**General information about the group, its vision and goals**

The research group on Computational Design and Fabrication is a reaction to the need for research at the interface of Technology and Design, two strongly coupled areas which are essential for the progress in modern societies. We perform cutting edge research in an area that involves Mathematics, Computer Science, Architecture and Design. Hence, our multidisciplinary group is composed of experts from all these fields. Unlike other groups working in a similar direction, we are probably more strongly rooted in Mathematics, especially in Geometry. We try to achieve our goals within the right mathematical framework. This continues to result in surprising new applications of classical geometry, but also in the development of new geometric concepts and structures, for example in Discrete Differential Geometry. Our research delivered solutions of practical problems, but also attracted the attention of researchers in pure geometry. We aim at a fruitful interplay of theory and practice.

Our current research evolves around the following projects:

**Projects in detail**

**1. Architectural Geometry**

**Why dealing with architecture?** Our research in architectural geometry is motivated by a trend in contemporary architecture towards geometrically complex structures, especially freeform shapes. Mastering this new type of geometric complexity on the big architectural scale is a big challenge. 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 are far from straightforward and 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. 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 would be powerless without geometric understanding.

**Main achievements.** Helmut Pottmann’s groups at KAUST and at TU Wien produced a series of key contributions to Architectural Geometry, starting with the invention of conical meshes and the realization of the close connection between Discrete Differential Geometry and the fabrication of freeform structures with planar quadrilateral panels [1]. We identified new relations between meshes with various types of discrete offsets and multilayer structures and torsion-free support structures [3], and investigated a way of accessing the shape space of constrained meshes [6]. A contribution towards optimal paneling of freeform surfaces [4] turned out to be of great value for a series of large scale and prestigious architectural projects. More recently, we put our focus on the integration of aspects of statics and structural analysis into design tools [8,13]. For more details, we refer to the survey [10]. Our monograph on Architectural Geometry [2] is widely recognized as the leading text in the field.

**Impact in architecture.** Our research found a lot of attention in the architectural community and by construction companies. This led to the foundation of a company, Evolute (www.evolute.at) which specializes in software development and consulting for mastering geometric complexity in architecture and manufacturing.

Figure 1: Projects by Evolute GmbH, Vienna, a company which evolved from our research in architectural geometry: The Eiffel Tower Pavilions (left) and the Yas Island Marina Hotel in Abu Dhabi (right).

**Ongoing and Future Research.** A major goal is the development of novel digital design tools which incorporate key aspects of function and fabrication. Unfortunately, this results in constraints which are especially hard to grasp for a user who lacks a deep understanding of the underlying principles. Hence, we are interested in interactive tools which require less understanding of the designer than the currently available ones.

We are also very much interested in architectural structures from simple elements. In most cases, this research leads to new results in Discrete Differential Geometry. In a recent paper [16], we showed how to arrange rectangles in a checkerboard pattern to obtain shapes which approximate curved surfaces (Fig. 2). This led to a new type of discrete surfaces which is particularly easy to deal with from a computational perspective. Applications include novel computational tools for surface deformation via pure bending, paneling of architectural facades and modeling with developable surfaces [20,23]. We also provided a careful study of architectural structures from planar long range beams and further constraints such as planar panels, controlled node angles and static equilibrium [28,30].

Figure 2 Left: Structure from glass rectangles and nearly rectangular white planar panels. Right: the checkerboard approach yields efficient algorithms for surface bending and paneling solutions for architectural skins. One needs only very few molds when panels can be bent after being formed over a mold (panels with the same color are formed over the same mold; white = bending a flat sheet).

Another important research direction is the combination of geometric design and structural analysis as well as minimal material usage (see Project 3).

**2. Fabrication-aware design and computational fabrication**

**Importance of the topic. **Materials which bend easily, but hardly stretch, such as paper, sheet metal and certain types of plastics, naturally assume shapes which are known as developable surfaces. Due to the importance of these surfaces in design and fabrication, a lot of research has been devoted to them. Developable surface modeling is still an active area of research. Most CAD systems are very weak when it comes to modeling with developable surfaces. Thus, there is a demand in industry to develop better design tools for modeling with developable surfaces, but also beyond that. There is a wide class of materials which allow for a little bit of stretching. Glass, wood, leather, and other materials fall into this category. Geometrically, this leads into the unexplored area of nearly developable surfaces. Overall, we want to develop design tools which do not just generate beautiful shapes, but incorporate the physical behavior of the material and key aspects of the fabrication process. This allows one to streamline the process from design to production and avoids costly feedback loops between design, engineering, and fabrication.

