Examinando por Autor "Ruíz, Óscar E."
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Ítem Adaptative cubical grid for isosurface extraction(2009) Congote, John; Moreno, Aitor; Barandiaran, Iñigo; Barandiaran, Javier; Ruíz, Óscar E.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEThis work proposes a variation on the Marching Cubes algorithm, where the goal is to represent implicit functions with higher resolution and better graphical quality using the same grid size -- The proposed algorithm displaces the vertices of the cubes iteratively until the stop condition is achieved -- After each iteration, the difference between the implicit and the explicit representations are reduced, and when the algorithm finishes, the implicit surface representation using the modified cubical grid is more detailed, as the results shall confirm -- The proposed algorithm corrects some topological problems that may appear in the discretisation process using the original gridÍtem Algebraic geometry and group theory in geometric constraint satisfaction for computer-aided design and assembly planning(Taylor & Francis, 1996) Ruíz, Óscar E.; Ferreira, Placid M.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEMechanical design and assembly planning inherently involve geometric constraint satisfaction or scene feasibility (GCS/SF) problems -- Such problems imply the satisfaction of proposed relations placed between undefined geometric entities in a given scenario -- If the degrees of freedom remaining in the scene are compatible with the proposed relations or constraints, a set of entities is produced that populate the scenario satisfying the relations -- Otherwise, a diagnostic of inconsistency of the problem is emitted -- This problem appears in various forms in assembly planning (assembly model generation), process planning, constraint driven design, computer vision, etc -- Previous attempts at solution using separate numerical, symbolic or procedural approaches suffer serious shortcomings in characterizing the solution space, in dealing simultaneously with geometric (dimensional) and topological (relational) inconsistencies, and in completely covering the possible physical variations of the problem -- This investigation starts by formulating the problem as one of characterizing the solution space of a set of polynomials -- By using theories developed in the area of algebraic geometry, properties of Grobner Bases are used to assess the consistency and ambiguity of the given problem and the dimension of its solution space -- This method allows for die integration of geometric and topological reasoning -- The high computational cost of Grobner Basis construction and the need for a compact and physically meaningful set of variables lead to the integration of known results on group theory -- These results allow the characterization of geometric constraints in terms of the subgroups of the Special Group of Euclidean displacements in E^3, SE(3) -- Several examples arc developed which were solved with computer algebra systems (MAPLE and Mathematica) -- They are presented to illustrate the use of the Euclidean group-based variables, and to demonstrate the theoretical completeness of the algebraic geometry analysis over the domain of constraints expressible as polynomialsÍtem Aspect ratio-and size-controlled patterned triangulations of parametric surfaces(2007-02) Ruíz, Óscar E.; Peña, Sebastián; Duque, Juan; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEA method to produce patterned, controlled size triangulation of Boundary Representations is presented -- Although the produced patterned triangulations are not immediately suited for fast visualization, they were used in Fixed Grid Finite Element Analysis, and do provide a control on the aspect ratio or shape factor of the triangles produced -- The method presented first calculates a triangulation in the parameter space of the faces in which the B-Rep is partitioned and then maps it to 3D space -- Special emphasis is set in ensuring that the triangulations of neighboring faces meet in a seamless manner, therefore ensuring that a borderless C2 2-manifold would have as triangulation a C0 borderless 2-manifold -- The method works properly under the conditions(i) the parametric form of the face is a 1-1 function, (ii) the parametric pre-image of a parametric face is a connected region, and (iii) the user-requested sampling frequency (samples per length unit ) is higher than twice the spatial frequency of the features in the B-Rep ( thus respecting the Nyquist principle ) -- As the conditions (i) and (ii) are possible under face reparameterization or sub-division and the condition (iii) is the minimum that a triangulation should comply with, the method is deemed as generally applicableÍtem Boolean 2D Shape Similarity For Surface Reconstruction(2001) Ruíz, Óscar E.; Cadavid, Carlos A.