a b s t r a c tAround 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural geometry, meanwhile contains a great wealth of individual contributions which are relevant in various fields. For mathematicians, the relation to discrete differential geometry is significant, in particular the integrable system viewpoint. Besides, new application contexts have become available for quite some old-established concepts. Regarding graphics and geometry processing, architectural geometry yields interesting new questions but also new objects, e.g. replacing meshes by other combinatorial arrangements. Numerical optimization plays a major role but in itself would be powerless without geometric understanding. Summing up, architectural geometry has become a rewarding field of study. We here survey the main directions which have been pursued, we show real projects where geometric considerations have played a role, and we outline open problems which we think are significant for the future development of both theory and practice of architectural geometry.

The emergence of large-scale freeform shapes in architecture poses big challenges to the fabrication of such structures. A key problem is the approximation of the design surface by a union of patches, socalled panels, that can be manufactured with a selected technology at reasonable cost, while meeting the design intent and achieving the desired aesthetic quality of panel layout and surface smoothness. The production of curved panels is mostly based on molds.Since the cost of mold fabrication often dominates the panel cost, there is strong incentive to use the same mold for multiple panels. We cast the major practical requirements for architectural surface paneling, including mold reuse, into a global optimization framework that interleaves discrete and continuous optimization steps to minimize production cost while meeting user-specified quality constraints. The search space for optimization is mainly generated through controlled deviation from the design surface and tolerances on positional and normal continuity between neighboring panels. A novel 6-dimensional metric space allows us to quickly compute approximate inter-panel distances, which dramatically improves the performance of the optimization and enables the handling of complex arrangements with thousands of panels. The practical relevance of our system is demonstrated by paneling solutions for real, cutting-edge architectural freeform design projects.

The emergence of large-scale freeform shapes in architecture poses big challenges to the fabrication of such structures. A key problem is the approximation of the design surface by a union of patches, socalled panels, that can be manufactured with a selected technology at reasonable cost, while meeting the design intent and achieving the desired aesthetic quality of panel layout and surface smoothness. The production of curved panels is mostly based on molds.Since the cost of mold fabrication often dominates the panel cost, there is strong incentive to use the same mold for multiple panels. We cast the major practical requirements for architectural surface paneling, including mold reuse, into a global optimization framework that interleaves discrete and continuous optimization steps to minimize production cost while meeting user-specified quality constraints. The search space for optimization is mainly generated through controlled deviation from the design surface and tolerances on positional and normal continuity between neighboring panels. A novel 6-dimensional metric space allows us to quickly compute approximate inter-panel distances, which dramatically improves the performance of the optimization and enables the handling of complex arrangements with thousands of panels. The practical relevance of our system is demonstrated by paneling solutions for real, cutting-edge architectural freeform design projects.

The emergence of large-scale freeform shapes in architecture poses big challenges to the fabrication of such structures. A key problem is the approximation of the design surface by a union of patches, socalled panels, that can be manufactured with a selected technology at reasonable cost, while meeting the design intent and achieving the desired aesthetic quality of panel layout and surface smoothness. The production of curved panels is mostly based on molds.Since the cost of mold fabrication often dominates the panel cost, there is strong incentive to use the same mold for multiple panels. We cast the major practical requirements for architectural surface paneling, including mold reuse, into a global optimization framework that interleaves discrete and continuous optimization steps to minimize production cost while meeting user-specified quality constraints. The search space for optimization is mainly generated through controlled deviation from the design surface and tolerances on positional and normal continuity between neighboring panels. A novel 6-dimensional metric space allows us to quickly compute approximate inter-panel distances, which dramatically improves the performance of the optimization and enables the handling of complex arrangements with thousands of panels. The practical relevance of our system is demonstrated by paneling solutions for real, cutting-edge architectural freeform design projects.

We propose a framework for 3D geometry processing that provides direct access to surface curvature to facilitate advanced shape editing, filtering, and synthesis algorithms. The central idea is to map a given surface to the curvature domain by evaluating its principle curvatures, apply filtering and editing operations to the curvature distribution, and reconstruct the resulting surface using an optimization approach. Our system allows the user to prescribe arbitrary principle curvature values anywhere on the surface. The optimization solves a nonlinear least-squares problem to find the surface that best matches the desired target curvatures while preserving important properties of the original shape. We demonstrate the effectiveness of this processing metaphor with several applications, including anisotropic smoothing, feature enhancement, and multi-scale curvature editing.

Figure 1: Designing caustics: the photograph shows an aluminum piece that was optimized such that, when lit by a simple flashlight from a certain angle, it produces a caustic image of a wave from the Hokusai print "The Great Wave off Kanagawa" shown on the left. The image on the wall is produced by the aluminum surface alone, focusing and diverting light onto the wall. See also the video on lgg.epfl.ch/caustics. This paper describes how to create such controlled caustic patterns and explores their application in architectural design.

We present a geometry processing framework that allows direct manipulation or preservation of positional, metric, and curvature constraints anywhere on the surface of a geometric model. Target values for these properties can be specified point-wise or as integrated quantities over curves and surface patches embedded in the shape. For example, the user can draw several curves on the surface and specify desired target lengths, manipulate the normal curvature along these curves, or modify the area or principal curvature distribution of arbitrary surface patches. This user input is converted into a set of non-linear constraints. A global optimization finds the new deformed surface that best satisfies the constraints, while minimizing adaptable measures for metric and curvature distortion that provide explicit control of the deformation semantics. We illustrate how this approach enables flexible surface processing and shape editing operations not available in current systems.

Abstract.Paneling an architectural freeform surface refers to an approximation of the design surface by a set of panels that can be manufactured using a selected technology at a reasonable cost, while respecting the design intent and achieving the desired aesthetic quality of panel layout and surface smoothness. Eigensatz and co-workers [Eigensatz et al. 2010] have recently introduced a computational solution to the paneling problem that allows handling large-scale freeform surfaces involving complex arrangements of thousands of panels. We extend this paneling algorithm to facilitate effective design exploration, in particular for local control of tolerance margins and the handling of sharp crease lines. We focus on the practical aspects relevant for the realization of large-scale freeform designs and evaluate the performance of the paneling algorithm with a number of case studies. Figure 1: Given a reference surface (top row), our paneling algorithm produces a rationalization of the the input. The solution (middle row) employs a small set of molds that can be reused for cost-effective panel production (bottom row), while preserving surface smoothness and respecting original design intent. The shown paneling solution (using metal) is 40% cheaper than the production alternative of using custom molds for each individual panels. The solution shown in Figure 10 is 60% cheaper compared to using custom molds for each individual panels.