Discrete S-CMC meshes
Soap films subjected to air pressure have a form mathematically referred to as a constant mean curvature surface (CMC). We developed a method to transform CMC smooth surfaces into meshes that have many properties for gridshell fabrication:
- They have quadrangular planar faces and torsion-free nodes.
- They admit an edge offset edges. This property enables a near-perfect alignment of the beams at the node while using only beam cross section.
- Each face has an inscribed circle. As a result, faces are « roughly square », which provides aesthetic value to the mesh, and also minimizes material loss if panels are cut out of a larger sheet.
- These meshes inherit the mechanical properties from the smooth CMC they are derived from.
- There is a sphere packing associated with the vertices.
- Architectural Geometry
Discrete CMC surfaces for doubly-curved building envelopes – X. Tellier et al. (2018)
Xavier Tellier, Cyril Douthe, Olivier Baverel, Laurent Hauswirth
Constant mean curvature surfaces (CMCs) have many interesting properties for use as a form for doubly curved structural envelopes. The discretization of these surfaces has been a focus of research amongst the discrete differential geometry community. Many of the proposed discretizations have remarkable properties for envelope rationalization purposes. However, little attention has been paid to generation methods intended for designers.
This paper proposes an extension to CMCs of the method developed by Bobenko, Hoffmann and Springborn (2006) to generate minimal S-isothermic nets. The method takes as input a CMC (smooth or finely triangulated), remeshes its Gauss map with quadrangular faces, and rebuilds a CMC mesh via a parallel transformation. The resulting mesh is S-CMC, a geometric structure discovered by Hoffmann (2010). This type of mesh have planar quads and offset properties, which are of particular interest in the fabrication of gridshells.
Architectural geometry is an active topic of research for architects, engineers and mathematicians. Some books or articles can be recommended on the topic of polyhedral surfaces.