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Este documento cómo principal objetivo pretende disminuir la accidentalidad vial, junto con la cantidad de pérdidas, tanto económicas como humanas por causa de la perdida de carril de conductores debido a desviación involuntaria por diferentes causas como el microsueño u otras causas como el uso de dispositivos electrónicos en autopistas con rectas de más de 1.5 km en la doble calzada Buga-Tulua-La Paila-La Victoria. Para esto se pretende la promoción del uso de tecnologías pasivas, como las bandas alertadoras longitudinales cómo solución a la problemática planteada. Para esto se revisa legislación en el país, diferentes tipologías para determinar su idoneidad, calificando su visibilidad, sonoridad, vibración, durabilidad y costo inicial. En esta búsqueda también se tienen en cuenta las condiciones de la vía y el uso de los diferentes actores viales para que esta solución llegue a aportar ideas que se puedan llegar a poner en funcionamiento esta solución que se encuentra autorizado en nuestro manual de señalización vial vigente, pero que no se utiliza al no tener mercado actualmente. This document, as its main objective, aims to reduce road accidents, along with the amount of losses, both economic and human, due to drivers losing lane due to involuntary deviation due to different causes such as microsleep or other causes such as the use of electronic devices. on highways with straight lines of more than 1.5 km on the Buga-Tulua-La Paila-La Victoria dual carriageway. For this, the promotion of the use of passive technologies is intended, such as longitudinal warning bands as a solution to the problem raised. For this, legislation in the country is reviewed, different typologies to determine their suitability, qualifying their visibility, sound, vibration, durability and initial cost. In this search, the conditions of the road and the use of the different road actors are also taken into account so that this solution comes to contribute ideas that can be put into operation this solution that is authorized in our current road signaling manual, but which is not used as it currently does not have a market. Profesional en Construcción en Arquitectura e Ingeniería Pregrado

The purpose of this study is to design a reliable methodology that facilitates the learning and calculation of the derivative used in graph construction, in the determination of the tangent and normal lines to a curve and its slopes, in the calculation of velocity and acceleration. snapshots, and problems of optimization in the various fields of science, with which, it is expected, the perception and learning of the engineering student will be increased and, consequently, his academic performance will rise. All this, in view of the fact that the general intelligence factor, on which this performance depends largely, is innate and usually does not increase substantially with the development of the individual. For the current progress of this research, two algebraic assistants extracted from the internet and a program created in Excel have been used, through algorithms previously designed for this purpose. In this sense, the use of important texts outlined in the References of this article was used, concluding, finally, that the methodology created in this way, became a significant improvement in student performance. El presente estudio tiene por objetivo diseñar una metodología confiable que facilite el aprendizaje y el cálculo de la derivada usada en la construcción de gráficos, en la determinación de las rectas tangente y normal a una curva y sus pendientes, en el cálculo de velocidad y aceleración instantáneas, y en problemas de optimización en los diversos campos de las ciencias, con lo cual, se espera, se incremente la percepción y el aprendizaje del alumno de ingeniería y, se eleve, en consecuencia, su rendimiento académico. Todo ello, en vista de que el factor inteligencia general, del cual depende en gran medida dicho rendimiento, es innato y no suele incrementarse sustancialmente con el desarrollo del individuo. Para el desarrollo del trabajo se han utilizado dos asistentes algebraicos extraídos de internet y un programa creado en Excel, a través de algoritmos previamente diseñados para este efecto. En tal sentido se hizo uso de textos de importancia reseñados en las referencias del presente artículo, concluyendo, finalmente, que la metodología así creada, devino en la mejora significativa del rendimiento de los alumnos.