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAESurface reconstruction problem (SRP) from planar samples has been traditionally approached by either (i) using local proximity between data points in adjacent layers, or by(ii) classifying the topological transitions that may explain the evolution of the cross sections -- Strategy (i) is robust in the sense that it has answers for every possible case, although in some scenarios renders counterintuitive surfaces, commented below -- Approach (ii) has mainly remained in the theoretical terrain -- The present work follows on aspect (ii), by using a Morse-based topological classification of the transitions, and complementing it with reasoning based on the geometry of the evolving cross sections to determine a high level description of the transitions from m to n contours (m:n transitions) -- This reasoning of shape similarity is performed by boolean operators -- Finally, the surface is synthesized using the m:n transitions -- This conjunction of topological and geometrical reasoning renders highly intuitive results, and allows for the incorporation of methods derived from the area of machine visionÍtem Boundary Representation of Anatomical Features(2002) Ruíz, Óscar E.; Karangelis, Grigorios; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEIn Simulationen für Bestrahlungstherapieplanung und andere medizinischen und bioingenieur wissenschaftlichen Anwendungen hat man traditionellerweise die anatomischen Eigenschaften mittels volumenbasierten Techniken angezeigt -- Trotzdem, während der Bedarf an Behandlung, Organsimulation und Erzeugung von Prothese präziser wird, reicht es nicht die Menge und die Position von solchen Features zu wissen, sondern auch Detailstufen, die vergleichbar sind zu denjenigen, die man für ingenieurwissenschaftliche Techniken verwendet (finite element analysis, rapid prototyping, Herstellung, usw.) -- Dieser Artikel präsentiert Bemühungen und Ergebnisse in der Anwendung von computerunterstützten geometrischen Algorithmen auf biomedizinischen Applikationen, wobei diese Algorithmen sich ursprünglich auf die Begrenzung von Formen gerichtet haben, die auf Planarmustern von anatomischen Zielregionen basiertenÍtem Design of computer experiments applied to modeling of compliant mechanisms for real-time control(Springer London, 2013-07) Acosta, Diego A.; Restrepo, David; Durango, Sebastián; Ruíz, Óscar E.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEThis article discusses the use of design of computer experiments (DOCE) (i.e., experiments run with a computer model to find how a set of inputs affects a set of outputs) to obtain a force–displacement meta-model (i.e., a mathematical equation that summarizes and aids in analyz-ing the input–output data of a DOCE) of compliant mechanisms (CMs) -- The procedure discussed produces a force–displacement meta-model, or closed analytic vector function, that aims to control CMs in real-time -- In our work, the factorial and space-filling DOCE meta-model of CMs is supported by finite element analysis (FEA) -- The protocol discussed is used to model the HexFlex mechanism functioning under quasi-static conditions -- The HexFlex is a parallel CM for nano-manipulation that allows six degrees of freedom (x, y, z, hx, hy, hz) of its moving platform -- In the multi-linear model fit of the HexFlex, the products or inter-actions proved to be negligible, yielding a linear model (i.e.,linear in the inputs) for the operating range -- The accuracy of the meta-model was calculated by conducting a set of computer experiments with random uniform distribution of the input forces -- Three error criteria were recorded comparing the meta-model prediction with respect to the results of the FEA experiments by determining: (1) maximum of the absolute value of the error, (2) relative error, and (3) root mean square error -- The maximum errors of our model are lower than high-precision manufacturing tolerances and are also lower than those reported by other researchers who have tried to fit meta-models to the HexFlex mechanismÍtem DigitLAB, an Environment and Language for Manipulation of 3D Digitizations(Presses internationales Polytechnique, 2000) Ruíz, Óscar E.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEIn Computer Aided Geometric Design the fitting of surfaces to massive series of data points has many applications, ranging from medicine to aerophotogrametry -- However, even the mathematical meaning of fitting a surface to a set of points is dependent on functional considerations, and not only on the geometric properties of the point set -- Also, characteristics of some parts of the data set must be interpreted as stochastic in nature, while others must be taken as literal and therefore they become constraints of the surface -- For these reasons, among others, automated surface fitting alone does not produce results usable at industrial level -- At the same time, it does not take advantage of sampling patterns, particular shapes of the cross sections, functionally different