Algebraic 3D Graphic Statics: reciprocal constructions

TL;DR

This paper introduces an algebraic framework for 3D graphic statics, providing rigorous definitions, numerical solution methods, and insights into form-finding and manipulation of reciprocal polyhedral diagrams.

Contribution

It offers a novel algebraic formulation for 3D graphic statics, enabling precise construction, analysis, and manipulation of reciprocal diagrams with multiple solution techniques.

Findings

Developed equilibrium equations around edges

Provided numerical methods for solving equations

Explored degrees of (in)determinacy and manipulation

Abstract

The recently developed 3D graphic statics (3DGS) lacks a rigorous mathematical definition relating the geometrical and topological properties of the reciprocal polyhedral diagrams as well as a precise method for the geometric construction of these diagrams. This paper provides a fundamental algebraic formulation for 3DGS by developing equilibrium equations around the edges of the primal diagram and satisfying the equations by the closeness of the polygons constructed by the edges of the corresponding faces in the dual/reciprocal diagram. The research provides multiple numerical methods for solving the equilibrium equations and explains the advantage of using each technique. The approach of this paper can be used for compression-and-tension combined form-finding and analysis as it allows constructing both the form and force diagram based on the interpretation of the input diagram. Besides, the paper expands on the geometric/static degrees of (in)determinacies of the diagrams using the algebraic formulation and shows how these properties can be used for the constrained manipulation of the polyhedrons in an interactive environment without breaking the reciprocity between the two.