Algebraic 3D Graphic Statics with Edge and Vertex Constraints: A Comprehensive Approach to Extend the Solution Space for Polyhedral Form-Finding

Built upon a previous algebraic framework for polyhedron-based 3D Graphic Statics (3DGS) that can numerically solve for a variety of dual diagrams given an input force or form diagram, this paper introduces an improved algebraic formulation that integrates edge lengths and vertex location constraints for better control over the results. Those constraints are realized by additional edge and vertex constraint equations to previously established closing equations. The entire system of equations can be solved using the Moore-Penrose inverse (MPI) method, and each solution represents a set of compatible edge lengths for the dual diagram to be constructed. The whole solution space of the equation system provides a wide range of dual diagrams, including forms with both tensile and compressive members, which can be easily explored and was not possible using iterative methods or previous algebraic formulations. This improved formulation has been computationally implemented and released as part of a plug-in software program within the environment of Rhino3D (R) and Grasshopper3D (R), enriching the structural form-finding toolset for designers, engineers, researchers, and educators. The tool's performance and accuracy are demonstrated through a series of comparative studies with iterative methods. Various case studies are also presented to showcase the application of this method.