The fabrication of many products requires the production of molds, which is mostly done by CNC machining. Despite a huge amount of literature on algorithms for CNC machining, the used algorithms are still quite simple and do not exploit the many degrees of freedom in this process. Hence, we want to develop geometric computing solutions for increased efficiency of CNC machining. Likewise, we are interested in geometric computing for additive manufacturing.

**Main achievements.** We have developed an interactive design tool for modeling developable NURBS surfaces [12] and with quad meshes [20]. Another remarkable way of getting curved shapes from flat sheets is the use of curved folds. We made a remarkable contribution to this area, linking it to discrete differential geometry and solving the hard problem of approximating a target shape by one achieved from a flat sheet with curved folds [17]; see Fig. 3.

In the area of geometric computing for CNC machining our focus is on 5-axis machining. This includes tool selection and tool motion planning. Our contributions have mostly dealt with the so-called finishing process where one aims at a high surface quality through a reduction of the scallop-heights, which appear between neighboring tool paths. The key is a good contact between cutter and target surface [9,14,25].

**Ongoing and Future Research.** In extension of our research on interactive design of modeling developable surfaces, we are working on new methods for designing with materials that are easily bent, but also allow for a bit of stretching. Glass, wood, leather, and other materials fall into this category. Geometrically, this leads into the unexplored area of nearly developable surfaces, where we are currently still lacking useful geometric insight and practical algorithms.

We are working on improved algorithms for 5-axis CNC machining and are also interested in related path planning algorithms for additive manufacturing processes.

**3. Geometry and mechanics**

**Motivation.** When one deals with architectural structures, issues of statics and structural analysis play an important role. For our research in material-aware design, one must couple geometry and the physical material behavior. Since we are highly interested in a strong theoretical background, we aim at approaches and theories, where geometry and mechanics are paired in a natural way.

**Main achievements.** Our first work in this direction dealt with the design of self-supporting surfaces based on the so-called thrust network method and on novel insights into the differential geometric background [7]. The latter has been of great importance for the problem of representing a self-supporting surface by a structure from planar quads which is in static equilibrium. Such structures are of great interest in architectural geometry. An interactive form-finding tool for their direct generation has been developed with C. Tang at KAUST [8]. Figure 4 shows a recent extension of that work, where we combine the classical work of Maxwell, Michell and Airy with differential-geometric considerations and obtain a geometric understanding of “optimality” of surface-like lightweight structures [13].

**Ongoing and Future Research.** We continue our research on the geometry of equilibrium forces and its direct connection to design. Our research on material-aware modeling includes a thorough study of nearly developable surfaces and efficient ways for computing them. We are also interested in new materials which exhibit a close relation to geometry and mechanics [26]. This is related to folding patterns, including those with cuts [22].

**4. Discrete differential geometry motivated by applications**

**Motivation.** Differential geometry provides fundamental methodology for analysis and understanding of curved geometries (curves, surfaces, manifolds in higher dimensions). However, in its classical form it requires differentiable geometry representations, which are rarely available in practice. In most applications, one works with different kinds of meshes or even just point clouds. Discrete differential geometry (DDG) is a modern branch of mathematics which extends the classical theory of differential geometry to non-differentiable representations such as meshes. Here, one tries to discretize the theory, not the equations, and to maintain fundamental invariances and structures. DDG has numerous applications, for example in physical simulation, geometry processing and architectural geometry.

**Main achievements. **Our path towards DDG led through the solution of a fundamental problem in Architectural Geometry, namely representing a freeform shape by a structure from planar quadrilateral panels. Such a planar quad mesh or Q-net is a discrete analogue of a conjugate surface parametrization. This insight is essential for successfully re-meshing a given shape with a Q-net. In architecture, nearly rectangular panels are often desired. This leads to discrete principal parameterizations, where we invented one of the two major types, the so-called conical meshes [1]. Conical meshes possess offsets at a constant face-face distance, which leads to multi-layer structures and so-called torsion-free support structures (Fig. 5, left). More generally, we studied meshes with various types of offsets [3], which led to the formulation of a new curvature theory for polyhedral meshes based on parallel meshes [5]. Conical meshes are a concept of Laguerre sphere geometry and the same is true for meshes which possess offsets at a constant edge-edge distance (Fig. 5, middle).

