina a amar, a dividir, a somar, tornando nossos dias melhores, alegres e felizes. RESUMO As cônicas foram objeto de investigação de vários estudiosos ao longo dos séculos, passando pelas geometrias sintética (do grego), analítica e projetiva. No entanto, o currículo para as curvas cônicas na Educação Básica foi reduzido no Brasil, na França, Espanha e México, por exemplo e embora continue presente em diversos cursos universitários, sua abordagem ocorre de maneira fragmenta, com ênfase nas representações algébricas. Com o propósito de enfrentar as dificuldades presentes no ensino das curvas cônicas, utilizamos a engenharia didático-informática para estruturarmos o estudo considerando as dimensões epistemológica, cognitiva, didática e informática. Foi apontada a necessidade de privilegiar uma proposta integrativa das abordagens sintética e analítica, em oposição à fragmentação, com articulação entre as várias representações, relacionando as variáveis visuais às unidades simbólicas correspondentes, por meio das transformações por conversão e tratamento, fazendo uso de um recurso computacional. Para atender a esses requisitos, não podemos nos concentrar apenas no campo matemático ou em sua história. Uma abordagem cognitiva é necessária para especificar a articulação dinâmica de várias representações das curvas cônicas por meio de recursos computacionais, para que o ensino possa contribuir para o desenvolvimento global das habilidades de raciocínio, análise e visualização de estudantes e professores. Desse modo, desenvolvemos o protótipo do Conics Studium 3D, oferecendo a possibilidade de exploração do que denominamos de "registros semióticos dinâmicos", isto é, de articular de maneira dinâmica suas equações (e suas várias formas entre sitransformação por tratamento), sua representação gráfica cartesiana e a sua representação figural espacial, de modo que, ao realizar deslocamentos na representação figural espacial, obtenhamos modificação no gráfico cartesiano e nas expressões algébricas, permitindo o acesso rápido e contínuo a várias situações (Transformação por conversão).

This article presents a careful comparative evaluation of two techniques for numerical curvature estimation of 2D closed contours (more specifically closed, regular and simple parametric curves). The considered methods are: (a) a 1-D Fourier-based approach; and (b) a 2-D Fourier-based approach involving the embedding of the contour into a 2-D regular surface (presented for the first time in this article). Both these techniques employ Gaussian smoothing as a regularizing condition in order to estimate the first and second derivatives needed for curvature estimation. These methods are considered according to a multiresolution approach, where the standard deviation of the Gaussians are used as scale parameters. The methods are applied to a standard set of curves whose analytical curvatures are known in order to estimate and compare the errors of the numerical approaches. Three kinds of parametric curves are considered: (i) curves with analytical description; (ii) curves synthesized in terms of Fourier components of curvature; and (iii) curves obtained by splines. A precise comparison methodology is devised which includes the adoption of a common spatial quantization approach (namely square box quantization) and the explicit consideration of the influence of the related smoothing parameters. The obtained results indicate that the 1-D approach is not only faster, but also more accurate. However, the 2-D approach is still interesting and reasonably accurate for applications in situations where the curvature along the whole 2-D domains is needed.

In this paper, the popular ®nite forward differencing algorithm for fast evaluation of points on a cubic parametric curve is modi®ed for its application to CNC systems so that the inherent accumulative error in the calculation of one piece of curve will not propagate to the whole composite curve. Using speci®ed tolerances, a computationally non-intensive method of analysis for the computation of step size has been developed. Tolerance, velocity, acceleration and jerk analyses are conducted based on computer simulation results. The interpolation scheme method developed in this paper has been analyzed and proven to be accurate, adaptive to the curvature of a curve, reduces jerk, and can be easily implemented in CNC systems. Finally, the paper explains why the algorithm developed here is natural for cubic parametric curve interpolations.

Stabilization by a parametric distributed control function plays a very important role in aeronautics, aerospace and physics. Choosing the right parameters is necessary for handling the distributed control. In the current paper, we introduce stabilization criteria for an n-order functional-differential equation with a parametric distributed control function in n-term integrals and 2n parameter sets. In our article, we use properties of unimodal and log-concave polynomials.

