NURBS

Abstract. The increasing demand for products with high accuracy and geometric dimensional measures are required high accuracy measurement system. Additionally, parts of complex geometry and free form parts have been a common practice in industries such as: automotive, aeronautics, bioengineering, among others. It is necessary inspection systems more accurate and flexible to meet these demands. The Coordinate Measuring Machines (CMMs) are an important tool in the design, fabrication and inspection of manufactured products. These machines are used by the engineer, whose main purpose is to produce an accurate digital model in a virtual space for later use in CAD / CAE / CAM / CAI. It is known, however, that the accuracy of the modeling process of a workpiece depends on the number of points which are measured on the surface of the inspected part. Thus, CMMs which incorporated Laser system are the best tool for reverse engineering process. However, this type of machine is very expensive....

Over the last decades, the new technical means and methods, and in particular computing technology, have become more and more popular and efficient and their capabilities evolved 
exponentially in due time. This development has led to the present situation, where structural analysis has stopped being a hindrance to the development of projects with geometrical and construction complexity. 
This paper explores the impact of the available design tools, especially in the field of structural analysis, on the built forms in each historical period. This process is studied through a detailed time-line, relating the evolution of built forms with the historical developments in the fields of construction materials, structural analysis methods, etc.

This paper introduces a method to extract "Shape-DNA", a numerical fingerprint or signature, of any 2d or 3d manifold (surface or solid) by taking the eigenvalues (i.e. the spectrum) of its Laplace-Beltrami operator. Employing the Laplace-Beltrami spectra (not the spectra of the mesh Laplacian) as fingerprints of surfaces and solids is a novel approach. Since the spectrum is an isometry invariant it is independent of the object's representation including parametrization and spatial position. Additionally the eigenvalues can be normalized, so that uniform scaling factors for the geometric objects can be obtained easily. Therefore, checking if two objects are isometric needs no prior alignment (registration/localization) of the objects but only a comparison of their spectra. In this paper we describe the computation of the spectra and their comparison for objects represented by NURBS or other parametrized surfaces (possibly glued to each other), polygonal meshes as well as solid polyhedra. Exploiting the isometry invariance of the Laplace-Beltrami operator, we succeed in computing eigenvalues for smoothly bounded objects without discretization errors caused by approximation of the boundary. Furthermore, we present two non-isometric but isospectral solids that cannot be distinguished by the spectra of their bodies and present evidence that the spectra of their boundary shells can tell them apart. Moreover, we show the rapid convergence of the heat trace series and demonstrate that it is computationally feasible to extract geometrical data such as the volume, the boundary length and even the Euler characteristic from the numerically calculated eigenvalues. This fact not only confirms the accuracy of our computed eigenvalues, but also underlines the geometrical importance of the spectrum. With the help of this "Shape-DNA" it is possible to support copyright protection, database retrieval and quality assessment of digital data representing surfaces and solids.

Derivatives are a very important tool of computations in an engineering practice on the graphics structures. B-Spline functions are defined recursively, so the direct computation is very difficult. In this article new direct proof of the formula used for simpler direct computation is shown. The paper also presents own method of programming derivatives of NURBS curves by means of the proven formula which is implemented in German engineering software RFEM 3D.

"Il saggio è dedicato alla realizzazione di un repertorio dei problemi classici della Geometria Descrittiva per mezzo degli strumenti informatici, alla ricerca di soluzioni più generali e innovative, facendo riferimento al manuale di F. G. M. (Frère Gabriel Marie, al secolo Edmond Brunhes) che ha conosciuto varie edizioni tra il 1877 e il 1909, con una postuma del 1920 e che, recentemente, è stato riproposto in edizione anastatica dall’editore Jacques Gabay. Si tratta di un manuale particolarmente significativo per due ordini di motivi: il primo è senz’altro la mole dell’opera, che raccoglie più di settecento problemi, non elementari, spesso corredati di note relative alla loro evoluzione storica; il secondo è il carattere innovativo, rispetto ad altre opere coeve, carattere che riguarda non solo le soluzioni proposte criticamente ma anche i contenuti. I metodi della Rappresentazione: in pianta e alzato, matematica, numerica, vengono qui approfonditi e messi a confronto nella risoluzione dei problemi proposti da Frère Gabriel Marie: 505, 506, 507, 508, 509, 509I, 509II, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520"

