César Otero

Elitist clonal selection algorithm for optimal choice of free knots in B-spline data fitting

Graphical abstractDisplay Omitted HighlightsIn this paper we introduce an adapted elitist clonal selection algorithm for automatic knot adjustment of B-spline curves.Our method determines the number and location of knots automatically in order to obtain an extremely accurate fitting of data.In addition, our method minimizes the number of parameters required for this task.Our approach performs very well and in a fully automatic way even for the cases of underlying functions requiring identical multiple knots, such as functions with discontinuities and cusps.Our experimental results show that our approach outperforms previous approaches in terms of accuracy and flexibility. Data fitting with B-splines is a challenging problem in reverse engineering for CAD/CAM, virtual reality, data visualization, and many other fields. It is well-known that the fitting improves greatly if knots are considered as free variables. This leads, however, to a very difficult multimodal and multivariate continuous nonlinear optimization problem, the so-called knot adjustment problem. In this context, the present paper introduces an adapted elitist clonal selection algorithm for automatic knot adjustment of B-spline curves. Given a set of noisy data points, our method determines the number and location of knots automatically in order to obtain an extremely accurate fitting of data. In addition, our method minimizes the number of parameters required for this task. Our approach performs very well and in a fully automatic way even for the cases of underlying functions requiring identical multiple knots, such as functions with discontinuities and cusps. To evaluate its performance, it has been applied to three challenging test functions, and results have been compared with those from other alternative methods based on AIS and genetic algorithms. Our experimental results show that our proposal outperforms previous approaches in terms of accuracy and flexibility. Some other issues such as the parameter tuning, the complexity of the algorithm, and the CPU runtime are also discussed.

Information management and improvement of citation indices

Bibliographic databases lack uniformity and accuracy due to duplications, misspellings, name variants, etc.Dataset pre-processing is very important to obtain more accurate, realistic and valid bibliometric analysis.STICCI.eu is a software tool for improving and converting citation indices - enhancing uniformity.STICCI.eu is able to convert bibliographical citation formats (WoS, Scopus, Google Scholar, CSV, BibTex, RIS).It corrects mistakes in databases, detecting duplications, misspellings, name variants, toponymical names, etc. Bibliometrics and citation analysis have become important sets of methods for library and information science, as well as exceptional sources of information and knowledge for many other areas. Their main sources are citation indices, which are bibliographic databases like Web of Science, Scopus, Google Scholar, etc. However, bibliographical databases lack perfection and standardization. There are several software tools that perform useful information management and bibliometric analysis importing data from them. A comparison has been carried out to identify which of them perform certain pre-processing tasks. Usually, they are not strong enough to detect all the duplications, mistakes, misspellings and variant names, leaving to the user the tedious and time-consuming task of correcting the data. Furthermore, some of them do not import datasets from different citation indices, but mainly from Web of Science (WoS).A new software tool, called STICCI.eu (Software Tool for Improving and Converting Citation Indices - enhancing uniformity), which is freely available online, has been created to solve these problems. STICCI.eu is able to do conversions between bibliographical citation formats (WoS, Scopus, CSV, BibTex, RIS), correct the usual mistakes appearing in those databases, detect duplications, misspellings, etc., identify and transform the full or abbreviated titles of the journals, homogenize toponymical names of countries and relevant cities or regions and list the processed data in terms of the most cited authors, journals, references, etc.

Planar subdivisions by radical axes applied to structural morphology

In previous articles the relation between Lattice and Plate structural systems to Delaunay and Voronoi planar diagrams has been demonstrated. The present contribution shows how Geotangent Mesh designs can also be formulated as a bi-dimensional problem stated as the Planar Subdivision of Radical Axes arising from a packing of circles. This way the origin of all of the Spatial Mesh Structural Typologies can be formulated by means of the basic elements of Computational Geometry.

Novel Technique for Obtaining Double-Layer Tensegrity Grids

Double-layer tensegrity grids (DLTGs) may be defined as tensegrity spatial systems containing two parallel horizontal networks of members in tension forming the top and bottom layers, whose nodes are linked by vertical and/or inclined bracing members in compression and/or tension. In this paper, a new approach is described. Conventional double-layer grids (DLGs) are composed of three layers: top, bottom and bracing members. This paper shows new rules for generating original DLGs following a recent methodology for their composition, from the mosaic of the bracing members and additional laws. Finally, from them, a new technique, known as Rot-Umbela manipulation, is applied to obtain their tensegrity form, opening and endless catalogue of DLTGs.