The development of Open Architecture (PC-based) Numerical Control (OAPC-NC) Interpolator has enhanced the possibilities of data communication between the systems. In specialty domain, such as design, manufacturing, and machining, the STEP AP238 (STEP-NC) is developed. A STEP-NC interface to OAPC-NC interpolator using STEP-NC tool path data is proposed and developed. A hierarchical-based algorithm is used to extract the tool path data from STEP-NC tool path file of a product model. The generated output of the interpolated data is computer simulated for 2D circular-arc paths and 3D linear paths to verify the validity of the interpolator input generated by the proposed interface.

In this paper, we study the position vectors of rectifying slant helices in $E^3$. First, we have found the general equations of the curvature and the torsion of rectifying slant helices. After that, we have constructed a second order linear differential equation and by solving the equation, we have obtained a family of rectifying slant helices which lie on cones.

In path planning design, potential fields can introduce force constraints to ensure curvature continuity of trajectories and thus facilitate path-tracking design. The parametric thrift of fractional potentials permits smooth variations of the potential in function of the distance to obstacles without requiring design of geometric charge distribution. In the approach we use, the fractional order of differentiation is the risk coefficient associated to obstacles. A convex danger map towards a target and a convex geodesic distance map are defined. Real-time computation can also lead to the shortest minimum danger trajectory, or to the least dangerous of minimum length trajectories.

An epitoniid specimen under the genus Cycloscala is mathematically modeled using parametric equations which allow comparison of growth functions and parameter values with other specimens of the same genus. This mathematical modeling approach may supplement the currently used genetic and microscopy methods in the taxonomic classification of species.

In this article, we justify the use of parametric planar Pythagorean Hodograph spline curves in path planning. The elegant properties of such splines enable us to design an efficient interpolator algorithm, more precise than the classical Taylor interpolators and faster than an interpolator based on arc length computations.

In this paper, a new form of 2D-plane curves is produced and graphically studied. The name of my daughter "Noor" has been given to this curve; therefore, Noor term describes this curve whenever it is used in this paper. This curve is a form of these opened curves as it extends in the infinity along both sides from the origin point. The curve is designed by a circle/ ellipse which are drawing curvatures that tangent at the origin point, where its circumference is passed through the (0,2a). By sharing two vertical lined points of both the circle diameter and the major axis of the ellipse, the parametric equation is derived. In this paper, a set of various cases of Noor curve are graphically studied by two curvature cases; a circle and an ellipse, and all figures and obtained rigour measurements are checked by AutoCAD program. With its simple, symmetric form, the future predictions are tuned for the Noor's curve to be usefully engaged in important practical applications. ...

Modalidad: Taller (TA) Nivel educativo: Formación y actualización docente Palabras clave: GeoGebra, GeoGebra Book Resumo Esta oficina tem como proposta orientar os participantes para a criação de um GeoGebra Book ou para a utilização de um GeoGebra Book já construído e disponível, por exemplo, em http://tube.geogebra.org/student/bI57WCUya Nos últimos anos, o software GeoGebra tem significado uma importante revolução,

Modalidad: Taller (TA) Nivel educativo: Formación y actualización docente Palabras clave: GeoGebra, GeoGebra Book Resumo Esta oficina tem como proposta orientar os participantes para a criação de um GeoGebra Book ou para a utilização de um GeoGebra Book já construído e disponível, por exemplo, em http://tube.geogebra.org/student/bI57WCUya Nos últimos anos, o software GeoGebra tem significado uma importante revolução,

We present an algorithm for computing a minimal set of generators for the ideal of a rational parametric projective curve in polynomial time. The method exploits the availability of polynomial algorithms for the computation of minimal generators of an ideal of points and is an alternative to the existing Gröbner bases techniques for the implicitization of curves. The termination criterion is based on the Castelnuovo-Mumford regularity of a curve. The described computation also yields the Hilbert function and, hence, the Hilbert polynomial and the Poincaré series of the curves. Moreover, it can be applied to unions of rational curves. We have compared the implementation of our algorithm with the Hilbert driven elimination algorithm included in CoCoA 3.6 and Singular 1.2, obtaining, in general, significant improvements in timings.