l'apparenza, la superficie, la pelle... l'involucro come sistema di comunicazione veste architettura.
Il libro di Massimiliano Ciammaichella osserva gli artefatti del presente, indagando le forme e le superfici da diverse angolazioni, esaltando la contaminazione concettuale e formale tra ambiti molteplici. le "interferenze" si possono stratificare mescolando contenuti simbolici, tecnologia digitale e saperi consolidati.
Effimero, leggero, apparente: sono gli aggettivi adatti agli oggetti del presente, ma dietro la costruzione delle sfavillanti superfici viene svelato il trucco: in questo testo si può scoprire come la stupefacente leggerezza e malleabilità siano insieme figlie della scienza della rappresentazione, della geometria, della tecnologia digitale 3D, e della fantasia di chi possiede la maestria nell'uso di quanto oggi è a disposizione dell'architetto, coniugando i saperi offerti dalla trattatistica storica, la modellazione tridimensionale, le tecniche di reverse engineering ecc.

The ability of managing big amounts of metric information coming from a LiDAR survey and the ability to reproduce high quality 3D models from them are still vivid problems to solve. Is it possible to create detailed models, geometrically and metrically correct, without using a large amount (often redundant) of metric data, such as massive point clouds? Obviously yes, but there are several ways to create a fitting 3D model for a specific research. A good solution is given by NURBS based algorithms that ensure high details of modelling. However, NURBS models can't be used directly on BIM platforms, because they need to be parametrized. In this sense, a parametric model is based on real measurements but each object could be interpreted and approximated based on objective and subjective (critic) view and also based on LODs (levels of detail or development) concerning a particular analysis. This kind of modelling of Cultural Heritage assets, fundamental for HBIM creation, need to be correctly planned especially for classification and definition of its historical features connected to an informative system, because nowadays information and then the semantic dimension are a necessary key points towards documentation analysis. Established this brief introduction, this schematic work will focus on the analysis of FreeCAD open BIM software and Rhinoceros as NURBS 3D modeller for Cultural Heritage is concerned, and whether and how they could integrate their tools for the purpose of managing dynamic high detailed data for the creation of an HBIM platform.

"Ripercorrendo le fasi progettuali dall’ideazione all’esecuzione dell’artefatto, è centrale il ruolo della geometria descrittiva
che si relaziona con strumenti digitali del disegno, nell’ottica di rilanciare consolidati metodi esperibili nelle diverse soluzioni poste dai problemi della trattatistica di settore. L’articolo analizza l’opera di Coop Himmelb(l)au, per approfondire gli studi sulle applicazioni dell’architettura contemporanea".

"Descriptive geometry plays a key role in the design process, from ideation to construction. It has to relate to digital design tools in order to revive consolidated methods which can be used to provide solutions to the problems posed by treatises on this issue. The paper analyses the work of the Coop Himmelb(l)au to study
how it is used in contemporary architecture."

Stockhausen’s graphically notated "Pole für 2" was created
for the multi-channel diffusion system in the spherical auditorium at the German pavilion at the Osaka World’s Fair in 1970. The work is remarkable for the way in which spatiality is explored through new developments in graphic notation – both for three-dimensional and sound projection, and to indicate musical imitation and transformation of short-wave radio signals. These two notational systems unite through the synchronous unfolding of time in Pole, and similarly through the spatial structure of auditorium’s architecture that housed over 50 loudspeakers in 6 layers. By using the sound projection score as the basis for the construction of a NURBS (non-uniform rational B-spline) model, each of the player’s sound projection geometries within the auditorium can be transmediated, thus allowing for a visual representation of the spatial design of the work. Examining the topology of the models produced from the work’s six sections further allows for emergent architectonic characteristics of volume, manifold and envelope to be tracked, thus pointing to Stockhausen’s characterisation of design space. The strong connection of the work to the auditorium’s spherical structure also points to the nature of the composer’s sound design as one predicated on the ability of patterns of sound projection to articulate tangible space, and therefore produce ephemeral aural architectures.

