Discrete Differential Geometry (DDG) concerns discrete counterparts of notions and methods in differential geometry. This thesis deals with a core subject in DDG, discrete curvature theories on various types of polyhedral surfaces that are practically important for free-form architecture, sunlight-redirecting shading systems, and face recognition. Modeled as polyhedral surfaces, the shapes of free-form structures may have to satisfy different geometric or physical constraints. We study a combination of geometry and physics – the discrete surfaces that can stand on their own, as well as having proper shapes for the manufacture. These proper shapes, known as circular and conical meshes, are closely related to discrete principal curvatures. We study curvature theories that make such surfaces possible. Shading systems of freeform building skins are new types of energy-saving structures that can re-direct the sunlight. From these systems, discrete line congruences across polyhedral surfaces can be abstracted. We develop a new curvature theory for polyhedral surfaces equipped with normal congruences – a particular type of congruences defined by linear interpolation of vertex normals. The main results are a discussion of various definitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula. In addition to architecture, we consider the role of discrete curvatures in face recognition. We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold, which is an extension of the classical notion of asymptotic directions. We get a simple expression of these cones for polyhedral surfaces, as well as convergence and approximation theorems. We use the asymptotic cones as facial descriptors and demonstrate the practicability and accuracy of their applications in face recognition.

The emergence of freeform architecture provides interesting geometric challenges with regards to the design and manufacturing of large-scale structures. To design these architectural structures, we have to consider two types of constraints. First, aesthetic constraints are important because the buildings have to be visually impressive. Sec- ond, functional constraints are important for the performance of a building and its e cient construction. This thesis contributes to the area of architectural geometry. Specifically, we are interested in the geometric rationalization of freeform architec- ture with the goal of combining aesthetic and functional constraints and construction requirements. Aesthetic requirements typically come from designers and architects. To obtain visually pleasing structures, they favor smoothness of the building shape, but also smoothness of the visible patterns on the surface. Functional requirements typically come from the engineers involved in the construction process. For exam- ple, covering freeform structures using planar panels is much cheaper than using non-planar ones. Further, constructed buildings have to be stable and should not collapse. In this thesis, we explore the geometric rationalization of freeform archi- tecture using four specific example problems inspired by real life applications. We achieve our results by developing optimization algorithms and a theoretical study of the underlying geometrical structure of the problems. The four example problems are the following: (1) The design of shading and lighting systems which are torsion-free structures with planar beams based on quad meshes. They satisfy the functionality requirements of preventing light from going inside a building as shad- ing systems or reflecting light into a building as lighting systems. (2) The Design of freeform honeycomb structures that are constructed based on hex-dominant meshes with a planar beam mounted along each edge. The beams intersect without torsion at each node and create identical angles between any two neighbors. (3) The design of polyhedral patterns on freeform surfaces, which are aesthetic designs created by planar panels. (4) The design of space frame structures that are statically-sound and material-e cient structures constructed by connected beams. Rationalization of cross sections of beams aims at minimizing production cost and ensuring force equilibrium as a functional constraint.

