Darboux cyclides and webs from circles

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Moebius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure

Rib-reinforced Shell Structure

Shell structures are extensively used in engineering due to their efficient load-carrying capacity relative to material volume. However, large-span shells require additional supporting structures to strengthen fragile regions. The problem of designing optimal stiffeners is therefore becoming a major challenge for shell applications. To address it, we propose a computational framework to design and optimize rib layout on arbitrary shell to improve the overall structural stiffness and mechanical performance. The essential of our method is to place ribs along the principal stress lines which reflect paths of material continuity and indicates trajectories of internal forces. Given a surface and user-specified external loads, we perform a Finite Element Analysis. Using the resulting principal stress field, we generate a quad-mesh whose edges align with this cross field. Then we extract an initial rib network from the quad-mesh. After simplifying rib network by removing ribs with little contribution, we perform a rib flow optimization which allows ribs to swing on surface to further adjust rib distribution. Finally, we optimize rib cross-section to maximally reduce material usage while achieving certain structural stiffness requirements. We demonstrate that our rib-reinforced shell structures achieve good static performances. And experimental results by 3D printed objects show the effectiveness of our method

Planar quad meshes from relative principal curvature lines

Zsfassung in dt. SpracheThis 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.7

Packing circles and spheres on surfaces

Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces' incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate circle packings, hexagonal meshes which carry a torsion-free support structure, hybrid tri-hex meshes, and others. We show how triangle meshes can be optimized so as to have the incircle packing property. We explain their relation to conformal geometry and implications on solvability of optimization. The examples we give confirm that this kind of meshes is a rich source of geometric structures relevant to architectural geometry