Computational Geometry and Spatial Meshes

This paper establishes a relation between the field of study of Geometric Design of the Structures known as “Spatial Meshes” and Computational Geometry. According to some properties of Voronoi Diagrams and Convex Hulls, and by means of two transformations: one inversive, the other projective, a way for generating polyhedra approximating spheres and other second order surfaces (ellipsoids, paraboloids and hyperboloids), starting from a simple Voronoi Diagram, is shown. The paper concludes with the geometric design of C-TANGENT domes: a new way for generating Spatial Meshes that can become a possible field of application for Computational Geometry.

An Application of Computer Graphics for Landscape Impact Assessment

This communication shows a procedure aimed at assisting in the assessment of the landscape impact caused by new lineal infrastructure services, such as motorways. The computer simulation has demanded the idealisation of the phenomenon to be graphically represented, its theoretical model and the functional analysis of the computational tool that can aid in finding a solution. The core of the problem is that there is a non finite set of landscapes to be considered (and analysed); however, according to the model proposed, if some suitable Computer Graphics libraries are used, it is possible to automatically produce a reduced catalogue of realistic images simulating, each of them, the effect that the new infrastructure will cause on the environment, just at the sites where the landscape results to be more vulnerable.

Innovative Families Of Double-Layer Tensegrity Grids: Quastruts and Sixstruts

AbstractDouble-layer tensegrity grids (DLTGs) are spatial reticulated systems based on tensegrity principles, which have been studied in detail over recent years. The most important investigations have been carried out focusing on a short list of tensegrity grids. This paper explains, using real examples, how to use Rot-Umbela manipulations, a unique technique developed for generating innovative typologies of tensegrity structures. It is applied to two already existing tensegrity grids to obtain two new DLTGs. Their analysis permits us to identify, inside these novel grids, the modules that compose them, which were unknown until now. A brief description of these components and some information about their static analysis, for example states of self-stress and internal mechanisms, is provided. These novel modules belong to a family, all of them with similar characteristics in terms of geometry and topology, and can be used to generate a wide catalog of DLTGs. Some examples of new grids are presented, descr...

Generation and Nomenclature of Tessellations and Double-Layer Grids

AbstractThe aim of this work is to establish a systematic methodology for generating automatically different tessellations and double-layer grids (DLGs) following a newly defined, specific nomenclature. This particular nomenclature defines the notation of mosaics and DLGs in a synthesized and unique manner, and can generate and design them after the parameters expressed in their own names. By means of an algorithm and some computational codes, it is possible to recreate in 3D any DLGs directly from their own names. The current nomenclature for tessellations is also analyzed and found to have severe disadvantages, such as the excessive length of their notations and their nonunique character. A new nomenclature is proposed to define and generate n-uniform mosaics consistently and unequivocally in a manner consistent with the current nomenclature used for Archimedean (regular and semiregular) tessellations.

Matlab Toolbox for a First Computer Graphics Course for Engineers

This paper describes a new Matlab toolbox designed for teaching a first Computer Graphics course for Engineering students. The aim of the toolbox is to provide the students with a tool for a comprehensive overview on the fundamentals of Computer Graphics in terms of basic algorithms and techniques. Firstly, the paper discusses the design of such a course taking into account its contents, goals, duration and other constraints. Then, the toolbox architecture is described. Finally the application of this toolbox to educational issues is discussed through some illustrative examples. The system has been successfully used by the authors during the last three years at the University of Cantabria. The main conclusions from this experience are also reported.

