While the notion of isometric deformations of surfaces is straightforward for surfaces with Euclidean metric, a corresponding notion in isotropic space has been missing. By making Gauss' Theorema Egregium a necessary condition we develop a sensible notion of isometric surfaces in isotropic space. The well-known examples in Euclidean space, like isometries within the associated family of minimal surfaces, Bour's theorem, and Minding isometries, find their natural analogues in isotropic space. We also include an extensive treatment of infinitesimal flexibility, or infinitesimal deformation, of surfaces. We prove results for the isotropic displacement diagrams in analogy to its well-known counterparts in Euclidean space culminating in the existence of an isotropic Darboux wreath consisting of six surfaces. We show several interesting relations for special parametrizations involving Koenigs and Voss nets of smooth and discrete surfaces within the Darboux wreath and we encounter surfaces of constant Gaussian and mean curvature. At several occasions, we point to connections to statics as the isotropic space is a natural language to describe the Airy stress function.

A self-Airy membrane shell is a special type of shell structure whose shape coincides with the shell's Airy stress surface. It provides the convenient property that any polyhedral discretization of such a surface will automatically generate a mesh in funicular equilibrium. A self-Airy shell designed for a uniform vertical load would simply have a constant isotropic Gaussian curvature. However, a challenge in implementing a self-Airy shell in architecture is the lack of a design method, especially in designing unreinforced boundaries. Those are singular planar curves, where the two principal curvatures approach 0 and infinity individually. This paper presents methods for designing unreinforced boundaries of self-Airy shells, including both smooth and discrete methods. These methods work for both positively and negatively curved surfaces. The proposed methods work linearly without iteration. The preliminary results show that the seemingly very restrictive conditions admit a variety of non-trivial surfaces.

We study analogs of planar-quadrilateral meshes in Laguerre sphere geometry and the approximation of smooth surfaces by them. These new Laguerre meshes can be viewed as watertight surfaces formed by planar quadrilaterals, strips of right circular cones (representing the edges), and spherical panels. In the smooth limit, we obtain an analog of conjugate nets in Laguerre geometry, which we call Laguerre conjugate nets with respect to an attached sphere congruence. We introduce the notion of Laguerre conjugate directions, provide a method for computing them, and apply them to approximate surfaces by L-meshes with prescribed radii of spherical panels.

Many problems in Euclidean geometry, arising in computational design and fabrication, amount to a system of constraints, which is challenging to solve. We suggest a new general approach to the solution, which is to start with analogous problems in isotropic geometry. Isotropic geometry can be viewed as a structure-preserving simplification of Euclidean geometry. The solutions found in the isotropic case give insight and can initialize optimization algorithms to solve the original Euclidean problems. We illustrate this general approach with three examples: quad-mesh mechanisms, composite asymptotic-geodesic gridshells, and asymptotic gridshells with constant node angle.

We study flexible polyhedral nets in isotropic geometry. This geometry has a degenerate metric, but there is a natural notion of flexibility. We study infinitesimal and finite flexibility, and classify all finitely flexible polyhedral nets of arbitrary size. We show that there are just two classes, in contrast to Izmestiev's rather involved classification in Euclidean geometry, for size 3x3 only. Using these nets to initialize the optimization algorithms, we turn them into approximate Euclidean mechanisms. We also explore the smooth versions of these classes.

Complex architectural structures may be built in a simple and cost-effective way if their geometry respects the fabrication constraints. Examples of such structures are provided by gridshells that are built from straight and flat slats which are bent on site so that they become tangential or normal to the design surface. Tangential slats follow geodesic curves on the surface, while normal slats are attached along asymptotic curves. Extending work by Frei Otto, Julius Natterer and others, who placed the slats tangentially, Eike Schling proposed structures which also contain slats normal to the reference surface. In the present paper we address those gridshells that consist of three families of bent elements, either tangential or normal to the design surface, and are arranged in a triangular web. We propose algorithms for the computational design of such webs that start from a boundary strip and propagate it, partially under additional guidance, to an entire web.

We introduce a novel class of quadrilateral gridshell structures in axial force equilibrium where rods are aligned symmetrically with the principal stress directions of a limit membrane shell. These structures exhibit a distinctive property where the axial forces in the four connected rods at each node are nearly equal. This characteristic enables a more uniform distribution of forces within the structure, particularly in cases where stresses exhibit significant anisotropy. In contrast, conventional gridshells often result in numerous rods remaining nearly unloaded in such scenarios. We begin by studying the equilibrium of rod networks that are symmetric to principal stress directions. Next, we explore the geometric properties of these networks in relation to the isotropic geometry of their Airy stress surface. We introduce then a computational pipeline for designing principal symmetric structures through quadrilateral remeshing of a surface in membrane equilibrium and subsequent optimization. Finally, we present some of the achieved results.