Our curvature theory for discrete surfaces turned out to be very useful for a geometric interpretation of static equilibrium in our study of self-supporting surfaces [7]. We have been able to formulate the force balance of a mesh under vertical nodal loads in a purely differential geometric way involving the discrete Airy stress surface of the horizontal force distribution and discrete curvatures. This provides the key for re-meshing a self-supporting surface with a Q-net in static equilibrium.

Again, motivated by architecture, we became interested in polyhedral surfaces which are more general than triangle and quad meshes. These polyhedral patterns [11] exhibit an interesting phenomenon: some of the patterns adapt the shapes of tiles according to the curvature of an underlying smooth reference surface (Fig. 6, left) while others lack this behavior (Fig. 6, right). However, this second type leads to a rougher surface. This motivated a study of smoothness concepts for polyhedral surfaces, both from a practical perspective and a fundamental geometric viewpoint [15].

Our work on material minimizing structures [13] included theoretical insight on the so-called Airy stress potential. While this is realized in a simple non-Euclidean geometry, the Euclidean counterpart is of high interest as well: We could solve the problem of finding the visually smoothest polyhedral surfaces [15] under various constraints (Fig. 7). This research also let to a new type of geometric energy (total absolute curvature) whose minimization can be performed computationally, but where we are still lacking a solid theoretical understanding.

Smoothness is an important issue in discrete differential geometry. The intentional lack of smoothness completely changes the picture, as has been shown recently in our work on curve pleated structures [17]. Conical meshes appear there as well, but now no longer discretize principal curvature parameterizations, but an important class of curved folded structures.

Another very recent contribution towards DDG is the study of discrete geodesic parallel coordinates [18]. They provide a simple computational access to families of shortest paths on arbitrary surfaces and to surfaces which are isometric to rotational surfaces. Applications are shown in Figure 8.

**Ongoing and Future Research.** Most of our ongoing research is in some way related to DDG. We recently extended the classical concept of asymptotic nets to principal symmetric nets [19], which provides computational access to certain types of Weingarten surfaces [27]. Those and more general Weingarten surfaces are useful for paneling in architecture [24]. We are also working on non-smooth structures, their relation to novel materials [26], and on transformable design. The latter concern structures which are mechanisms and link motion design with art and architecture. In the spirit of a fruitful interplay of theory and applications, we also pursue research in pure geometry, inspired by applications [31].

