Minimal Surfaces through Skeletal Graphs (MSSG)

This research develops a computational design method and design interface that allows anyone to generate, explore and visualize minimal surfaces intuitively in real-time instead of having to manipulate mathematical expressions. This method generates a minimal surface in a 3D Euclidean space for any given two sets of curves, which represent the minimal surface medial graph.  

Most methods use mathematical expressions or computational physics simulation with mesh relaxation to generate minimal surfaces in 3D design space, which can be difficult to use when a high degree of formal control and articulation is needed. This newly developed approach uses skeletal graphs to generate, alter, and manipulate minimal surfaces intuitively, enabling their geometric adaptation to functional and program requirements. A skeletal graph of a minimal surface can be viewed as a curve resulting from the shrinking of a minimal surface along the direction of its normal vectors while avoiding any pinching off that would change the topology of the surface. The MSSG algorithm takes two sets of curves, representing “skeletal graphs” of a minimal surface, as inputs and then generates a surface on which any point is equidistant from the closest curve in the two sets. A Marching Cube algorithm (Lorensen and Cline 1987) was employed to create triangle meshes by iterating or “marching” over a uniform grid of cubes. Generating minimal surfaces through MSSG allows for intuitive interaction and a higher degree of control in the design process, as the designer can parametrically manipulate sets of curves that are in a lower dimension than the surface yet generate live updates in the 3D topology of the minimal surfaces. 

Minimal surfaces are advantageous for application in lightweight structures in architecture as weight, and material consumption are reduced to a minimum. Moreover, the aggregation of triply periodic minimal surfaces, which is a continuous non-self-intersecting surface, produces cellular structures like in bone microstructure and can be used as an effective infill between sandwich composite building elements to improve both mechanical and thermal properties.