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Helmut Pottmann

Helmut Pottmann
TU Wien | TU Wien · Faculty of Mathematics and Geoinformation

About

313
Publications
113,667
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11,371
Citations
Citations since 2016
57 Research Items
4557 Citations
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20162017201820192020202120220200400600
20162017201820192020202120220200400600

Publications

Publications (313)
Preprint
Full-text available
There are different ways to capture the property of a surface being developable, i.e., it can be mapped to a planar domain without stretching or tearing. Contributions range from special parametrizations to discrete-isometric mappings. So far, a local criterion expressing the developability of general quad meshes has been lacking. In this paper, we...
Article
In this paper we investigate geometric properties and modeling capabilities of quad meshes with planar faces whose mesh polylines enjoy the additional property of being contained in a single plane. This planarity is a major benefit in architectural design and building construction: if a structural element is contained in a plane, it can be manufact...
Article
Full-text available
Fabrication and assembly of freeform shells can be simplified significantly when controlling the curvature of structural elements during the design phase. This approach has produced fundamental insights to bending-active construction, using the elastic property of elements to form efficient load-bearing structures. This paper is focused on gridshel...
Preprint
Full-text available
We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio $a:=\kappa_1/\kappa_2$ of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. A key ingredient in the successful determination of these surfaces is the proper choice of generating...
Preprint
Full-text available
Design decisions in urban planning have to be made with particular carefulness as the resulting constraints are binding for the whole architectural design that follows. In this context, investigating and optimizing the airflow in urban environments is critical to design comfortable outdoor areas as unwanted effects such as windy areas and the forma...
Article
Motivated by applications in architectural geometry, we study and compute surfaces with a constant ratio of principal curvatures (CRPC surfaces) based on their characteristic parameterizations. For negative Gaussian curvature K, these parameterizations are asymptotic. For positive K they are conjugate and symmetric with respect to the principal cur...
Article
Full-text available
Small-scale cut and fold patterns imposed on sheet material enable its morphing into three-dimensional shapes. This manufacturing paradigm has been receiving much attention in recent years and poses challenges in both fabrication and computation. It is intimately connected with the interpretation of patterned sheets as mechanical metamaterials, typ...
Chapter
Möbius geometry (signature \((n+1,1)\), see Sect. 5.4), hyperbolic Laguerre geometry (signature (n, 2), see Sect. 6.2), elliptic Laguerre geometry (signature \((n+1,1)\), see Sect. 6.3), as well as Euclidean Laguerre geometry (signature (n, 1, 1), see Sect. A.4) can all be lifted to Lie geometry (signature \((n+1,2)\)) using the methods from Chaps....
Chapter
In this section we study the general construction of central projection of a quadric from a point onto its polar hyperplane, see, e.g., [Kle1928, Bla1954, Gie1982]. This leads to a double cover of a Cayley-Klein space in the hyperplane such that the spheres in that Cayley-Klein space correspond to hyperplanar sections of the quadric. Vice versa, a...
Chapter
We begin our general discussions with the introduction of quadrics in projective space, see, e.g., [Kle1928, Bla1954, Gie1982].
Chapter
In this section, as an application of two-dimensional Lie and Laguerre geometry, we present new research results. While incircular nets and their Laguerre geometric generalization to checkerboard incircular nets have been studied in great detail [Böh1970, AB2018, BST2018], we introduce their generalization to Lie geometry, and show that they may be...
Chapter
The primary objects in Möbius geometry are points on \(\mathcal {S}\), which yield a double cover of the points in hyperbolic/elliptic space, and spheres, which yield a double cover of the spheres in hyperbolic/elliptic space. The primary incidence between these objects is a point lying on a sphere.
Chapter
In this section we present two-dimensional Laguerre geometry in an elementary way, without reference to the following more general discussion, which begins in Chap. 3. We first introduce the most basic concepts of these geometries in the Euclidean plane and then turn to the elliptic and hyperbolic plane. The intention here is to enable the reader t...
Chapter
In Klein’s Erlangen program Euclidean and non-Euclidean geometries are considered as subgeometries of projective geometry. Projective models for, e.g., hyperbolic, deSitter, and elliptic space can be obtained by using a quadric to induce the corresponding metric [Kle1928]. In this section we introduce the corresponding general notion of Cayley-Klei...
Conference Paper
We solve the task of representing free forms by an arrangement of panels that are manufacturable by precise isometric bending of surfaces made from a small number of molds. In fact we manage to solve the paneling task with surfaces of constant Gaussian curvature alone. This includes the case of developable surfaces which exhibit zero curvature. Our...
Article
In this paper we study Weingarten surfaces and explore their potential for fabrication-aware design in freeform architecture. Weingarten surfaces are characterized by a functional relation between their principal curvatures that implicitly defines approximate local congruences on the surface. These symmetries can be exploited to simplify surface pa...
