Examinando por Materia "Shape theory"
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Ítem 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 PL-Geodesics on PL-continuous partial meshes(2001) Ruíz, Óscar E.; Cadavid M., Carlos A.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEGeometric characteristics of 2-manifolds embedded in R3 space have been analyzed from the point of view of differential geometry and topology -- In the past, results relevant to these areas have been found for C∞ curves and surfaces -- However, current scientific, industrial, entertainment and medical applications, and availability of more powerful point sampling systems, press for characterization of discrete counterparts for the continuous properties and characteristics evaluated previously in C∞ curves and surfaces -- Recent works have presented estimation methods for properties such as the principal and rotated quadrics of point sampled surfaces -- The present article uses the findings of previous investigations to propose and implement a method for evaluation of planarity of surfaces -- It is based on:(i) Estimation of a C0 partial mesh fitting sets of planar or grid sample points -- (ii) Evaluation of the piecewise - linear (PL) version of families of geodesic curves on the mesh -- (iii) Diagnostic of the property of planarity based on the behavior of the families of geodesic curves -- The present work can be applied in the area of design and manufacturing of products based on sheet materials, such as apparel, metal stamping, thin structures, etcÍtem Topologically consistent partial surface reconstruction from range pictures(2000) Ruíz S., Óscar E.; Neugebauer, Peter; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAESurface or shape reconstruction from 3D digitizations, range pictures play an important role as the sizes and accessibility of the sampled object become intractable -- Range pictures, however, present challenges regarding: (i) recovery of topological structure from geometrical information of a partial view: (ii) inclusion of several or self obstructing objects on the same picture, and (iii) conciliation of partial topological and geometrical information from the individual views into a main model -- Issue (i) and (ii) require introduction of data structures and algorithms able to consistently represent incompleteness and discontinuities in the surface -- Aspect (iii) demands the application of statistical methods to sort redundant e inconsistent information in the overlaps between the individual views -- In this investigation, tasks (i) and (ii) have been undertaken by designing and populating an extended Boundary Representation (B-rep), using equivalence relations that induce partitions on the data sets -- Task (iii) has been carried out by using data processing tools (DigitLAB) that filter, resample, and recover shape from planar digitizations, by applying formalisms of differential topology