The student who access the University understands a curve as a function represented by an equality involving the Cartesian variables x and y. But he does not know another great set of curves such as those described by parametric equations and polar coordinates. We show an experience developed in two consecutive courses in the first course of a bachelor's degree, which allows the students to discover and create many curves in parametric or polar representation, working in group. Also the experience facilitate the students to achieve several generic skills.

Subdivision schemes (SSs) have been the heart of computer-aided geometric design almost from its origin, and several unifications of SSs have been established. SSs are commonly used in computer graphics, and several ways were discovered to connect smooth curves/surfaces generated by SSs to applied geometry. To construct the link between nonstationary SSs and applied geometry, in this paper, we unify the interpolating nonstationary subdivision scheme (INSS) with a tension control parameter, which is considered as a generalization of 4-point binary nonstationary SSs. The proposed scheme produces a limit surface having $C^{1}$ C 1 smoothness. It generates circular images, spirals, or parts of conics, which are important requirements for practical applications in computer graphics and geometric modeling. We also establish the rules for arbitrary topology for extraordinary vertices (valence ≥3). The well-known subdivision Kobbelt scheme (Kobbelt in Comput. Graph. Forum 15(3):409–420, 199...

This paper proposes a new technique based on double parametric form of fuzzy numbers to handle the uncertain vibration equation for very large membrane for different particular cases. Uncertainties present in the initial condition and the wave velocity of free vibration are modelled through Gaussian convex normalised fuzzy set. Using the single parametric form of fuzzy number, the original fuzzy vibration equation is converted first to an interval fuzzy vibration equation. Next this equation is transformed to crisp form by applying double parametric form of fuzzy numbers. Finally the same governing equation is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds. The present methods are very simple and effective. Obtained results are depicted in terms of plots to show the efficiency and powerfulness of the present analysis. Results obtained by the methods with new techniques are compared with existing results in special cases.

A novel state-space realization for parametric macromodeling is proposed in this paper. A judicious choice of the state-space realization is required to account for the generally assumed smoothness of the state-space matrices with respect to the design parameters. This is used in combination with suitable interpolation schemes to interpolate a set of state-space matrices, and hence the poles and residues indirectly, in order to build accurate parametric macromodels. The key points are the choice of a proper pivot matrix and the solution of a Sylvester equation for pole placement. Pertinent numerical examples validate the proposed state-space realization for parametric macromodeling.

A judicious choice of the state-space realization is required in order to account for the assumed smoothness of the state-space matrices with respect to the design parameters. The direct parameterization of poles and residues may be not appropriate, due to their possible nonsmooth behavior with respect to design parameters. This is avoided in the proposed technique, by converting the a pole-residue description to a Sylvester description which is computed for each root macromodel.This technique is used in combination with suitable parameterizing schemes for interpolating a set of state-space matrices, and hence the poles and residues indirectly, in order to build accurate parametric macromodels. The key features of the present approach are first the choice of a proper pivot matrix and second, finding a well-conditioned solution of a Sylvester equation. Stability and passivity are guaranteed by construction over the design space of interest. Pertinent numerical examples validate the proposed Sylvester technique for parametric macromodeling.

This paper proposes trajectory planning strategies for online reconfiguration of a multi-agent formation on a Lissajous curve. In our earlier work [2], a multi-agent formation with constant parametric speed was proposed in order to address multiple objectives such as repeated collision-free surveillance and guaranteed sensor coverage of the area with ability for rogue target detection and trapping. This work addresses the issue of formation reconfiguration within this context. In particular, smooth parametric trajectories are designed for the purpose using calculus of variations. These trajectories have been employed in conjunction with a simple local cooperation scheme so as to achieve collision-free reconfiguration between different Lissajous curves. A detailed theoretical analysis of the proposed scheme is provided. These surveillance and reconfiguration strategies have also been validated through simulations in MATLAB ® for agents performing parametric motion along the curves, and by Software-In-The-Loop simulation for quadrotors. In addition, they are validated experimentally with a team of quadrotors flying in a motion capture environment.