Over the years, the installed capacity of offshore wind turbines has increased rapidly. However, the LCOE (Levelized Costs Of Energy, average price per kWh) is still higher than the LCOE of traditional energy production methods like nuclear power or energy from coals or gas.
This research focuses on further decreasing the LCOE, by minimizing the weight of the support structure of the wind turbine. This is done in a so called integrated way: By optimizing the design of the tower and the foundation at the same time. Due to the interaction between wave loads and structure, many local optima are expected. Therefore an optimization strategy is proposed using a Particle Swarm Optimization (PSO) algorithm followed by a Gradient Based Algorithm on an objective which includes the mass and penalized constraints.
To reduce the optimization time, the evaluations of the objective function are done by using load estimations instead of extensive load calculations. To select reliable estimation methods, several methods are compared on a theoretical basis, after which the performance of a selection of these methods is examined on the problem. To improve the accuracy of the estimations, interaction of PSO and the estimators is proposed via estimator updating.
The proposed optimization approach is compared with the traditional design approach, by the optimization of several realistic case studies. This shows the potentials of using the estimator and optimization tool, developed during this project, in engineering practice.

Hull lines are not only a representation or a designing tool for ships, but also the fundamental analyzing tool for Ship Reconstruction.


Recommended Drawing Order
1. Baselines on the sheer, the h-b (half-breadth), and the body plans
2. A keel and posts on the sheer and the body plans
3. Sheer lines on the h-b and the sheer plans
4. Section lines on the sheer and the h-b plans
5. Midship section on the body plan
6. Sheer lines on the body plan
7. Transom lines on the body plan
8. Transom lines on the h-b plan
9. Waterlines on the sheer and the body plan
10. Tentative section lines on the body plan
11. Tentative waterlines on the h-b plan
12. Fix the h-b plan’s waterlines
13. Fix the body plan’s waterlines
14. Buttock lines on the body and the h-b plans
15. Buttock lines on the sheer plan
16. Fix section lines on the body plan if needed
17. Diagonal lines on the body plan
18. Diagonal lines on the h-b plan
19. Deck lines on the sheer, the body, and h-b plans

In this article, we will follow the history of splines, digital methods that have characterized the architectural design process since the last decade of the twentieth century.  We will describe their evolutions since the beginning as analogical tools for boat construction in the eighteenth century, to its passage to digital within the automotive industry. Then the relationship with the aerospace industry, and the evolution which from digital animation, have led nurbs, subdivision surfaces and t-splines to become everyday computational tools within architecture workflow. Recent developments on t-splines, following their acquisition by Autodesk, will lead us to a reflection on the relationship between software houses, which develop tools, and architects, who are now dependent on them. Therefore, we will consider the need to develop open-source methods, as Pixar did with Opensubdiv, as knowledge and responsibility shared solution.

Merging NURBS surfaces together and extending existing ones are common problems in CAD systems. Extending existing surface is usually done by creating a new separate surface and applying some merging function – result is one continuous surface. In this paper, full description of a method of direct extending existing surfaces is introduced. The extension of the surface is recomputed during its construction, and the NURBS surfaces are immediately C 1 connected

Yago Conde based his 1988 concept design for the Barcelona
Olympic Village water fountain on John Cage’s indeterminate graphic work "Fontana Mix." In this paper, a critique of Conde’s work serves as the departure point for an examination of Cagean notions of indeterminacy, an interrogation of the original musical context of Fontana Mix, and the presentation of a distinct methodology (employing NURBS modeling) for proto-architectural representation of Cage’s graphic score "Variations III.