regions within the object, etc -- The latest literature reviews show the need for utilities to process point data sets that must be asynchronous, (applicable at any time and upon any region of the point set) -- Addressing this need, this article reports new tools developed within DigitLAB, a language that allows topological traversal, retrieval and statistical modifications to the data, and surface fitting -- They can handle arbitrary topology, as case studies in medicine, mathematics, landscaping, etc discussed here demonstrateÍtem Edge and corner identification for tracking the line of sight(Fondo Editorial EAFIT, 2005) Orozco, María S.; Ruíz, Óscar E.; Jasnoch, Uwe; Kretschmer, Ursula; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEThis article presents an edge-corner detector, implemented in the realm of the GEIST project (an Computer Aided Touristic Information System) to extract the information of straight edges and their intersections (image corners) from camera-captured (real world) and computer-generated images (from the database of Historical Monuments, using observer position and orientation data) -- Camera and computer-generated images are processed for reduction of detail, skeletonization and corner-edge detection -- The corners surviving the detection and skeletonization process from both images are treated as landmarks and fed to a matching algorithm, which estimates the sampling errors which usually contaminate GPS and pose tracking data (fed to the computer-image generatator) -- In this manner, a closed loop control is implemented, by which the system converges to exact determination of position and orientation of an observer traversing a historical scenario (in this case the city of Heidelberg) -- With this exact position and orientation, in the GEIST project other modules are able to project history tales on the view field of the observer, which have the exact intended scenario (the real image seen by the observer) -- In this way, the tourist “sees” tales developing in actual, material historical sites of the city -- To achieve these goals this article presents the modification and articulation of algorithms such as the Canny Edge Detector, SUSAN Corner Detector, 1-D and 2-D filters, etceteraÍtem EGCL: an extended G-Code Language with flow control, functions and mnemonic variables(World Academy of Science, Engineering and Technology (WASET), 2012) Ruíz, Óscar E.; Arroyave, S.; Cardona, J.F.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEIn the context of computer numerical control (CNC) and computer aided manufacturing (CAM), the capabilities of programming languages such as symbolic and intuitive programming, program portability and geometrical portfolio have special importance -- They allow to save time and to avoid errors during part programming and permit code re-usage -- Our updated literature review indicates that the current state of art presents voids in parametric programming, program portability and programming flexibility -- In response to this situation, this article presents a compiler implementation for EGCL (Extended G-code Language), a new, enriched CNC programming language which allows the use of descriptive variable names, geometrical functions and flow-control statements (if-then-else, while) -- Our compiler produces low-level generic, elementary ISO-compliant Gcode, thus allowing for flexibility in the choice of the executing CNC machine and in portability -- Our results show that readable variable names and flow control statements allow a simplified and intuitive part programming and permit re-usage of the programs -- Future work includes allowing the programmer to define own functions in terms of EGCL, in contrast to the current status of having them as library built-in functionsÍtem Erratum to “A novel technique for position analysis of planar compliant mechanisms” [Mech. Machine Theory 40 (2005)1224–1239](Elsevier, 2010-05) Correa, Jorge E.; Ruíz, Óscar E.; Durango, Sebastián; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEÍtem Finite Element Modeling of Composite Materials using Kinematic Constraints(Universidad EAFIT, 2009-12) Barschke, Merlin; Uribe, David; Ruíz, Óscar E.; Jensen, Jens; López, Carlos; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEEl propósito de este artículo es presentar simulaciones del comportamiento de materiales compuestos basado en restricciones cinemáticas entre las mismas fibras y entre las fibras y la resina circundante -- En la revisión de literatura, los autores han encontrado que las restricciones cinemáticas no han sido plenamente explotadas para modelar materiales compuestos, probablemente debido a su alto costo computacional -- El propósito de este artículo es exponer la implementación y resultados de tal modelo, usando Análisis por Elementos Finitos de restricciones geométricas prescritas a los nodos de la resina y las fibras -- Las descripciones analíticas del comportamiento de materiales