Geometry Applied to Designing Spatial Structures: Joining Two Worlds

The usefulness that Computational Geometry can reveal in the design of building and engineering structures is put forward in this article through the review and unification of the procedures for generating C-Tangent Space Structures, which make it possible to approximate quadric surfaces of various types, both as lattice and panel structures typologies. A clear proposal is derived from this review: the possibility of synthesizing a great diversity of geometric design methods and techniques by means of a classic Computational Geometry construct, the power diagram, deriving from it the concept of Chordal Space Structure. 1 Definition and Typology of Space Structures A space frame is a structural system assembled of linear elements so arranged that forces are transferred in a three-dimensional manner. In some cases, the constituent elements may be two-dimensional. Macroscopically a space frame often takes the form of a flat or curved surface [15]. That classical definition can be extended following the next classification of space structures [16]: − Lattice archetype: frames composed by bars (one-dimensional elements) interconnected at nodes (zero-dimensional point objects). The structure is stabilized by the interaction of axial forces that concur at the nodes (fig.1). − Plate archetype: plates (bi-dimensional elements) that conform a polyhedron’s faces stabilized by the shear forces acting along its edges (one-dimensional hinges) (fig. 2 and 3). − Solid archetype: structures composed by three-dimensional elements which are stabilized by the action of forces transferred between the plane facets of the solids. 2 Geometric Generation of Space Structures The design of space structures can be approached in different ways. We now review three methods developed during the second half of the 20 century that suggest different ways for approximating a quadric surface taken as reference. Geometry Applied to Designing Spatial Structures: Joining Two Worlds 159 − Geodesic Dome: lattice type structure with a configuration derived from regular or semi-regular polyhedra in which the edges are subdivides into equal number of parts (“frequency” [3]); making use of these subdivisions, a three-way grid can be induced upon the faces of the original polyhedron. The central projection of this grid’s vertices on the polyhedron’s circumscribed sphere (see fig. 1), leads to a polyhedron approximating the sphere in which only the lattice’s nodes lie on the sphere’s surface (more details in [7]). Fig. 1. Left: Generation of the Geodesic Dome through the projection of the three-way grid on the circumscribed sphere. Right: U.S. Pavilion, Montreal Universal Exposition (1967) [5] − Geotangent Dome: it’s a plate type polyhedral structure in which the edges are tangent to a sphere. Such a sphere is sectioned by the polyhedron’s faces in such a way (fig. 2) that the faces’ inscribed circles are tangent to the inscribed circles of neighboring faces. Following this rule it is possible to determine the planes containing the circles generating the polyhedron’s edges from their intersection [17]. Fig. 2. Geotangent Polyhedron elevation (left). Nine meter diameter geotangent dome crowning Canopy Tower, Cerro Semáforo, Panamá (1963) [4] The procedure is involved and its calculations imply the solution of a non-linear equation system through an iterative process base on successive approximations. − Panel Structure: these plate type structures derives from lattice type geometries by applying the principle of structural and geometric duality (based on the concept of 160 J.A. Díaz, R. Togores, and C. Otero a point’s polarity regarding a sphere). Taking as a starting point the geodesic dome’s circumsphere, it is possible to transform the lattice’s nodes in the faces of its dual structure (fig. 3 and 5); the primitive sphere remaining as the new structure’s insphere. Fig. 3. Panel structure (left), derived as the dual polyhedron of a Schwedler type dome. (Right) Structures suggesting the plate typology. Eden Project, Cornwall, UK. [6] If in this procedure the sphere on which the polarity is applied is displaced in relation to the polyhedron that is to be transformed, the resulting panel structure no longer approximates a sphere, it approximates an ellipsoid instead. The first of these procedures is known as the Dual Transformation (DuT), while the second is the Dual Manipulation (DuM) [16]. 