Graphology-based handwriting analysis to identify human behavior, irrespective of applications, is interesting. Unlike existing methods that use characters, words and sentences for behavioural analysis with human intervention, we propose an automatic method by analysing a few handwritten English lowercase characters from a to z to identify person behaviours. The proposed method extracts structural features, such as loops, slants, cursive, straight lines, stroke thickness, contour shapes, aspect ratio and other geometrical properties, from different zones of isolated character images to derive the hypothesis based on a dictionary of Graphological rules. The derived hypothesis has the ability to categorise the personal, positive, and negative social aspects of an individual. To evaluate the proposed method, an automatic system is developed which accepts characters from a to z written by different individuals across different genders and age groups. This automatic privacy projected system is available on the website (http://subha.pythonanywhere.com). For quantitative evaluation of the proposed method, several people are requested to use the system to check their characteristics with the system automatic response based on his/her handwriting by choosing to agree or disagree options. The automatic system receives 5300 responses from the users, for which, the proposed method achieves 86.70% accuracy.

Submitted by Ana Breda (ambreda@ua.pt) on 2019-09-20T10:29:16Z No. of bitstreams: 1 Editorial.dvi - Breda-DosSantos2019_Article_SphericalTilingWithGeoGebra.pdf: 1676219 bytes, checksum: e33a8a12e98133e5cb8ccc118a33fe5b (MD5) Approved for entry into archive by Rita Gonçalves (ritaisabel@ua.pt) on 2019-09-20T11:05:19Z (GMT) No. of bitstreams: 1 Editorial.dvi - Breda-DosSantos2019_Article_SphericalTilingWithGeoGebra.pdf: 1676219 bytes, checksum: e33a8a12e98133e5cb8ccc118a33fe5b (MD5) Made available in DSpace on 2019-09-20T11:05:20Z (GMT). No. of bitstreams: 1 Editorial.dvi - Breda-DosSantos2019_Article_SphericalTilingWithGeoGebra.pdf: 1676219 bytes, checksum: e33a8a12e98133e5cb8ccc118a33fe5b (MD5) Previous issue date: 2019-08 This work was supported in part by the Portuguese Foundation for Science and Technology (FCT - Fundacäo para a Ciência e a Tecnologia), through CIDMA - Centre for Research and Development in Mathematics and Applications, within project UID/-MAT/04106/2019. published

Neste trabalho é proposto um método automático para o controle da acurácia relativa de dados LiDAR (Light Detection and Ranging) aerotransportados usando superfícies planas, linhas retas e pontos. Primeiramente, é realizada uma filtragem dos dados LiDAR, através do filtro morfológico progressivo e o filtro RANSAC (Random Sample Consensus). Em seguida, as superfícies planas são extraídas automaticamente sobre os telhados encontrados nas faixas adjacentes através do algoritmos crescimento de região. Posteriormente, as correspondências entre as feições extraídas em cada faixa são estabelecidas e os parâmetros de transformação são estimados através de uma variante do método ICP. Finalmente, para avaliar simultaneamente a acurácia plani-altimétrica dos dados LiDAR são empregadas linhas retas extraídas a partir da interseção de planos adjacentes. Para avaliar o método proposto foi conduzido um experimento com dados LiDAR obtidos numa região urbanizada recoberta por 2 faixas com densidade de 1 ponto/m². A avaliação da acurácia relativa desses dados é realizada automaticamente e os resultados obtidos com o método proposto mostraram que a média das distâncias ponto-a-plano é de -0,0019 cm e a raiz quadrada do erro médio quadrático é de 6,04 cm. In this paper an automatic method for assessment of relative accuracy of airborne LiDAR (Light Detection and Ranging) data using surface planes, straight lines and points is presented. Firstly, the LiDAR data is filtered using the progressive morphological filter and RANSAC (Random Sample Consensus) filter. Then, the roof planes are automatically extracted on both strips by region growing algorithm. After, the feature correspondences are established and the transformation parameters are estimated using a variant of ICP algorithm. Finally, in order to evaluate simultaneously planimetric and altimetric relative accuracy, straight lines obtained by intersection of two adjacent surface planes are used. An experiment was conducted to evaluate the proposed method using two overlapping LiDAR strips, whose density is 1 point / m². The assessment of relative accuracy of these data was performed automatically and the results obtained by proposed method, showing values equal -0,0019 cm to mean of the point-to-plane distance and a rmse around 6,04 cm