compuestos raramente aparecen -- Muchas aproximaciones para describir materiales compuestos en capas son basadas en la teoría de funciones C1 Z y C0Z, tal como la Teoría Clásica de Capas (CLT) -- Estas teorías de funciones contienen significativas simplificaciones del material, especialmente para compuestos tejidos -- Una aproximación hibrida para modelar materiales compuestos con Elementos Finitos (FEA) fue desarrollada por Sidhu y Averill [1] y adaptada por Li y Sherwood [2] para materiales compuestos tejidos con polipropileno de vidrio -- Este artículo presenta un método para obtener valores para las propiedades de los materiales compuestos -- Tales valores son usados para simular las fibras reforzadas tejidas aplicando elementos de capas en el software ANSYS -- El presente modelo requiere menos simplificaciones que las teorías C1Z y C0Z -- En el artículo presente, a diferencia del modelo Li–Sherwood, el tejido es modelado geométricamente -- Una Representación por la Frontera (B-Rep del modelo “Hand”) con genus 1 (con geometría compleja) fue usada para aplicar restricciones geométricas a las capas de resina, fibra, etcétera, mostrando que es apropiada para simular estructuras complejas -- En el futuro, las propiedades no–lineales de los materiales deben ser consideradas, y el trabajo experimental requerido debe ser realizadoÍtem Fixed grid finite element analysis for 3D structural problems(World Scientific Publishing Co., 2005) García, Manuel J.; Henao, Miguel A.; Ruíz, Óscar E.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEFixed Grid (FG) methodology was first introduced by García and Steven as an engine for numerical estimation of two-dimensional elasticity problems -- The advantages of using FG are simplicity and speed at a permissible level of accuracy -- Two dimensional FG has been proved effective in approximating the strain and stress field with low requirements of time and computational resources -- Moreover, FG has been used as the analytical kernel for different structural optimisation methods as Evolutionary Structural Optimisation, Genetic Algorithms (GA), and Evolutionary Strategies -- FG consists of dividing the bounding box of the topology of an object into a set of equally sized cubic elements -- Elements are assessed to be inside (I), outside (O) or neither inside nor outside (NIO) of the object -- Different material properties assigned to the inside and outside medium transform the problem into a multi-material elasticity problem -- As a result of the subdivision NIO elements have non-continuous properties -- They can be approximated in different ways which range from simple setting of NIO elements as O to complex noncontinuous domain integration -- If homogeneously averaged material properties are used to approximate the NIO element, the element stiffness matrix can be computed as a factor of a standard stiffness matrix thus reducing the computational cost of creating the global stiffness matrix. An additional advantage of FG is found when accomplishing re-analysis, since there is no need to recompute the whole stiffness matrix when the geometry changes -- This article presents CAD to FG conversion and the stiffness matrix computation based on non-continuous elements -- In addition inclusion/exclusion of O elements in the global stiffness matrix is studied -- Preliminary results shown that non-continuous NIO elements improve the accuracy of the results with considerable savings in time -- Numerical examples are presented to illustrate the possibilities of the methodÍtem Gabriel-constrained Parametric Surface Triangulation(2008-10) Ruíz, Óscar E.; Cadavid, Carlos; Lalinde, Juan G.; Serrano, Ricardo; Peris-Fajarnés, Guillermo; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEThe Boundary Representation of a 3D manifold contains FACES (connected subsets of a parametric surface S : R2−R3) -- In many science and engineering applications it is cumbersome and algebraically difficult to deal with the polynomial set and constraints (LOOPs) representing the FACE -- Because of this reason, a Piecewise Linear (PL) approximation of the FACE is needed, which is usually represented in terms of triangles (i.e. 2-simplices) -- Solving the problem of FACE triangulation requires producing quality triangles which are: (i) independent of the arguments of S, (ii) sensitive to the local curvatures, and (iii) compliant with the boundaries of the FACE and (iv) topologically compatible with the triangles of the neighboring FACEs -- In the existing literature there are no guarantees for the point (iii) -- This article contributes to the topic of triangulations conforming to the boundaries of the FACE by applying the concept of parameter independent Gabriel complex, which improves the correctness of the triangulation regarding aspects (iii) and (iv) -- In addition, the article applies the geometric concept of tangent ball to a surface at a point to address points (i) and (ii) -- Additional research is needed in algorithms that (i) take advantage of the concepts presented in the heuristic algorithm proposed and (ii) can be proved correctÍtem Geodesic-based manifold learning for parameterization of triangular meshes(Springer Verlag, 2014) Acosta, Diego A.; Ruíz, Óscar E.; Arroyave, Santiago; Ebratt, Roberto; Cadavid, Carlos; Londono, Juan J.