This paper proposes trajectory planning strategies for online reconfiguration of a multi-agent formation on a Lissajous curve. In our earlier work [2], a multi-agent formation with constant parametric speed was proposed in order to address multiple objectives such as repeated collision-free surveillance and guaranteed sensor coverage of the area with ability for rogue target detection and trapping. This work addresses the issue of formation reconfiguration within this context. In particular, smooth parametric trajectories are designed for the purpose using calculus of variations. These trajectories have been employed in conjunction with a simple local cooperation scheme so as to achieve collision-free reconfiguration between different Lissajous curves. A detailed theoretical analysis of the proposed scheme is provided. These surveillance and reconfiguration strategies have also been validated through simulations in MATLAB ® for agents performing parametric motion along the curves, and by Software-In-The-Loop simulation for quadrotors. In addition, they are validated experimentally with a team of quadrotors flying in a motion capture environment.

We develop an algorithm to detect geometric collisions between pairs of time-dependent parametric surfaces. The algorithm works on surfaces that are continuous and have bounded derivatives, and includes objects that move or deform as a function of time. The algorithm numerically solves for the parametric values corresponding to coincident points and near-misses between the surfaces of two parametric functions.Upper bounds on the parametric derivatives make it possible to guarantee the successful detection of collisions and near-misses; we describe a method to find the derivative bounds for many surface types. To compute collisions between new types of surfaces, the mathematical collision analysis is needed only once per surface type, rather than analyzing for each pair of surface types.The algorithm is hierarchical, first finding potential collisions over large volumes, and then refining the solution to smaller volumes. The user may specify the desired accuracy of the solution. A C-...

Los estudios histórico-epistemológicos en torno al análisis algebraico que Viète y Descartes han profundizado poco tanto en el tipo de problemas que ambos matemáticos resolvieron, en su intento por renovar el análisis geométrico griego, como en las implicaciones de estos en la creación de la ecuación paramétrica. Por esta razón, en este trabajo reportamos un estudio histórico-epistemológico con sustento socioepistemológico cuyo foco de atención se centra en los aspectos epistemológicos que permearon la construcción de la ecuación paramétrica en Viète y Descartes. Los resultados arrojan consideraciones epistemológicas relativas al trabajo con problemas geométricos de cierta complejidad, lo cual permite vislumbrar una vía didáctica hasta ahora no explorada que pudiera explotar el potencial de la actividad algebraica centrada en
el uso de parámetros e incógnitas con la intención de construir un álgebra para la geometría.

The problem of perfect cuboids is one of the old unsolved problems in number theory. By means of various methods it can be reduced to finding a solution of some single Diophantine equation of high degree obeying certain restrictions in the form of inequalities. Each such Diophantine equation is called a characteristic equation of a perfect cuboid. In this paper we present the results obtained by applying asymptotic metods to one of the characteristic equations of a perfect cuboid in the case of the second cuboid conjecture. This results shrink the domain of the integer parameters of the considered characteristic equation and thus make more effective the computer search of perfect cuboids based on this equation.

Purpose – Curve fitting from unordered noisy point samples is needed for surface reconstruction in many applications. In the literature, several approaches have been proposed to solve this problem. However, previous works lack formal characterization of the curve fitting problem and assessment on the effect of several parameters (i.e. scalars that remain constant in the optimization problem), such as control points number (m), curve degree (b), knot vector composition (U), norm degree (k), and point sample size (r) on the optimized curve reconstruction measured by a penalty function (f). The paper aims to discuss these issues. Design/methodology/approach – A numerical sensitivity analysis of the effect of m, b, k and r on f and a characterization of the fitting procedure from the mathematical viewpoint are performed. Also, the spectral (frequency) analysis of the derivative of the angle of the fitted curve with respect to u as a means to detect spurious curls and peaks is explored. ...