This work was partially supported by the Turkish Academy of Sciences. In this work, treating the arteries as a thin walled, prestressed elastic tube and the blood as an incompressible viscous fluid of variable viscosity, we have studied the interactions of two nonlinear waves, in the long wave approximation, through the use of extended PLK perturbation method, and the evolution equations are shown to be the Korteweg-deVries-Burgers equation. The results show that, Up to O(is an element of(3/2)), the head-on-collision of two nonlinear progressive waves is elastic and the nonlinear progressive waves preserve their original properties after the collision. The phase functions for each wave are derived explicitly and it is shown that they are not straight lines anymore, they are rather some curves. Turkish Academy of Sciences Publisher's Version Q1 WOS:000268060100004

We raise two questions on Euclid's Elements: How to explain that Propositions 16 and 27 in his first book do not follow, strictly speaking, from his postulates (or are perhaps meaningless)? and: What are the mathematical consequences of the meanings of the term eutheia, which we today often prefer to consider as different? The answer to the first question is that orientability is a tacit assumption. The answer to the second is rather a discussion on efforts to avoid actual infinity, and having to (in some sense or another) construct equivalence classes of segments to achieve uniqueness.

Mazurkiewicz proved the existence of a subset of the Euclidean plane E 2 with the property that every straight line intersects it in exactly two points. A set with this property is called a Mazurkiewicz set . A nondegenerate subset X of E 2 is a generalized Mazurkiewicz set if each line that separates two points of X intersects X in exactly two points. We prove that a generalized Mazurkiewicz set must be a simple closed curve if it contains an arc. From this we deduce that a closed, generalized Mazurkiewicz set is a simple closed curve. Simple closed curves in E 2 are generalized Mazurkiewicz sets if and only if they bound convex disks.

Diplomsko delo opisuje postopek oblikovanja enoročajne mešalne kopalniške armature za umivalnik. Oblikovanje je bilo izvedeno okoli že obstoječih notranjih komponent, zunanja oblika pa temelji na ravnih linijah, ki predstavljajo nove smernice pri oblikovanju armatur. Pred oblikovanjem je bila izvedena obširna tržna raziskava, na podlagi katere je bilo ugotovljeno kateri tipi armatur prevladujejo in kaj trg že ponuja. Predstavljeni so tudi izdelovalni postopki in njihove omejitve, na katere je bilo treba paziti. Rezultati so prikazani z virtualnim modelom v obliki slik in s fizičnim modelom, ki je narejen v dejanski velikosti, vendar ni funkcionalen. This diploma work presents the process of designing a single-handle mixing bathroom fitting. The design was executed around already existing internal components. External features are based on straight lines which represent a new trend in designing bathroom fittings. Before the process of designing started, an extensive market research was carried out. Based on this research it was found which types of faucets are already available and which ones dominate the market at this moment. Manufacturing process is also presented along with limitations which had to be taken into consideration. Final results are displayed by a virtual model in the form of images and by physical model in actual size which thoug is not functional.