**References**

- Y. Liu, H. Pottmann, J. Wallner, Y.-L. Yang, and W. Wang. Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graphics, 25(3):681-689, 2006. Proc. SIGGRAPH.
- H. Pottmann, A. Asperl, M. Hofer, and A. Kilian. Architectural Geometry. Bentley Institute Press, 2007.
- H. Pottmann, Y. Liu, J.Wallner, A. Bobenko, and W.Wang. Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics, 26(3):#65, 1-11, 2007. Proc. SIGGRAPH.
- M. Eigensatz, M. Kilian, A. Schiftner, N. Mitra, H. Pottmann, and M. Pauly. Paneling architectural freeform surfaces. ACM Trans. Graphics, 29(4):#45,1-10, 2010. Proc. SIGGRAPH.
- Bobenko, H. Pottmann, and J. Wallner. A curvature theory for discrete surfaces based on mesh parallelity. Math. Annalen, 348:1-24, 2010.
- Y. Yang, Y. Yang, H. Pottmann, and N. Mitra. Shape space exploration of constrained meshes. ACM Trans. Graphics, 30:#124,1-11, 2011. Proc. SIGGRAPH Asia.
- E. Vouga, M. Höbinger, J. Wallner, and H. Pottmann. Design of self-supporting surfaces. ACM Trans. Graphics, 31:#87,1-11, 2012. Proc. SIGGRAPH.
- C. Tang, X. Sun, A. Gomes, J. Wallner, and H. Pottmann. Form-finding with polyhedral meshes made simple. ACM Trans. Graphics, 33(4):#70,1-9, 2014. Proc. SIGGRAPH.
- Y.-J. Kim, G. Elber, M. Barton, and H. Pottmann. Precise gouging-free tool orientations for 5-axis CNC machining. Computer Aided Design, 58:220-229, 2015.
- H. Pottmann, M. Eigensatz, A. Vaxman, and J. Wallner. Architectural geometry. Computers and Graphics, 47:145-164, 2015.
- C. Jiang, C. Tang, A. Vaxman, P. Wonka, and H. Pottmann. Polyhedral patterns. ACM Trans. Graphics, 34(6):#172,1-12, 2015. Proc. SIGGRAPH Asia.
- C. Tang, P. Bo, J. Wallner, and H. Pottmann. Interactive design of developable surfaces. ACM Trans. Graphics, 35(2):#12,1-12, 2016.
- M. Kilian, D. Pellis, J. Wallner, and H. Pottmann. Material-minimizing forms and structures. ACM Trans. Graphics, 36(6):article 173, 2017. Proc. SIGGRAPH Asia.
- P. Bo, M. Barton, and H. Pottmann. Automatic fitting of conical envelopes to freeform surfaces for flank CNC machining. Computer Aided Design, 91:84-94, 2017.
- D. Pellis, M. Kilian, F. Dellinger, J. Wallner, and H. Pottmann. Visual smoothness of polyhedral surfaces. ACM Trans. Graphics, 38(4):260:1-260:11, 2019. Proc. SIGGRAPH.
- C. Jiang, C.-H. Peng, P. Wonka, and H. Pottmann. Checkerboard patterns with black rectangles. ACM Trans. Graphics, 38(6):171:1-171:13, 2019. Proc. SIGGRAPH Asia.
- C. Jiang, K. Mundilova, F. Rist, J. Wallner, and H. Pottmann. Curve-pleated structures. ACM Trans. Graphics, 38(6):169:1-169:13, 2019. Proc. SIGGRAPH Asia.
- H. Wang, D. Pellis, F. Rist, H. Pottmann, and C. Müller. Discrete geodesic parallel coordinates. ACM Trans. Graphics, 38(6):173:1-173:13, 2019. Proc. SIGGRAPH Asia.
- D. Pellis, H. Wang, M. Kilian, F. Rist, H. Pottmann, and C. Müller. Principal symmetric meshes. ACM Trans. Graphics, 39(4):127:1-127:17, 2020. Proc. SIGGRAPH.
- C. Jiang, C.Wang, F. Rist, J.Wallner, and H. Pottmann. Quad-mesh based isometric mappings and developable surfaces. ACM Trans. Graphics, 39(4):128:1-128:13, 2020. Proc. SIGGRAPH.
- K. Gavriil, R. Guseinov, J. Perez, D. Pellis, P. Henderson, F. Rist, H. Pottmann, and B. Bickel. Computational design of cold bent glass facades. ACM Trans. Graphics 39(6):208:1-208:16, 2020. Proc. SIGGRAPH Asia.
- C. Jiang, F. Rist, H. Pottmann, and J. Wallner. Freeform quad-based kirigami. ACM Trans. Graphics, 39(6):209:1-209:11, 2020. Proc. SIGGRAPH Asia.
- C. Jiang, H. Wang, V. C. Inza, F. Dellinger, F. Rist, J. Wallner, and H. Pottmann. Using isometries for computational design and fabrication. ACM Trans. Graphics, 40(4):42:1-42:12, 2021. Proc. SIGGRAPH.
- D. Pellis, M. Kilian, H. Pottmann, and M. Pauly. Computational design of Weingarten surfaces. ACM Trans. Graphics, 40(4):114:1-114:11, 2021. Proc. SIGGRAPH.
- M. Bizzarri, M. Barton, F. Rist, O. Sliusarenko, and H. Pottmann. Geometry and tool motion planning for curvature adapted CNC machining. ACM Trans. Graphics, 40(4):180:1-180:16, 2021. Proc. SIGGRAPH.
- C. Jiang, F. Rist, H.Wang, J.Wallner, and H. Pottmann. Shape-morphing mechanical metamaterials. Computer Aided Design, 143, 2022.
- H. Wang and H. Pottmann. Characteristic parameterizations of surfaces with a constant ratio of principal curvatures. Comp.-Aided Geometric Design, 93, 2022.
- C.Wang, C. Jiang, X. Tellier, J.Wallner, and H. Pottmann. Planar panels and planar supporting beams in architectural structures. ACM Trans. Graphics, 41, 2022.
- E. Schling, H. Wang, S. Hoyer, and H. Pottmann. Designing asymptotic geodesic hybrid gridshells. Computer Aided Design, 152, 2022.
- C. Wang, C. Jiang, H. Wang, X. Tellier, and H. Pottmann. Architectural structures from quad meshes with planar parameter lines. Computer Aided Design, 156, 2023.
- Y. Liu, O. Pirahmad, H. Wang, D. Michels, and H. Pottmann. On helical surfaces with a constant ratio of principal curvatures. Contributions to Algebra and Geometry, 2023.