We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions (the latter two cases only in isotropic geometry). We use the interlacing of various methods of differential geometry, including line geometry and Lie sphere geometry, ordinary differential equations, and elementary algebraic geometry.

In this paper, we introduce and study a remarkable class of mechanisms formed by a 3 × 3 arrangement of rigid quadrilateral faces with revolute joints at the common edges. In contrast to the well-studied Kokotsakis meshes with a quadrangular base, we do not assume the planarity of the quadrilateral faces. Our mechanisms are a generalization of Izmestiev’s orthodiagonal involutive type of Kokotsakis meshes formed by planar quadrilateral faces. The importance of this Izmestiev class is undisputed as it represents the first known flexible discrete surface – T-nets – which has been constructed by Graf and Sauer. Our algebraic approach yields a complete characterization of all flexible 3 × 3 quad meshes of the orthodiagonal involutive type up to some degenerated cases. It is shown that one has a maximum of 8 degrees of freedom to construct such mechanisms. This is illustrated by several examples, including cases which could not be realized using planar faces. We demonstrate the practical realization of the proposed mechanisms by building a physical prototype using stainless steel. In contrast to plastic prototype fabrication, we avoid large tolerances and inherent flexibility.

This paper provides computational tools for the modeling and design of quad mesh mechanisms, which are meshes allowing continuous flexions under the assumption of rigid faces and hinges in the edges. We combine methods and results from different areas, namely differential geometry of surfaces, rigidity and flexibility of bar and joint frameworks, algebraic geometry, and optimization. The basic idea to achieve a time-continuous flexion is timediscretization justified by an algebraic degree argument. We are able to prove computationally feasible bounds on the number of required time instances we need to incorporate in our optimization. For optimization to succeed, an informed initialization is crucial. We present two computational pipelines to achieve that: one based on remeshing isometric surface pairs, another one based on iterative refinement. A third manner of initialization proved very effective: We interactively design meshes which are close to a narrow known class of flexible meshes, but not contained in it. Having enjoyed sufficiently many degrees of freedom during design, we afterwards optimize towards flexibility.

Meshes with spherical faces and circular edges are an attractive alternative to polyhedral meshes for applications in architecture and design. Approximation of a given surface by such a mesh needs to consider the visual appearance, approximation quality, the position and orientation of circular intersections of neighboring faces and the existence of a torsion free support structure that is formed by the planes of circular edges. The latter requirement implies that the mesh simultaneously defines a second mesh whose faces lie on the same spheres as the faces of the first mesh. It is a discretization of the two envelopes of a sphere congruence, i.e., a two-parameter family of spheres. We relate such sphere congruences to torsal parameterizations of associated line congruences. Turning practical requirements into properties of such a line congruence, we optimize line and sphere congruence as a basis for computing a mesh with spherical triangular or quadrilateral faces that approximates a given reference surface.

Motivated by the emergence of rough surfaces in various areas of design, we address the computational design of triangle meshes with controlled roughness. Our focus lies on small levels of roughness. There, roughness or smoothness mainly arises through the local positioning of the mesh edges and faces with respect to the curvature behavior of the reference surface. The analysis of this interaction between curvature and roughness is simplified by a 2D dual diagram and its generation within so-called isotropic geometry, which may be seen as a structure-preserving simplification of Euclidean geometry. Isotropic dihedral angles of the mesh are close to the Euclidean angles and appear as Euclidean edge lengths in the dual diagram, which also serves as a tool for visualization and interactive local design. We present a computational framework that includes appearance-aware remeshing, optimization-based automatic roughening, and control of dihedral angles.

We introduce a new class of quadrilateral gridshell structures in axial force equilibrium where beams run symmetrically to the principal stress directions of their limit membrane shell. This kind of structures have the property that, at each node, the axial forces in the four connected beams are approximately equal. This allows for a more homogeneous distribution of forces in the structure, particularly in shapes where stresses are significantly anisotropic, in which case a conventional gridshell typically results in numerous beams remaining nearly unloaded. In this work, we first discuss the properties of principal symmetric structures and evaluate their advantages relative to other types of gridshells. We introduce then a computational pipeline for the design of such structures based on a quadrilateral remeshing and a subsequent optimization, and show some results.

We introduce the new concept of C-mesh to capture kinetic structures that can be deployed from a collapsed state. Quadrilateral C-meshes enjoy rich geometry and surprising relations with differential geometry: A structure that collapses onto a flat and straight strip corresponds to a Chebyshev net of curves on a surface of constant Gaussian curvature, while structures collapsing onto a circular strip follow surfaces which enjoy the linear-Weingarten property. Interestingly, allowing more general collapses actually leads to a smaller class of shapes. Hexagonal C-meshes have more degrees of freedom, but a local analysis suggests that there is no such direct relation to smooth surfaces. Besides theory, this paper provides tools for exploring the shape space of C-meshes and for their design. We also present an application for freeform architectural skins, namely paneling with spherical panels of constant radius, which is an important fabrication-related constraint.

