Algebraic 3D graphic statics: Reciprocal constructions. (March 2019)

• Record Type: Journal Article
• Authors: Hablicsek, Márton; Akbarzadeh, Masoud; Guo, Yi
• Is Part Of: Computer aided design. Volume 108(2019)
• Page Start: 30
• Page End: 41
• Publication Date: 2019-03
• Subjects: Algebraic three-dimensional graphic statics -- Polyhedral reciprocal diagrams -- Geometric degrees of freedom -- Static degrees of indeterminacies -- Tension and compression combined polyhedra -- Constraint manipulation of polyhedral diagrams; Computer-aided design -- Periodicals; Engineering design -- Data processing -- Periodicals; Computer graphics -- Periodicals; Conception technique -- Informatique -- Périodiques; Infographie -- Périodiques; Computer graphics; Engineering design -- Data processing; Periodicals; Electronic journals; 620.00420285
• DOI: 10.1016/j.cad.2018.08.003
• Languages: English
• ISSNs: 0010-4485

Abstract

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 diagrams 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. Highlights: Provides an algebraic formulation to construct polyhedral reciprocal diagrams of 3DGS. Explains the equilibrium equations as closed polygons around the edges of the primal. Provides multiple numerical methods for solving the equilibrium equations. Can construct both form and force diagrams with both tension and compression members. Describes the geometric/static degrees of (in)determinacies of the polyhedral diagrams.