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEReverse Engineering (RE) requires representing with free forms (NURBS, Spline, Bézier) a real surface which has been pointsampled -- To serve this purpose, we have implemented an algorithm that minimizes the accumulated distance between the free form and the (noisy) point sample -- We use a dualdistance calculation point to / from surfaces, which discourages the forming of outliers and artifacts -- This algorithm seeks a minimum in a function that represents the fitting error, by using as tuning variable the control polyhedron for the free form -- The topology (rows, columns) and geometry of the control polyhedron are determined by alternative geodesicbased dimensionality reduction methods: (a) graphapproximated geodesics (Isomap), or (b) PL orthogonal geodesic grids -- We assume the existence of a triangular mesh of the point sample (a reasonable expectation in current RE) -- A bijective composition mapping allows to estimate a size of the control polyhedrons favorable to uniformspeed parameterizations -- Our results show that orthogonal geodesic grids is a direct and intuitive parameterization method, which requires more exploration for irregular triangle meshes -- Isomap gives a usable initial parameterization whenever the graph approximation of geodesics on be faithful -- These initial guesses, in turn, produce efficient free form optimization processes with minimal errors -- Future work is required in further exploiting the usual triangular mesh underlying the point sample for (a) enhancing the segmentation of the point set into faces, and (b) using a more accurate approximation of the geodesic distances within , which would benefit its dimensionality reductionÍtem Geometric constraint subsets and subgraphs in the analysis of assemblies and mechanisms(Universidad EAFIT, 2006-03) Ruíz, Óscar E.; Ferreira, Placid M.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEGeometric Reasoning ability is central to many applications in CAD/CAM/CAPP environments -- An increasing demand exists for Geometric Reasoning systems which evaluate the feasibility of virtual scenes specified by geometric relations -- Thus, the Geometric Constraint Satisfaction or Scene Feasibility (GCS/SF) problem consists of a basic scenario containing geometric entities, whose context is used to propose constraining relations among still undefined entities -- If the constraint specification is consistent, the answer of the problem is one of finitely or infinitely many solution scenarios satisfying the prescribed constraints -- Otherwise, a diagnostic of inconsistency is expected -- The three main approaches used for this problem are numerical, procedural or operational and mathematical -- Numerical and procedural approaches answer only part of the problem, and are not complete in the sense that a failure to provide an answer does not preclude the existence of one -- The mathematical approach previously presented by the authors describes the problem using a set of polynomial equations -- The common roots to this set of polynomials characterizes the solution space for such a problem -- That work presents the use of Groebner basis techniques for verifying the consistency of the constraints -- It also integrates subgroups of the Special Euclidean Group of Displacements SE(3) in the problem formulation to exploit the structure implied by geometric relations -- Although theoretically sound, these techniques require large amounts of computing resources -- This work proposes Divide-and-Conquer techniques applied to local GCS/SF subproblems to identify strongly constrained clusters of geometric entities -- The identification and preprocessing of these clusters generally reduces the effort required in solving the overall problem -- Cluster identification can be related to identifying short cycles in the Spatial Con straint graph for the GCS/SF problem -- Their preprocessing uses the aforementioned Algebraic Geometry and Group theoretical techniques on the local GCS/SF problems that correspond to these cycles -- Besides improving theefficiency of the solution approach, the Divide-and-Conquer techniques capture the physical essence of the problem -- This is illustrated by applying the discussed techniques to the analysis of the degrees of freedom of mechanismsÍtem Geometry simplification for modeling of porous materials(2015) Ruíz, Óscar E.; Cadavid, Esteban; Osorno, María C.; Uribe, David; Steeb, Holger; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEPorous and lattice materials have become everpresent in applications such as