Article
CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow...
Conference Paper
Full-text available
The realization of architectural freeform skins is a big challenge, in particular if one desires a smooth appearance and uses curved panels. These have to be brought into shape by special manufacturing technologies, most of which require the costly production of molds. Previous approaches to mold re-use relied on optimization algorithms which play...
Article
Full-text available
CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow...
Article
Full-text available
Kirigami, the traditional Japanese art of paper cutting and folding generalizes origami and has initiated new research in material science as well as graphics. In this paper we use its capabilities to perform geometric modeling with corrugated surface representations possessing an isometric unfolding into a planar domain after appropriate cuts are...
Article
Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces...
Preprint
Cold bent glass is a promising and cost-efficient method for realizing doubly curved glass façades. They are produced by attaching planar glass sheets to curved frames and require keeping the occurring stress within safe limits. However, it is very challenging to navigate the design space of cold bent glass panels due to the fragility of the materi...
Preprint
Full-text available
Classical (Euclidean) Laguerre geometry studies oriented hyperplanes, oriented hyperspheres, and their oriented contact in Euclidean space. We describe how this can be generalized to arbitrary Cayley-Klein spaces, in particular hyperbolic and elliptic space, and study the corresponding groups of Laguerre transformations. We give an introduction to...
Article
Full-text available
The isolines of principal symmetric surface parametrizations run symmetrically to the principal directions. We describe two discrete versions of these special nets/quad meshes which are dual to each other and show their usefulness for various applications in the context of fabrication and architectural design. Our discretization of a principal symm...
Article
Full-text available
We discretize isometric mappings between surfaces as correspondences between checkerboard patterns derived from quad meshes. This method captures the degrees of freedom inherent in smooth isometries and enables a natural definition of discrete developable surfaces. This definition, which is remarkably simple, leads to a class of discrete developabl...
Conference Paper
Full-text available
We present our recent results on the geometric and static optimization of freeform load-bearing architectural skins. An efficient design strategy is the use of planar cladding panels supported by a prismatic framework substructure with optimized static performance. We show how these structures can be achieved discretizing membranes where principal...
Article
Full-text available
We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter rectangles. For discrete planar quadrilateral nets, circular nets, $Q^*$-nets and conical nets we obtain a characteriza...
Article
Full-text available
Motivated by applications in architecture, we study surfaces with a constant ratio of principal curvatures. These surfaces are a natural generalization of minimal surfaces, and can be constructed by applying a Christoffel-type transformation to appropriate spherical curvature line parametrizations, both in the smooth setting and in a discretization...
Preprint
Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address developable surfaces and ruled surfaces. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable su...
Article
Full-text available
In this paper we study pleated structures generated by folding paper along curved creases. We discuss their properties and the special case of principal pleated structures. A discrete version of pleated structures is particularly interesting because of the rich geometric properties of the principal case, where we are able to establish a series of a...
Article
Full-text available
Geodesic parallel coordinates are orthogonal nets on surfaces where one of the two families of parameter lines are geodesic curves. We describe a discrete version of these special surface parameterizations and show that they are very useful for specific applications, most of which are related to the design and fabrication of surfaces in architectur...
Article
Full-text available
Checkerboard patterns with black rectangles can be derived from quad meshes with orthogonal diagonals. First, we present an initial theoretical analysis of these quad meshes. The analysis reveals many possible applications in geometry processing and also motivates the numerical optimization for aesthetic and functional checkerboard pattern design....
Article
Full-text available
Representing smooth geometric shapes by polyhedral meshes can be quite difficult in situations where the variation of edges and face normals is prominently visible. Especially problematic are saddle-shaped areas of the mesh, where typical vertices with six incident edges are ill suited to emulate the more symmetric smooth situation. The importance...
Article
Full-text available
We present a system for automatic reassembly of broken 3D solids. Given as input 3D digital models of the broken fragments, we analyze the geometry of the fracture surfaces to find a globally consistent reconstruction of the original object. Our reconstruction pipeline consists of a graph-cuts based segmentation algorithm for identifying potential...
Chapter
Wood as structural bearing material is often encountered with skepticism and, therefore, it is not used as extensively as its very good material properties would suggest. Beside building physics and construction reasons, the main cause of this skepticism is its quite complex material behavior, which is the reason that design concepts for wood have...