Closing the control loop of a manipulator robot with vision feedback is widelyknown. It concerns nowadays all areas of robotics. Such a return can make a comparison between a desired state and current state, using visual measurements. The main objective of this doctoral thesis is to design several types of kinematic control laws for stereo visual servoing. It strongly involves the formalism of the task function which is a well-known and useful mathematical tool to express the visual error as a function of state vectors.We have investigated the decoupling between the rotational and translational velocities control laws together with the epipolar constraint with a stereo visual feedback.That is why, the visual measurements and features used in this thesis are the 3Dstraight lines.The interests of this type of visual features rely on the robustness against the noise, and the possibility to represent straight lines or other features like points or planes pairs by the Plücker coordinates, as a 3D straight line can be represented as well by two points or the intersection of two planes. This makes all the control laws designed in this thesis valid for another visual features like points, L'emploi d'un retour visuel dans le but d'effectuer une commande en boucle fermée de robot s'est largement répandu et concerne de nos jours tous les domaines de la robotique. Un tel retour permet d'effectuer une comparaison entre un état désiré et l'état actuel, à l'aide de mesures visuelles. L'objectif principal de cette thèse consiste à concevoir plusieurs types de lois de commande cinématiques par vision stéréo. Ceci concerne aussi l'étude de la stabilité du système en boucle fermée et la convergence des fonctions de tâche. C'est essentiellement le découplage des lois de commandes cinématiques en rotation et en translation qui est recherché ici, selon le nombre d'indices visuels considérés.Les mesures visuelles utilisées dans cette thèse sont les lignes droites 3D. Les intérêts apportés à ce type de mesures visuelles sont la robustesse contre le bruit, et la possibilité de représenter d'autres primitives comme des couples de points ou de plans par la modélisation de Plücker.

Cataloged from PDF version of article. In general, feature points and camera parameters can only be estimated with limited accuracy due to noisy images. In case of collinear feature points, it is possible to benefit from this geometrical regularity by correcting the feature points to lie on the supporting estimated straight line, yielding increased accuracy of the estimated camera parameters. However, regarding Maximum-Likelihood (ML) estimation, this procedure is incomplete and suboptimal. An optimal solution must also determine the error covariance of corrected features. In this paper, a complete theoretical covariance propagation analysis starting from the error of the feature points up to the error of the estimated camera parameters is performed. Additionally, corresponding Fisher Information Matrices are determined and fundamental relationships between the number and distance of collinear points and corresponding error variances are revealed algebraically. To demonstrate the impact of collinearity, experiments are conducted with covariance propagation analyses, showing significant reduction of the error variances of the estimated parameters. (C) 2010 Elsevier Inc. All rights reserved.

This paper uses the vector field approach to represent both the desired ground track heading and the vertical flight path angle to navigate unmanned aerial vehicles onto the desired path. The path following is realized in 3-dimensions, thus allowing the aircraft to navigate longitudinally as well as laterally, unlike the current approaches which require the aircraft to maintain a constant altitude. This results in an increase in the maneuverability (like overflying a hill) of the vehicle allowing it to navigate over various terrains and terrain following. Methods for following straight lines in 3-dimensional space are demonstrated through numerical simulations.

Made available in DSpace on 2013-09-27T14:53:51Z (GMT). No. of bitstreams: 1 WOS000278763500009.pdf: 137450 bytes, checksum: 5a8e46ca9b0ebb6c9f9a53a2cd255460 (MD5) Previous issue date: 2010-01-01 Made available in DSpace on 2013-09-30T18:08:57Z (GMT). No. of bitstreams: 1 WOS000278763500009.pdf: 137450 bytes, checksum: 5a8e46ca9b0ebb6c9f9a53a2cd255460 (MD5) Previous issue date: 2010-01-01 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T13:22:46Z No. of bitstreams: 1 WOS000278763500009.pdf: 137450 bytes, checksum: 5a8e46ca9b0ebb6c9f9a53a2cd255460 (MD5) Made available in DSpace on 2014-05-20T13:22:46Z (GMT). No. of bitstreams: 1 WOS000278763500009.pdf: 137450 bytes, checksum: 5a8e46ca9b0ebb6c9f9a53a2cd255460 (MD5) Previous issue date: 2010-01-01 In this work a method is proposed to allow the indirect orientation of images using photogrammetric control extracted through integration of data derived from Photogrammetry and Light Detection and Ranging (LiDAR) system. The photogrammetric control is obtained by using an inverse photogrammetric model, which allows the projection of image space straight lines onto the object space. This mathematical model is developed based on the intersection between the collinearity-based straight line and a DSM of region, derived from LiDAR data. The mathematical model used in the indirect orientation of the image is known as the model of equivalent t planes. This mathematical model is based on the equivalence between the vector normal to the projection plane in the image space and to the vector normal to the rotated projection plane in the object space. The goal of this work is to verify the quality, efficiency and potential of photogrammetric control straight lines obtained with proposed method applied to the indirect orientation of images. The quality of generated photogrammetric control was statistically available and the results showed that proposed method is promising and it has potential for the indirect orientation of images. Universidade Federal do Paraná (UFPR), Setor Ciencias Terra, Programa Posgrad Ciencias Geodes, BR-81531 Curitiba, Parana, Brazil Univ Estadual Paulista, Fac Ciencias & Tecnol, Programa Posgrad Ciencias Cartograf, BR-19060900 Presidente Prudente, SP, Brazil Univ Estadual Paulista, Fac Ciencias & Tecnol, Programa Posgrad Ciencias Cartograf, BR-19060900 Presidente Prudente, SP, Brazil