Este trabalho tem como objetivo analisar diferentes documentos que forneçam informações sobre o currículo da disciplina de Geometria Analítica em cursos de licenciatura em Matemática, ofertados por Instituições de Ensino Superior públicas no estado do Ceará, observando se a noção de parábola é contemplada no estudo desta disciplina e de que forma ocorre sua abordagem com os graduandos. Para tal, a metodologia adotada ao estruturar este trabalho consiste na primeira fase da Engenharia Didática, que é a análise preliminar. Esta primeira análise foi documental, realizada a partir de ementários da disciplina e de programas curriculares dos cursos de licenciatura em Matemática. Os documentos analisados foram disponibilizados por sete instituições públicas, sendo seis Universidades e um Instituto Federal. Neste estudo foram observados aspectos como objetivos da disciplina, forma de avaliação, procedimentos de ensino, bibliografia utilizada, entre outras informações consideradas relevantes, quando disponíveis, almejando compreender as lacunas durante a formação inicial dos professores de matemática no referido tema. Os resultados apontam para a demanda de materiais de apoio direcionados a uma abordagem da parábola que vincule sua aplicação tanto no contexto da Geometria Analítica quanto das funções, bem como para a necessidade de manter a conexão destes tópicos de estudo no percurso entre a educação básica e o ensino superior. Palavras-chave: parábolas; geometria analítica; formação inicial de professores; ensino de matemática. Preliminary analysis in the study of conics: the case of the parabola in the context of mathematics undergraduate courses in Ceará ABSTRACT. This work aims to analyze different documents that provide information about the curriculum of the discipline of Analytical Geometry in Mathematics Licentiate courses, offered by public Higher Education Institutions in the state of Ceara, observing whether the notion of parabola is contemplated in the study of this discipline and how it is approached with undergraduates. To this end, the methodology adopted when structuring this work consists of the first phase of Didactic Engineering, which is the preliminary analysis. This first analysis was documentary, carried out from the syllabus of the discipline and curricular programs of the degree courses in Mathematics. The documents analyzed were made available by seven public institutions, six of which are Universities and one Federal Institute. In this study, aspects such as objectives of the discipline, form of evaluation, teaching procedures, bibliography used, among other information considered relevant, when available, were observed aiming to understand the gaps during the initial training of mathematics teachers in that subject. The results point to the demand for support materials aimed at an approach to the parabola that links its application both in the context of Analytical Geometry and the functions, as well as for the need to maintain the connection between these topics of study in the path between basic education and higher education.

Bu çalışmada, Lorentzian 3-uzayında ekseni Killing vektör alanı olan null Cartan helisler araştırılmıştır. Bu uzayda sabit Killing ekseninin spacelike, timelike ve null (lightlike) olma durumları göz önünde bulundurularak helis eğrileri türetilmiştir. Daha sonra, bu eğrilerin Bishop eğrilikleri ve parametrik denklemleri elde edilmiştir. Son olarak, çeşitli örnekler verilmiş ve bu örnekler Mathematica programı yardımıyla görselleştirilmiştir.

Bu çalışmada, Lorentzian 3-uzayında ekseni Killing vektör alanı olan null Cartan helisler araştırılmıştır. Bu uzayda sabit Killing ekseninin spacelike, timelike ve null (lightlike) olma durumları göz önünde bulundurularak helis eğrileri türetilmiştir. Daha sonra, bu eğrilerin Bishop eğrilikleri ve parametrik denklemleri elde edilmiştir. Son olarak, çeşitli örnekler verilmiş ve bu örnekler Mathematica programı yardımıyla görselleştirilmiştir.