Discrete surfaces with spherical faces are interesting from a simplified manufacturing viewpoint when compared to other double curved face shapes. Furthermore, by the nature of their definition they are also appealing from the theoretical side leading to a Möbius invariant discrete surface theory. We therefore systematically describe so called sphere meshes with spherical faces and circular arcs as edges where the Möbius transformation group acts on all of its elements. Driven by aspects important for manufacturing, we provide the means to cluster spherical panels by their radii. We investigate the generation of sphere meshes which allow for a geometric support structure and characterize all such meshes with triangular combinatorics in terms of non-Euclidean geometries. We generate sphere meshes with hexagonal combinatorics by intersecting tangential spheres of a reference surface and let them evolve - guided by the surface curvature - to visually convex hexagons, even in negatively curved areas. Furthermore, we extend meshes with circular faces of all combinatorics to sphere meshes by filling its circles with suitable spherical caps and provide a remeshing scheme to obtain quadrilateral sphere meshes with support structure from given sphere congruences. By broadening polyhedral meshes to sphere meshes we exploit the additional degrees of freedom to minimize intersection angles of neighboring spheres enabling the use of spherical panels that provide a softer perception of the overall surface.

We address the computational design of architectural structures which are based on a grid of intersecting beams that are aligned with the parameter lines of a quad mesh. While previous work mainly put a planarity constraint onto the faces of the mesh, we focus on the planarity of long-range supporting beams which follow selected polylines in the underlying mesh. In addition to that, we impose further constraints including planarity of faces, right node angles and static equilibrium, and discuss in which way these may be combined. Some of the studied meshes are discrete counterparts of certain well-known surfaces in classical geometry, whose knowledge is helpful for initializing the proposed optimization algorithms.

The property of a surface being developable can be expressed in different equivalent ways, by vanishing Gauss curvature, or by the existence of isometric mappings to planar domains. Computational contributions to this topic range from special parametrizations to discrete-isometric mappings. However, so far a local criterion expressing developability of general quad meshes has been lacking. In this paper, we propose a new and efficient discrete developability criterion that is applied to quad meshes equipped with vertex weights, and which is motivated by a well-known characterization in differential geometry, namely a rank-deficient second fundamental form. We assign contact elements to the faces of meshes and ruling vectors to the edges, which in combination yield a developability condition per face. Using standard optimization procedures, we are able to perform interactive design and developable lofting. The meshes we employ are combinatorially regular quad meshes with isolated singularities but are otherwise not required to follow any special curves on a developable surface. They are thus easily embedded into a design workflow involving standard operations like remeshing, trimming, and merging operations. An important feature is that we can directly derive a watertight, rational bi-quadratic spline surface from our meshes. Remarkably, it occurs as the limit of weighted Doo-Sabin subdivision, which acts in an interpolatory manner on contact elements.

The geometry of webs has been investigated over more than a century driven by still open problems. In our paper we contribute to extending the knowledge on webs from the perspective of the geometry of webs on surfaces in three dimensional space. Our study of AGAG-webs is motivated by architectural applications of gridshell structures where four families of manufactured curves on a curved surface are realizations of asymptotic lines and geodesic lines. We describe all discrete AGAG-webs in isotropic space and propose a method to construct them. Furthermore, we prove that some sub-nets of an AGAG-web are timelike minimal surfaces in Minkowski space and can be embedded into a one-parameter family of discrete isotropic Voss nets.

Straight flat strips of inextensible material can be bent into curved strips aligned with arbitrary space curves. The large shape variety of these so-called rectifying strips makes them candidates for shape modeling, especially in applications such as architecture where simple elements are preferred for the fabrication of complex shapes. In this paper, we provide computational tools for the design of shapes from rectifying strips. They can form various patterns and fulfill constraints which are required for specific applications such as gridshells or shading systems. The methodology is based on discrete models of rectifying strips, a discrete level-set formulation and optimization-based constrained mesh design and editing. We also analyse the geometry at nodes and present remarkable quadrilateral arrangements of rectifying strips with torsion-free nodes.

In this article, 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 manufactured on the ground without scaffolding and put into place as a whole. Further, the plane it is contained in serves as part of a so-called support structure. We discuss design of meshes under the requirement that one half of mesh polylines are planar (“P meshes”), and we also investigate the geometry and design of meshes where all polylines enjoy this property (“PP meshes”). We work in the space of planes and with appropriate transformations of that space. We also incorporate further properties relevant for architectural design, such as near-rectangular panels and repetitive nodes. We provide geometric insights, give explicit constructions, and show an approach to geometric modeling of both P meshes and PP meshes, in particular, the case of nearly rectangular panels.