Double enveloping worm gearing is expected to have contact over larger number of teeth and higher load carrying capacity compared to single enveloping worm gearing. In this paper, contact in this gearing is analysed by geometrical simulation of worm gear tooth generation using intersection profiles of different axial sections of worm representing the hob tooth profile with transverse plane of worm gear. The analysis reveals that in the engaging zone a straight line contact always exists in the median plane and intermittent contact exists at the extreme end sections of worm. This has lead to the idea of using two fly cutters positioned at the location identical to the extreme end sections of worm to generate full worm gear tooth thereby eliminating the need for hobs of complex geometry. For a given worm, a mating worm gear is machined in a gear hobbing machine using fly tool in two settings and nature of contact with the worm is checked by a blue test. � 2009 Elsevier Ltd. All rights reserved.

In the present study we consider spherical product surfaces X = α⊗β of two 2D curves in E3. We prove that if a spherical product surface patch X = α⊗β has vanishing Gaussian curvature K (i.e. a flat surface) then either α or β is a straight line. Further, we prove that if α(u) is a straight line and β(v) is a 2D curve then the spherical product is a non-minimal and flat surface. We also prove that if β(v) is a straight line passing through origin and α(u) is any 2D curve (which is not a line) then the spherical product is both minimal and flat. We also give some examples of spherical product surface patches with potential applications to visual cyberworlds., IEEE Computer Society

Made available in DSpace on 2013-09-27T14:53:52Z (GMT). No. of bitstreams: 1 WOS000260625400008.pdf: 355389 bytes, checksum: 545ea30295f289c2330cc6cc6432c01b (MD5) Previous issue date: 2008-01-01 Made available in DSpace on 2013-09-30T18:08:49Z (GMT). No. of bitstreams: 1 WOS000260625400008.pdf: 355389 bytes, checksum: 545ea30295f289c2330cc6cc6432c01b (MD5) Previous issue date: 2008-01-01 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T13:22:42Z No. of bitstreams: 1 WOS000260625400008.pdf: 355389 bytes, checksum: 545ea30295f289c2330cc6cc6432c01b (MD5) Made available in DSpace on 2014-05-20T13:22:42Z (GMT). No. of bitstreams: 1 WOS000260625400008.pdf: 355389 bytes, checksum: 545ea30295f289c2330cc6cc6432c01b (MD5) Previous issue date: 2008-01-01 In the fields of Machine Vision and Photogrammetry, extracted straight lines from digital images can be used either as vector elements of a digital representation or as control entities that allow the determination of the camera interior and exterior orientation parameters. Applications related with image orientation require feature extraction with subpixel precision, to guarantee the reliability of the estimated parameters. This paper presents three approaches for straight line extraction with subpixel precision. The first approach considers the subpixel refinement based on the weighted average of subpixel positions calculated on the direction perpendicular to the segmented straight line. In the second approach, a parabolic function is adjusted to the grey level profile of neighboring pixels in a perpendicular direction to the segmented line, followed by an interpolation of this model to estimate subpixel coordinates of the line center. In the third approach, the subpixel refinement is performed with a parabolic surface adjustment to the grey level values of neighboring pixels around the segmented line. The intersection of this surface with a normal plane to the line direction generates a parabolic equation that allows estimating the subpixel coordinates of the point in the straight line, assuming that this is the critical point of this function. Three experiments with real images were made and the approach based on parabolic surface adjustment has presented better results. Univ Estadual Paulista, Programa Posgrad Ciencias Cartograf, BR-19060900 Presidente Prudente, SP, Brazil Univ Estadual Paulista, Fac Ciencias & Tecnol, Depto Cartog, BR-19060900 Presidente Prudente, SP, Brazil Univ Estadual Paulista, Programa Posgrad Ciencias Cartograf, BR-19060900 Presidente Prudente, SP, Brazil Univ Estadual Paulista, Fac Ciencias & Tecnol, Depto Cartog, BR-19060900 Presidente Prudente, SP, Brazil