medicine, aerospace, design, manufacturing, art, entertainment, robotics, material handling, etc -- However, their application is impeded by the uncertainty of their mechanical properties (elongation, torsion, compression moduli, etc.) -- Computational Mechanics of poorus materials is also hindered by the massive geometric data sets that they entail, if their full geometric representations are used -- In response to these limitations, this article presents a truss simplification of a porous material --This simplified representation is usable in computer simulations, instead of the full triangle- or freeform-based Boundary Representations (B-Rep), which produce intractable problems -- This article presents the simplification methodology, along with results of estimation of the stress - strain response of porous material (in this case, Aluminum) -- Our methodology presents itself as a possible alternative in contrast with impossible processing when full data is used -- Follow up work is needed in using the truss methodology for calculating macro-scaleequivalent Young or Poisson moduli, with applications on mechanical designÍtem Graphs of optimally fit features in assessment of geometric tolerances(2014) Ruíz, Óscar E.; Congote, John; Acosta, Diego A.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEThis article presents an industrial application case of geometric constraint graphs, whose nodes are statistically optimal instances of manufacturing or design features and whose edges are usual geometric relations used in tolerance applications -- The features might be virtual ones -- As a consequence, they may lie beyond the piece’s extents -- The geometric constraint graph may have cyclic topology -- Contrary to deterministic geometric constraint graphs, tolerance constraint graphs admit numerical slacks, due to their stochastic nature -- The methodology has been applied in industrial scenarios, showing superiority to traditional material features for the assessment of tolerancesÍtem Manifold Learning with Orthogonal Geodesic Grids(2014) Ruíz, Óscar E.; Cadavid, Carlos; Ebratt, Roberto; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEIn Reverse Engineering, it is capital to find a parametric trimmed surface which approximates a triangular mesh (2-manifold with border) M in R3 -- This article proposes and implements a quasi isometry f: M -> R2 which allows a parameterization of M -- We consider quasi - developable 2- manifolds M in R3 -- f(p) = (u,w) with (u,w) being the coordinates of p in M under a grid of geodesic curves Ci(u) and Cj(w) on M -- We seek that the geodesic curves Ci(u) and Cj(w) be orthogonal to each other on M -- This means, that the Ci(u) should not cross each other, and each Ci(u) should intersect each Cj(w) in perpendicular mannerÍtem A new evaluation framework and image dataset for keypoint extraction and feature descriptor matching(2013-02) Barandiaran, Iñigo; Cortes, Camilo; Nieto, Marcos; Graña, Manuel; Ruíz, Óscar E.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEKey point extraction and description mechanisms play a crucial role in image matching, where several image points must be accurately identified to robustly estimate a transformation or to recognize an object or a scene -- New procedures for keypoint extraction and for feature description are continuously emerging -- In order to assess them accurately, normalized data and evaluation protocols are required -- In response to these needs, we present a (1) new evaluation framework that allow assessing the performance of the state-of-the-art feature point extraction and description mechanisms, (2) a new image dataset acquired under controlled affine and photometric transformations and (3) a testing image generator -- Our evaluation framework allows generating detailed curves about the performance of different approaches, providing a valuable insight about their behavior -- Also, it can be easily integrated in many research and development environments -- The contributions mentioned above are available on-line for the use of the scientific communityÍtem On the critical point structure of eigenfunctions belonging to the first nonzero eigenvalue of a genus two closed hyperbolic surface(2012-05-30) Cadavid, Carlos A.; Osorno, María C.; Ruíz, Óscar E.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEWe develop a method based on spectral graph theory to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of a compact riemannian manifold -- The method is applied to a closed hyperbolic surface of genus two -- The results obtained agree with the ones obtained by other authors by different methods, and they serve as experimental evidence supporting the conjectured fact that the generic eigenfunctions belonging to the first nonzero eigenvalue of a closed hyperbolic surface of arbitrary genus are Morse functions having the least possible total number of critical points among all Morse functions admitted by such manifolds