Form-finding with polyhedral meshes made simple

Abstract

We solve the form-finding problem for polyhedral meshes in a way which combines form, function and fabrication; taking care of userspecified 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.

Interactive Modeling of Architectural Freeform Structures: Combining Geometry with Fabrication and Statics

“…Vouga et al [2012] show how to rationalize any selfsupporting shape by a self-supporting planar quad mesh, but do not directly design it. Form-finding with polyhedral meshes in static equilibrium is the topic of [Tang et al 2014], of which our paper is a follow-up.…”

“…In this way, the planarity constraint is quadratic. This approach motivated the systematic avoidance of constraints of degree ≥ 3 by Tang et al [2014] which forms the basis of the present paper. Extensive comparative tests showed that it clearly outperforms previous work in terms of the combination of speed and high accuracy of constraint satisfaction.…”

“…In § 2, we briefly outline the surprisingly effective method of Tang et al [2014] which is suitable for interactive design of constrained meshes and more general structures. § 3 discusses form-finding with polyhedral meshes.…”

“…For details and other ways to define the weight, e.g. proportional to area as shown by Figure 4a, see [Tang et al 2014].…”

“…Tang et al [2014] describe how to treat constraints of the kind described above (the restriction that constraints should be at most quadratic is not severe, since any equality or inequality expressible in polynomial form can be made quadratic, by introducing products of two existing variables as new variables, until the polynomial degree is down to 2). They describe how to quickly solve the system of constraints, and thus make interactive modeling of constrained geometry possible.…”