1,721,341 research outputs found
Quadrilateral meshes generated by fairing structure lines
Get PDFDas Generieren ästhetisch ansprechender Flächen ist in der architektonischen Geometrie von besonderem Interesse. Dieser Prozess bedingt mitunter das Anwenden geeigneter Glättungsverfahren. In der vorliegenden Diplomarbeit wird der spezielle Fall von diskreten 3-dimensionalen planaren Vierecksflächen betrachtet. Die Glätte eines Netzes wird - nicht wie im kontinuierlichen Fall durch die Differenzierbarkeit, sondern - anhand des Verlaufs einer Strukturlinie bestimmt; je geradliniger ein Polygon wird, umso glatter ist es.Einleitend werden dazu in dieser Arbeit bekannte Glättungsverfahren vorgestellt; wie zum Beispiel das Minimieren von Energiefunktionalen, das Verfahren von Laplace oder Taubin. Anschließend wird ein Fairing-Konzept präsentiert, welches mit Hilfe von Optimierungsmethoden die unerwünschten Zickzacklinien aus dem Netz "glättet". Das Verfahren untersucht vorab jeden Punkt auf dessen Glattheit und fixiert ihn gegebenenfalls. Mit Hilfe einer Gewichtsfunktion kann der Abstand der geglätteten Punkte zu den Ausgangspunkten gesteuert werden. Anhand von Beispielen werden die Ergebnisse der Glättung dieser Fairing-Methode mit denen der bekannten Verfahren von Laplace und Taubin verglichen.One of the main problems in architectural geometry is the generation of aesthetically appealing surfaces. This process occasionally conditions the application of appropriate smoothing methods. In this thesis a particular case of 3-dimensional planar quadrilateral meshes is observed. The smoothness of a quadrilateral mesh is defined, not through differentiability like in the continuous case, but based on the trend of a structure line; the more rectilineal a polygon becomes, the smoother it gets. This thesis also sets out to preliminary introduce well-known smoothing methods; such as the minimization of energy functionals, the Laplacian smoothing method or Taubin's smoothing method. Subsequently an alternative fairing concept is introduced, which 'smoothes' the undesirable zigzag lines with methods of optimization. This approach examines the smoothness of each point in advance and flattens it if necessary. The distance between the smoothed points can be directed towards the initial point via a penalty function. With reference to examples, the results of the smoothing through the alternative fairing method and that of the well-known methods of Laplace and Taubin can be compared
Interactive freeform architectural design with nearly developables and cold bent glass
Get PDFInteractive design of freeform architectural surface panelizations is at the coreof this PhD thesis. We provide the computational framework for dealing with two important types of paneling elements. Specifically, we focus on certain types of developable surfaces and cold bent glass panels, all relevant to contemporary freeform architecture.To this end, we initially present a novel method for increasing the developabilityof a B-spline surface. We use the property that the Gauss image of a developable surfaceis 1-dimensional and can be locally well approximated by circles. This is cast intoan algorithm for thinning the Gauss image by increasing the planarity of the Gaussimages of appropriate neighborhoods. A variation of the main method allows us totackle the problem of paneling a freeform architectural surface with developable panels,in particular enforcing rotational cylindrical, rotational conical and planar panels,which are the main preferred types of developable panels in architecture due to there duced cost of manufacturing. We are interested in near developability, rather than exact developability, so the optimization approach is sucient. The motivation behind this is the fact that most materials allow for a little bit of stretching and therefore developability needs not be satised to a high degree.One such material is glass which is the main focus of the second panelizationproblem of this thesis. Toughened glass can with stand higher stresses, and therefore allows initially planar glass panels to be elastically bent and xed at ambient temperatures to a curved frame. This process is called cold bending and it produces panels that can exhibit double curvature, providing a cost- and energy-ecient alternative of higher optical quality than traditional hot bent glass panels. However, it is very challenging to navigate the design space of cold bent glass panels due to the fragility of the material, which impedes the form-nding for practically feasible and aesthetically pleasing cold bent glass façades. We present an interactive, data-driven approachfor designing cold bent glass façades that can be seamlessly integrated into a typical architectural design pipeline. Our method allows non-expert users to interactively edit a parametric surface while providing real-time feedback on the deformed shape and maximum stress of cold bent glass panels. Designs are automatically rened to minimize several fairness criteria while maximal stresses are kept within glass limits.We achieve interactive frame rates by using a dierentiable mixture density network trained from more than a million simulations. Given a curved boundary, our regressionmodel is capable of handling multistable congurations and accurately predicting the equilibrium shape of the panel and its corresponding maximal stress. We show predictions are highly accurate and validate our results with a physical realization ofa cold bent glass surface. For both applications explored in this work, a plethora ofresults and examples are provided
Geometrie und interaktives Design gekrümmter Falten