Article
Full-text available
Concrete shells are fascinating structures. Even thin shells can span over large areas without requiring any columns. If a form-defining load case exists, the shape of the shell can be designed to ensure that the forces in the structure are transferred primarily by the membrane action, which leads to an even distribution of the stresses across the...
Conference Paper
Full-text available
Designing freeform architectural surfaces with due regard to economic and feasibility factors is a challenging task. Rationalizing such surfaces by means of quadrilateral meshes following principal curvature lines has proven to be beneficial for manufacturing reasons, such as planar cladding panels and simplified substructure connections. On the ot...
Conference Paper
Full-text available
The fabrication and construction of curved beams along freeform skins pose many challenges related to their individual and complex geometry. One strategy to simplify the fabrication process uses elastic deformation to construct curved beams from flat elements. Controlling the curvature of the design surface and beams has the additional potential to...
Preprint
Motivated by applications in architecture and design, we present a novel method for increasing the developability of a B-spline surface. We use the property that the Gauss image of a developable surface is 1-dimensional and can be locally well approximated by circles. This is cast into an algorithm for thinning the Gauss image by increasing the pla...
Preprint
Full-text available
Motivated by applications in architecture and design, we present a novel method for increasing the developability of a B-spline surface. We use the property that the Gauss image of a developable surface is 1-dimensional and can be locally well approximated by circles. This is cast into an algorithm for thinning the Gauss image by increasing the pla...
Poster
The computation and construction of curved beams along freeform skins pose many challenges. We show how to use surfaces of constant mean curvature (CMC) to compute beam networks with beneficial properties, both aesthetically and from a fabrication perspective. To explore variations of such networks we introduce a new discretization of CMC surfaces...
Article
We present a framework to generate mesh patterns that consist of a hybrid of both triangles and quads. Given a 3D surface, the generated patterns fit the surface boundaries and curvatures. Such regular and near regular triangle-quad hybrid meshes provide two key advantages: first, novel-looking polygonal patterns achieved by mixing different arrang...
Article
Three-dimensional structures in building construction and architecture are realized with conflicting goals in mind: engineering considerations and financial constraints easily are at odds with creative aims. It would therefore be very beneficial if optimization and side conditions involving statics and geometry could play a role already in early st...
Conference Paper
Freeform structures play an important role within contemporary architecture. While there is a wealth of excellent tools for the digital design of free-form geometry, the actual fabrication on the architectural scale is a big challenge. Key issues in this context are free-form surfaces composed of panels which can be manufactured at reasonable cost,...
Article
Full-text available
We propose a new algorithm to detect patches of free-form surfaces that can be well approximated by envelopes of a rotational cone under a rigid body motion. These conical envelopes are a preferable choice from the manufacturing point of view as they are, by-definition, manufacturable by computer numerically controlled (CNC) machining using the eff...
Article
Full-text available
Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the aim of our work is to find and to discuss suitable assessments of smoothness of polyhedral surfaces that only...
Conference Paper
Full-text available
The fairness of meshes which represent geometric shapes is a topic which has been studied extensively and thoroughly. However, the focus in such considerations often is not the mesh itself, but rather the smooth surface approximated by it, and fairness essentially expresses a mesh’s suitability for purposes such as visualization or simulation. This...
Chapter
Full-text available
This study contributes to the discrete differential geometry of triangle meshes, in combination with discrete line congruences associated with such meshes. In particular we discuss when a congruence defined by linear interpolation of vertex normals deserves to be called a ‘normal’ congruence. Our main results are a discussion of various definitions...
Presentation
We introduce a new method that approximates free-form surfaces by envelopes of one-parameter motions of surfaces of revolution. In the context of 5-axis computer numerically controlled (CNC) machining, we propose a flank machining methodology which is a preferable scallop-free scenario when the milling tool and the machined free-form surface meet t...
Article
Full-text available
We present a new approach to geometric modeling with developable surfaces and the design of curved-creased origami. We represent developables as splines and express the nonlinear conditions relating to developability and curved folds as quadratic equations. This allows us to utilize a constraint solver, which may be described as energy-guided proje...
Article
Full-text available
We introduce a new method that approximates free-form surfaces by envelopes of one-parameter motions of surfaces of revolution. In the context of 5-axis computer numerically controlled (CNC) machining, we propose a flank machining methodology which is a preferable scallop-free scenario when the milling tool and the machined free-form surface meet t...