Asia Pacific Microwave Conference -- DEC 11-14, 2007 -- Bangkok, THAILAND, WOS: 000262091700013, A new type of dual-mode microstrip square-loop resonator that produces asymmetrical frequency responses is proposed. The coupling between degenerate modes of the proposed dual-mode resonator is discussed depending on the perturbation size. The effect of non-orthogonal input/output feed lines located along a straight line on the frequency response is investigated using full-wave electromagnetic simulations. Two second-order bandpass filters using this type of dual-mode microstrip resonator are designed, fabricated and measured to demonstrate the dual-mode microstrip filters exhibiting transmission zero (TZ) shifting property. The one of the second-order filters produces a filter characteristic with two TZs in the upper stopband, whereas the other one has two TZs in the lower and upper stopbands. The shifting property of one of the TZs from one side of the passband to the other side is realised by only changing the perturbation size. The fourth-order filters of this type are also investigated. Both simulated and measured results are presented.

Made available in DSpace on 2022-04-28T20:41:57Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-01-01 In the fields of Machine Vision and Photogrammetry, extracted straight lines from digital images can be used either as vector elements of a digital representation or as control entities that allow the determination of the camera interior and exterior orientation parameters. Applications related with image orientation require feature extraction with subpixel precision, to guarantee the reliability of the estimated parameters. This paper presents three approaches for straight line extraction with subpixel precision. The first approach considers the subpixel refinement based on the weighted average of subpixel positions calculated on the direction perpendicular to the segmented straight line. In the second approach, a parabolic function is adjusted to the grey level profile of neighboring pixels in a perpendicular direction to the segmented line, followed by an interpolation of this model to estimate subpixel coordinates of the line center. In the third approach, the subpixel refinement is performed with a parabolic surface adjustment to the grey level values of neighboring pixels around the segmented line. The intersection of this surface with a normal plane to the line direction generates a parabolic equation that allows estimating the subpixel coordinates of the point in the straight line, assuming that this is the critical point of this function. Three experiments with real images were made and the aproach based on parabolic surface adjustment has presented better results. Universidade Estadual Paulista Programa de Pós-Graduação em Ciências Cartográficas, Rua Roberto Simonsen, 305, 19060-900 Presidente Prudente SP Universidade Estadual Paulista Faculdade de Ciências e Tecnologia Depto. de Cartografia, Rua Roberto Simonsen, 305, 19060-900 Presidente Prudente SP Universidade Estadual Paulista, Rua Roberto Simonsen, 305, 19060-900 Presidente Prudente SP Universidade Estadual Paulista Programa de Pós-Graduação em Ciências Cartográficas, Rua Roberto Simonsen, 305, 19060-900 Presidente Prudente SP Universidade Estadual Paulista Faculdade de Ciências e Tecnologia Depto. de Cartografia, Rua Roberto Simonsen, 305, 19060-900 Presidente Prudente SP Universidade Estadual Paulista, Rua Roberto Simonsen, 305, 19060-900 Presidente Prudente SP