In der vorliegenden Arbeit werden Geradenkongruenzen des projektiv abgeschlossenen dreidimensionalen euklidischen Raumes differentialgeometrisch untersucht. Nach J. PLÜCKER lassen sich Geraden in gleicher Weise als Grundelemente eines Geradenraumes auffassen wie die Punkte in einem Punktraum. Unter Beachtung dieser Überlegung scheint eine "natürliche" Behandlung der Geradenkongruenzen interessant und sinnvoll. Sie bildet den Gegenstand der vorliegenden Dissertation. Ein besonderes Augenmerk richtet sich dabei auf die Frage nach "kleinsten" Geradenkongruenzen ("Minimalkongruenzen") in der Geradenmenge des reellen projektiv abgeschlossenen dreidimensionalen euklidischen Raumes. Dahinter verbirgt sich eine gewisse Analogiebildung in der Liniengeometrie, die der klassischen Differentialgeometrie entstammt. Die Geradenkongruenzen bilden hierbei das liniengeometrische Analogon zu den Flächen des dreidimensionalen (Punkt-)Raumes. Das Wort "Kleinste" stellt im Geradenraum einen Bezug zu den Minimalflächen in der Differentialgeometrie her. Nun gestatten diese Fragestellungen in der Liniengeometrie eine anschauliche Interpretation, sobald man ein Punktmodell des Geradenraumes vorliegen hat. Einparametrige Geradenmannigfaltigkeiten (Regelflächen) lassen sich darin als Kurven und Geradenkongruenzen als zweidimensionale Flächen auffassen. Die vierparametrige Geradenmenge des reellen projektiven dreidimensionalen Raumes ist in diesem Modell eine Quadrik vom Index 2 in einem reellen projektiven fünfdimensionalen Raum, die so genannte KLEINsche Hyperquadrik. Der Modellwechsel wird durch die KLEINsche Abbildung vollzogen. / In the available work line congruences of the projectively extended three-dimensional euclidean space will be analysed. Following to J. PLÜCKER lines can be seen as basic elements of an line space like in the same way points in a point-space. Taking this fact in consideration a "natural" handling with line congruences might be interesting and reasonable. A special detail in the thesis is the question to minimal congruences in the set of lines of the projectively extended euclidean three-space. It can also be seen as an analogous problem in the geometry of lines which can be find in the differential geometry of surfaces. In this case the line congruences are similar to the surfaces of the three-dimensional (point-)space. The phrase "minimal" means in the line space the connection to the minimal surfaces in the differential geometry. These questions offer in line geometry demonstrative interpretation possibilities if a point-model in the line space exists. One-parameter manifolds of lines (rule surfaces) can be seen in this ambiance as curves and line congruences as two dimensional surfaces. The four-parametric set of lines in the projectively extended three-dimensional euclidian space is in this model a quadric of the index 2 in a real projective five-dimensional space, the so called KLEIN-quadric. The changing of the model is managed by the KLEIN-mapping.

In der vorliegenden Arbeit werden Geradenkongruenzen des projektiv abgeschlossenen dreidimensionalen euklidischen Raumes differentialgeometrisch untersucht. Nach J. PLÜCKER lassen sich Geraden in gleicher Weise als Grundelemente eines Geradenraumes auffassen wie die Punkte in einem Punktraum. Unter Beachtung dieser Überlegung scheint eine "natürliche" Behandlung der Geradenkongruenzen interessant und sinnvoll. Sie bildet den Gegenstand der vorliegenden Dissertation. Ein besonderes Augenmerk richtet sich dabei auf die Frage nach "kleinsten" Geradenkongruenzen ("Minimalkongruenzen") in der Geradenmenge des reellen projektiv abgeschlossenen dreidimensionalen euklidischen Raumes. Dahinter verbirgt sich eine gewisse Analogiebildung in der Liniengeometrie, die der klassischen Differentialgeometrie entstammt. Die Geradenkongruenzen bilden hierbei das liniengeometrische Analogon zu den Flächen des dreidimensionalen (Punkt-)Raumes. Das Wort "Kleinste" stellt im Geradenraum einen Bezug zu den Minimalflächen in der Differentialgeometrie her. Nun gestatten diese Fragestellungen in der Liniengeometrie eine anschauliche Interpretation, sobald man ein Punktmodell des Geradenraumes vorliegen hat. Einparametrige Geradenmannigfaltigkeiten (Regelflächen) lassen sich darin als Kurven und Geradenkongruenzen als zweidimensionale Flächen auffassen. Die vierparametrige Geradenmenge des reellen projektiven dreidimensionalen Raumes ist in diesem Modell eine Quadrik vom Index 2 in einem reellen projektiven fünfdimensionalen Raum, die so genannte KLEINsche Hyperquadrik. Der Modellwechsel wird durch die KLEINsche Abbildung vollzogen. / In the available work line congruences of the projectively extended three-dimensional euclidean space will be analysed. Following to J. PLÜCKER lines can be seen as basic elements of an line space like in the same way points in a point-space. Taking this fact in consideration a "natural" handling with line congruences might be interesting and reasonable. A special detail in the thesis is the question to minimal congruences in the set of lines of the projectively extended euclidean three-space. It can also be seen as an analogous problem in the geometry of lines which can be find in the differential geometry of surfaces. In this case the line congruences are similar to the surfaces of the three-dimensional (point-)space. The phrase "minimal" means in the line space the connection to the minimal surfaces in the differential geometry. These questions offer in line geometry demonstrative interpretation possibilities if a point-model in the line space exists. One-parameter manifolds of lines (rule surfaces) can be seen in this ambiance as curves and line congruences as two dimensional surfaces. The four-parametric set of lines in the projectively extended three-dimensional euclidian space is in this model a quadric of the index 2 in a real projective five-dimensional space, the so called KLEIN-quadric. The changing of the model is managed by the KLEIN-mapping.