NURBS (non-uniform rational b-spline) modelling has become a ubiquitous tool within architectural design praxis. In this article I examine three projects that utilise NURBS modelling as a means for which a musical system’s inherent spatiality is visualised. There are numerous precedents for which architectural form is a derivation of a musical system, or a musical system is proportionally informed by architectonic gesture. I propose in this article three NURBS modelling methodologies: for the spatial analysis of Karlheinz Stockhausen’s sound projection geometries in "Pole für 2"; for a spatial realisation of John Cage’s indeterminate work "Variations III"; and for the generation of a surface manifold informed by musically derived soundscape data from the Japanese garden Kyu Furukawa Teien. Rather than seeking to translate music into inhabitable architecture, or architectonic form into music, I highlight an approach that produces an interstitial territory between discourses on architecture and music analysis.

The implementation of historical information within BIM (Building Information Modelling) platforms has experienced great development processes during last years, generating excellent studies based on Historic Building Information Modelling (Murphy et al., 2009; 2013). The HBIM developing growth is certainly explained due to advantages concerning the documentation step as well as monitoring operations for Cultural Heritage assets. In this sense, information concerning historical architectures can be extracted directly from walls and masonries and it is related to stratigraphic information derived from archaeological analysis: this kind of analysis is fundamental in order to comprehend the evolution of the construction site through the identification of layers due to modifications and actions (Parenti R., 2000). The inclusion of stratigraphic analysis inside a HBIM workflow could be an innovative point as far as the management and monitoring is concerned. This kind of documentation, that was not designed to be included inside a common BIM platform, could be collected coupled with digital metric information derived from metric surveys even if it is still considered an ongoing research field, especially since Cultural Heritage assets have no BIM standard classification. For this reason, the main goal of this research is to adapt the possibilities of open source solutions concerning BIM methodologies to building archaeology documentation and analysis exploring unconventional strategies and also overcoming 3D modelling limitations of BIM software with free form modeler based on NURBS algorithm (Oreni et al., 2014), developing a particular scan-to-BIM process that, owing to the used opens source solutions and algorithm, can be renamed scan-to-openBIM via NURBS.

Ensayo que reivindica la generación y representación de la forma en la arquitectura, tomando en consideración basamentos teóricos recientemente revisados en lo que respecta a la topología en la arquitectura y uno de sus recursos de modelado, la NURBS. Se presentan antecedentes a las actuales técnicas generativas, desde la computación manual hasta la programación con lenguajes interpretados como el RhinoScript, del cual se expone un esbozo del estado de la cuestión dentro del área de la proyectación arquitectónica tanto a nivel profesional como académico, especialmente en el ámbito iberoamericano. Esto muestra los recursos con los que cuentan los diseñadores que están poniendo de manifiesto una nueva sensibilidad por la forma en la arquitectura de la era digital.

This article proposes a technique for the geometrically stable modeling of high-degree B-spline curves based on S-polygon in a float format, which will allow the accurate positioning of the end points of curves and the direction of the tangent vectors. The method of shape approximation is described with the purpose of providing geometrical proximity between the original and approximating curve. The content of the notion of a "harmonious, regular form" of B-spline curve's S-polygon in a float format is revealed as a factor in achieving a high-quality of fit for the generated curve. The expediency of the shape modeling method based on S-polygon in a float format at the end portions of the curve for quality control of curve modeling and editing is substantiated. The results of a comparative test are presented, demonstrating the superlative efficacy of using the Mineur-Farin configuration for constructing constant and monotone curvature curves based on an S-polygon in a float format. The findings presented in this article confirm that it is preferable to employ the principle of "constructing a control polygon of a harmonious form (or the Mineur-Farin configuration) of a parametric polynomial" to a B-spline curve's S-polygon in a float format, and not to a B-polygon of the Bézier curve. Recommendations are given for prospective studies in the field of applying the technique of constructing a high-quality B-spline curves to the approximation of log-aesthetic curves, Ziatdinov's superspirals, etc. The authors of the article developed a technique for constructing smooth connections of B-spline curves with ensuring a high order of smoothness of the composite curve. The proposed techniques are implemented in the FairCurveModeler program as a plug-in to engineering CAD systems.