3 C-Tangent Spatial Structures Three typologies seemingly so different as those presented in paragraph two can be integrated under a unifying proposal in the realm of Computational Geometry, through the generation of C-Tangent structures [11]: it is sufficient to apply to a set of points S = { P1, P2, ..., PN } lying on the plane z=1 the sequence of transformations (translation, scale and inversion) which can be expressed as matrices the following way: P’ = [ MTRA(-) · MESC(-) · MINV · MESC · MTRA ] · P (1) followed by a projective transformation [13] (MHOM matrix): P’’ = MHOM · P’ (2) which transforms the Voronoi Diagram of these points V (S) into the polyhedral structure that approximates any quadric surface (fig. 4). Accepting this definition for C-Tangent structures, it is feasible to perform the following interpretation of the previously defined structures: − Plate Structure: a C-Tangent structure in which the proposed point set in z=1 are related by their Voronoi Diagram [11]. Geometry Applied to Designing Spatial Structures: Joining Two Worlds 161 − Lattice Structure: a C-Tangent structure obtained from the z=1 point set’s Delaunay triangulation [10]. − Geotangent Structure: a C-Tangent structure generated from the subdivision induced by the arrangement of the radical axes obtained from a tangent circles packing on z=1 [14]. Fig. 4. Generation of C-Tangent space structures. Left: inversion transforms the z=1 point set’s Voronoi Diagram into the polyhedral structure circumscribed to the sphere. Right: A projective transformation converts the approximating polyhedron into one circumscribed to a quadric This dispersion in the starting arguments needed for the generation of C-Tangent structures is only apparent. It is enough to introduce the concept of power diagrams to confirm this. Fig. 5. Lattice mesh (left) and plate structure (right) generated from the same set of points in the z=1 plane 162 J.A. Díaz, R. Togores, and C. Otero 4 Metric and Computational Geometry Notions 4.1 Power Diagrams From the most elemental definition: Definition 1: the constant (signed) product of the distances from a point P to the two intersection points A and B of any line which passes through P with a circumference is called the power of a point with respect to a circle [12]. The power of a point P can be expressed as: Power = PA · PB = (d + r) · (d – r) = d – r (3) where d is the distance from a point P to the circle’s center and r is the circle’s radius (this expression is still valid for points that lie inside the circumference). Property 1: The locus of those points in the plane that have equal circle power with respect to two non-concentric circles is a straight line perpendicular to the line of centers. It is called radical axis or power line. The generalization of such a definition to an n-dimensional space requires that we consider hiperspheres, not circles, centered on two generator points, in which case we formulate the locus of points in space with equal power with respect to both hiperspheres as a hiperplane orthogonal to the spheres’ center line. This hiperplane is known as the chordale for both generator points. Fig. 6. Power diagrams for seven circumferences (left) and four spheres (right) Property 2: given a collection of circumferences lying on a plane (hiperspheres in E) it is possible to bring about its tessellation considering nothing else than the intersection of the power lines for each pair of properly chosen neighboring circles (chordales of neighboring hiperspheres) (fig. 6). To each circumference (hipersphere), a convex region of the plane (E space) is associated, which is defined by the intersection of half planes (half spaces) containing those points with the least circle power. This region is known as the power cell, and the set of cells for the said Geometry Applied to Designing Spatial Structures: Joining Two Worlds 163 collection of circumferences (hiperspheres) is known as its associated power diagram [1]. Power diagrams and the procedures for the generation of space structures can be related through the concept of polarity. 4.2 Polarity in E Definition 2: the polar plane for a point P (xP, yP, zP) with respect to a quadric [8] is the locus of those points in space that are harmonic conjugates to P with respect to the two points in which any line passing through P, which is known as this plane’s pole, intersects the given quadric (see fig. 7). Property 3: the contact curve of the cone circumscribed to a quadric from an exterior point P is the conic section generated by the polar plane of point P. If among all the possible quadrics we select the paraboloid Ω (z = x + y), it is also true that the orthogonal projection of this section on a plane z=const is a circle [12]. Fig. 7. Polar plane for a point P with respect to a quadric Fig. 8. Spatial interpretation of a chordale With the projection of two of these conic sections we obtain a power diagram in which the power line is the projection on the same plane of the intersection of the polar planes containing both conic sections [1] (fig. 8). An immediate consequence is that every power diagram is the equivalent of the orthogonal projection of the boundaries of a convex polyhedral surface (resulting from the intersection of the half-spaces defined by polar planes). This surface can be regarded as the polyhedron that approximates the quadric. 164 J.A. Díaz, R. Togores, and C. Otero 5 Revision of the Mechanism for the Definition of C-Tangent Structures. Chordal Space Structures We propose that it is possible tThe usefulness that Computational Geometry can reveal in the design of building and engineering structures is put forward in this article through the review and unification of the procedures for generating C-Tangent Space Structures, which make it possible to approximate quadric surfaces of various types, both as lattice and panel structures typologies.