Geometric Definition of Transformable Surfaces with Homogeneous ABC Reciprocal Meshes

Abstract Reciprocal frame structures are nonhierarchical stable systems formed by elements that are simply connected to each other by tangency with no interpenetration. They are systems composed of elementary spatial self-supporting interlocking units or reciprocal nodes of mutually supporting beams placed in closed structural circuits. Those nodes can be designed as transformable by allowing the modification of the position of the tangency or supporting points. This article addresses the characterization and geometric definition of surfaces of variable geometry (bidimensional objects in three-dimensional space that can transform its geometrical form) by means of meshes, with cylindrical equal beams, arranged by the repetition of a pattern consisting of three transformable nodes of n A, n B, and n C elements around a triangular one (transformable homogeneous ABC reciprocal meshes). The objective of this analytical study is to define the topological and geometrical characteristics of these types of meshes. In addition, the study aims to parameterize them, taking into account their transformability, so they can be applied to the design of surfaces of variable geometry by using a newly created geometric software. To do so, we use analytical and optimization methods to develop a new form-finding approach of transformable reciprocal frame structures with elements of given fixed dimensions and a parameterization of a given mesh based on the moving capability of each of the elements of the system considering them as part of a whole transformable entity and making it possible to transform the geometry of the mesh by modifying the numerical value of the parameters.