Get PDFThis thesis discusses curved creases from a theoretic and an applied point of view: On the one hand, we utilize differential geometry to describe curved creases between developable surfaces by five quantities. We conclude that defining three of them appropriately determines the remaining two except in some special cases. Apart from degenerated folds, we also address two special types of folds: the planar crease, i.e. the rulings of the corresponding developable surfaces are refected on a plane, and creases of constant angle. The latter crease curves are known as pseudo-geodesics in classical differential geometry. By combining these classical results with our approaches, we examine pseudo-geodesics on cylinders and cones. Furthermore, a connection between bi-cylindrical, bi-concial and cylindro-conical creases of constant angle and geodesics on quadrics can be established. The applied approach is based on the work made by Tang et al. on the interactive design of curved creases. We utilize their proposed guided projection algorithm to solve an optimization problem for B-spline representations of developable surfaces with creases, and discuss the needed variables and constraints. Finally, we present some examples obtained from the author's implementation
Planar quad meshes from relative principal curvature lines
Get PDFThis thesis proposes a technique for the approximation of surfaces by PQ meshes. These are meshes with planar and mostly quadrilateral faces. Relative differential geometry is used for the generation of conjugate curve networks. It is well known that a discrete choice of curves from these networks naturally leads to meshes with quadrilateral faces, which are in turn planarized using optimization algorithms. The possibility to choose a convex ob ject, defining the relative differential geometry, gives rise to bounding the minimum intersecting angle of conjugate curves from below. This is a requirement for practical applications. Methods from convex geometry and Fourier analysis on the unit sphere are utilized to allow an interactive layout of the conjugate curve networks. This is followed by a discussion of the possibility to influence singularities in the conjugate curve networks, and consequently in the resulting PQ meshes. In a new approach, non-flat isotropic subdomains can be given an anisotropy, which is a replacement for the smoothing techniques introduced in recent papers on quad-dominant meshing. Finally, examples from architecture are used for demonstrating the capabilities of these techniques.In dieser Diplomarbeit wird ein Verfahren zur Approximation von Flächen mit PQ Netzen vorgestellt. PQ Netze bestehen aus planaren und hauptsächlich viereckigen Flächenstücken. Relative Differentialgeometrie wird dazu benutzt um konjugierte Kurvennetze zu erzeugen, welche auf natürliche Weise zu Netzen mit viereckigen Flächenstücken führen. Die Flächenstücke werden danach mit Hilfe von Optimierungsmethoden planarisiert. Durch die Wahl einer entsprechenden konvexen Fläche, welche die relative Differentialgeometrie definiert, kann der minimale Schnittwinkel konjugierter Kurven nach unten beschränkt werden. Dies ist eine Forderung, die in praktischen Anwendungen auftaucht. Methoden der konvexen Geometrie, sowie der Fourieranalyse auf der Einheitssphäre, werden dazu verwendet um die Erzeugung von konjugierten Kurvennetzen interaktiv vorzunehmen. Darauf folgend wird beschrieben wie Singularitäten in den konjugierten Kurvennetzen, und dadurch auch in den resultierenden PQ Netzen, beeinflusst werden können. Darüber hinaus können isotrope Teilbereiche wie anisotrope behandelt werden. Dies führt zu einem Ersatz der Glättungstechniken, die in kürzlich erschienenen Veröffentlichungen zur Erzeugung von Vierecksnetzen vorgestellt wurden. Schlussendlich werden die Möglichkeiten der untersuchten Methoden an Beispielen aus der Architektur demonstriert
Constrained rhombic nets - discrete differential geometry and applications
Get PDFRhombennetze sind diskrete Vierecksnetze, deren Flächenkanten windschiefe Rhomben bilden. Nachdem besondere Klassen von ihnen diskrete Analogien gewisser differenzierbarer Flächen darstellen, ist es naheliegend, sie mit den Methoden der Differenzengeometrie anstatt der Differentialgeometrie zu untersuchen.Der erste Teil dieser Arbeit betrachtet Rhombennetze mit ebenen Knoten (die diskrete Analogien von Tschebyscheffnetzen sind) und eine Verallgemeinerung, Rhombennetze mit konischen Knoten. Es wird gezeigt, dass diese genau die Parallelverschiebungen der Rhombennetze mit ebenen Knoten sind. Es werden Algorithmen zur Konstruktion der Netze angegeben, und mehrere Methoden zur Definition diskreter Gauss'scher Krümmungen werden gezeigt und besprochen, davon eine, die auf Netzen mit ebenen Flächen beruht, die von den Rhombennetzen abgeleitet werden können.Der zweite Teil betrachtet Rhombennetze, deren Knoten auf einem gegebenen Dreiecksnetz liegen. Es werden Computeralgorithmen angegeben, um solche Netze unter der Vorraussetzung minimaler Verzerrung zu konstruieren. Der letzte Teil bespricht Anwendungen von Rhombennetzen, insbesondere in der Architekturgeometrie.Rhombic nets are discrete quadrilateral nets whose faces form skewed rhombi. As special classes of them can be considered discrete analoga of certain differentiable surfaces it suggests itself to investigate them with the methods of discrete differential geometry as opposed to normal differential geometry. The first part of this thesis considers rhombic nets with planar knots (which are discrete analoga of Chebyshev nets) and a generalization of them, rhombic nets with conical knots, which are shown to be precisely the offsets of those with planar knots. Algorithms for their construction are given and several methods of defining discrete Gaussian curvatures for them are shown and discussed, including one that uses nets with planar faces derived from the rhombic nets.The second part considers rhombic nets whose knots lie on a given triangle mesh. Computer algorithms are shown for constructing such nets with minimal deformation. The last part discusses application of rhombic nets, in particular in architectural geometry
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