In architectural freeform design, the relation between shape and fabrication poses new challenges and requires more sophistication from the underlying geometry. The new concept of conical meshes satisfies central requirements for this application: They are quadrilateral meshes with planar faces, and therefore particularly suitable for the design of freeform glass structures. Moreover, they possess a natural offsetting operation and provide a support structure orthogonal to the mesh. Being a discrete analogue of the network of principal curvature lines, they represent fundamental shape characteristics. We show how to optimize a quad mesh such that its faces become planar, or the mesh becomes even conical. Combining this perturbation with subdivision yields a powerful new modeling tool for all types of quad meshes with planar faces, making subdivision attractive for architecture design and providing an elegant way of modeling developable surfaces.

/pdf/geometric-modeling-with-conical-meshes-and-developable-1yxp9wwbnw.pdf

Geometric modeling with conical meshes and developable surfaces

Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semi-regular ones, have advantages for many applications, and significant progress was made in quadrilateral mesh generation and processing during the last several years. In this survey we discuss the advantages and problems of techniques operating on quadrilateral meshes, including surface analysis and mesh quality, simplification, adaptive refinement, alignment with features, parametrisation and remeshing.

Quad-Mesh Generation and Processing: A Survey

Building envelope shape design in early stages of the design process: Integrating architectural design systems and energy simulation

We solve the form-finding problem for polyhedral meshes in a way which combines form, function and fabrication; taking care of user-specified constraints like boundary interpolation, planarity of faces, statics, panel size and shape, enclosed volume, and last, but not least, cost. Our main application is the interactive modeling of meshes for architectural and industrial design. Our approach can be described as guided exploration of the constraint space whose algebraic structure is simplified by introducing auxiliary variables and ensuring that constraints are at most quadratic. Computationally, we perform a projection onto the constraint space which is biased towards low values of an energy which expresses desirable "soft" properties like fairness. We have created a tool which elegantly handles difficult tasks, such as taking boundary-alignment of polyhedral meshes into account, planarization, fairing under planarity side conditions, handling hybrid meshes, and extending the treatment of static equilibrium to shapes which possess overhanging parts.

/pdf/form-finding-with-polyhedral-meshes-made-simple-1n7qnrzbmm.pdf

Form-finding with polyhedral meshes made simple

We present a novel method to approximate a freeform shape with a planar quadrilateral (PQ) mesh for modeling architectural glass structures. Our method is based on the study of conjugate direction fields (CDF) which allow the presence of ±κ/4(κ e Z) singularities. Starting with a triangle discretization of a freeform shape, we first compute an as smooth as possible conjugate direction field satisfying the user's directional and angular constraints, then apply mixed-integer quadrangulation and planarization techniques to generate a PQ mesh which approximates the input shape faithfully. We demonstrate that our method is effective and robust on various 3D models.

/pdf/general-planar-quadrilateral-mesh-design-using-conjugate-4oidngfvq9.pdf

General planar quadrilateral mesh design using conjugate direction field

A parametric strategy for free-form glass structures using quadrilateral planar facets

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A parametric strategy for free-form glass structures using quadrilateral planar facets