Asia Pacific Microwave Conference -- DEC 11-14, 2007 -- Bangkok, THAILAND, WOS: 000261353300120, A new type of dual-mode resonator that produces asymmetrical frequency responses is proposed. The effect of nonorthogonal input/output feed lines located along a straight line on filter response is investigated for the dual-mode filter. The coupling between degenerate modes of the proposed dual-mode microstrip resonator is discussed depending on the perturbation size. The two samples of asymmetric dual-mode microstrip filter were designed, fabricated and measured to demonstrate the design of dual-mode microstrip filters exhibiting transmission zero shifting property. The one of them produces a filter characteristic with two TZs in the upper stopband the other has two TZs in the lower and upper stopbands. The shifting of the one of the TZs from one side of the passband to the other is realized by only changing the perturbation size., Scientific and Technological Research Council of Turkey (TUBITAK) [104E030], This work was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under Grant 104E030.

[EN] Radial and decentring distortion parameters have long been reported as fundamental elements in camera calibration, especially in digital close range photogrammetry. This paper reports on an investigation for the determination of both radial and decentring distortion parameters using analytical plumb-line calibration. Straight line patterns, ranging from 1, 2, 4, 10 and up to 20 lines, not strictly plumb, are examined on a set of distorted imagery. Additionally, not only the number of straight lines but also the linear image distribution is analysed. For that purpose, linear features have been distributed in different positions and orientations over the whole of the imagery. Furthermore, the relationship between the parameters mentioned and the location of the principal point is also analysed, and shows how sensitive they are to deviations from the centre of the sensor. The motivation for this paper is the measurement of the minimum number of linear features for proper calibration of non-metric digital cameras. Both radial and decentring distortion parameters are examined. Finally, some line patterns are recommended for making the plumb-line calibration technique a reliable, easy and fast procedure. Photogrammetrists could use it as a first step in the procedure of camera calibration; users who are not experts in photogrammetry could employ it as a final step., [ES] Los para´metros de distorsión radial y tangencial se han considerado desde hace mucho tiempo para´metros fundamentales en la calibración de cámaras, especialmente en fotogrametría de objeto cercano. Este artículo expone los resultados de una investigación que utiliza la calibración analítica mediante líneas de plomada para determinar los para´metros de distorsión radial y tangencial. Para conseguir dicho propósito se han examinado distintos patrones de líneas rectas en imágenes distorsionadas. Dichos patrones constan de una, dos, cuatro, diez y veinte rectas, no necesariamente de plomada. Adicionalmente, se han analizado no solo el nu´mero de rectas sino también su distribución en la imagen. Para dicha tarea, se han distribuido entidades lineales en diferentes posiciones y orientaciones sobre toda la imagen. Además, se ha estudiado la relación entre los para´metros mencionados y la posición del punto principal, poniendo de manifiesto la sensibilidad de los mismos al descentrado con respecto al centro del sensor. El propósito de este artículo consiste en medir el mínimo número de entidades lineales que aseguren una adecuada calibración de cámaras digitales no métricas. Para ello se determinan los para´metros de distorsión radial y tangencial. Finalmente, se recomiendan patrones lineales que hagan fiable, fácil de aplicar y rápida la técnica de calibración mediante líneas de plomada. Los expertos en fotogrametría podían usar este me´todo como paso previo al procedimiento de calibración de cámaras; los expertos no fotogramétricos podrían usarlo como un paso final., This research was partly funded by the Spanish Ministry of Education and Culture through the project HUM2005-03152. The valuable comments from anonymous reviewers are gratefully acknowledged

In this note we show that every affine transformation in the Euclidean space Rn , which has no fixed points and fulfils the inequality |/(x)/(y)| < \xy\ for any x and y has invariant straight line.