A Deus, criador dos céus e da terra, por mais essa graça alcançada. Obrigado, Senhor! Ao Professor Doutor Franck Bellemain, pelo trabalho de orientação desenvolvido com dedicação, empenho e amizade. Às Professoras Doutoras Verônica Gitirana e Paula Bautar, pelas sugestões, comentários e críticas que tanto contribuíram para a elaboração e evolução desta tese. Aos Professores: Doutora Jana Trgalová, Doutor Vincenzo Bongiovanni e Doutor Vladimir Andrade, que muito nos honraram em aceitar o convite para a participação da banca de defesa. Aos professores do Programa de Pós-Graduação em Educação Matemática e Tecnológica da UFPE e do Departamento de Expressão Gráfica da UFPE, por todo o incentivo dado durante o curso. Aos estudantes José Barbosa, Jorge Alexandre e Igor Melo, que contribuíram no processo de prototipação do Conics Studium 3D. Aos colegas professores que participaram da fase experimental. Aos amigos do Doutorado, pelo companheirismo e sugestões e, em especial, aos integrantes do LEMATEC (Laboratório de Ensino da Matemática e Tecnologia), pelas contribuições fornecidas durante as discussões nos seminários. Aos meus pais, Mariluce Beserra e José de Almeida (In memorian), irmãos, e demais familiares, os quais sempre forneceram apoio e estímulo em todas as fases acadêmicas. Em especial a minha querida esposa, Nívia Mendes, que em todos os momentos me incentivou, demonstrando compreensão, carinho e respeito pelo trabalho. Ao querido filho José Guilherme, um anjo que Deus colocou em nossas vidas. Ele nos ensina a amar, a dividir, a somar, tornando nossos dias melhores, alegres e felizes. RESUMO As cônicas foram objeto de investigação de vários estudiosos ao longo dos séculos, passando pelas geometrias sintética (do grego), analítica e projetiva. No entanto, o currículo para as curvas cônicas na Educação Básica foi reduzido no Brasil, na França, Espanha e México, por exemplo e embora continue presente em diversos cursos universitários, sua abordagem ocorre de maneira fragmenta, com ênfase nas representações algébricas. Com o propósito de enfrentar as dificuldades presentes no ensino das curvas cônicas, utilizamos a engenharia didático-informática para estruturarmos o estudo considerando as dimensões epistemológica, cognitiva, didática e informática. Foi apontada a necessidade de privilegiar uma proposta integrativa das abordagens sintética e analítica, em oposição à fragmentação, com articulação entre as várias representações, relacionando as variáveis visuais às unidades simbólicas correspondentes, por meio das transformações por conversão e tratamento, fazendo uso de um recurso computacional. Para atender a esses requisitos, não podemos nos concentrar apenas no campo matemático ou em sua história. Uma abordagem cognitiva é necessária para especificar a articulação dinâmica de várias representações das curvas cônicas por meio de recursos computacionais, para que o ensino possa contribuir para o desenvolvimento global das habilidades de raciocínio, análise e visualização de estudantes e professores. Desse modo, desenvolvemos o protótipo do Conics Studium 3D, oferecendo a possibilidade de exploração do que denominamos de "registros semióticos dinâmicos", isto é, de articular de maneira dinâmica suas equações (e suas várias formas entre sitransformação por tratamento), sua representação gráfica cartesiana e a sua representação figural espacial, de modo que, ao realizar deslocamentos na representação figural espacial, obtenhamos modificação no gráfico cartesiano e nas expressões algébricas, permitindo o acesso rápido e contínuo a várias situações (Transformação por conversão).

The study of arrangements of geometric objects is a problem of great interest in combinatorial and computational geometry, and in CAGD, with applications in computer-aided manufacturing, geometric modeling, robotics, NC-machining, molecular modeling, etc. [2, 3] We present a new algebraic method for computing the topology of the arrangement of a finite family of planar algebraic curves. The curves are given either by their implicit equations or by polynomial parametrizations. As far as we know, none of the previous works on this problem include the treatment of parametrized curves. The input of the method is assumed to be given as Lagrange data, i.e., by the values of the polynomials defining the curves, at a suitable set of nodes. The output is a graph representing critical points of the curves, all the intersection points between curves, and segments joining some of these points isotopic to the arcs of the curves. The method follows a sweep-line algorithm. To find critical points, intersections and self-intersections we use Bezout matrices. We take advantage of the existence of a basis of the nullspace of a Bezout matrix that has a nice and useful structure (see [4]). We use algebra by values techniques, which means that all computations (Bezout matrices, polynomial derivatives, etc.) are done in the Lagrange basis [5, 1]. This approach reduces the computational cost of manipulating high degree polynomials, and avoids conversion between different basis, which is known to be a numerically unstable procedure. To compute the roots of the determinant of a Bezout matrix we apply linearization, i.e., we use a straightforward procedure to build (in the Lagrange basis) a companion matrix pencil whose generalized eigenvalues are the roots of the aforementioned determinant. The algorithm has been tested using Maple 17, showing that it is accurate and quite efficient.