TY - JOUR
T1 - Algebraic 3D graphic statics
T2 - Reciprocal constructions
AU - Hablicsek, Márton
AU - Akbarzadeh, Masoud
AU - Guo, Yi
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
KW - Algebraic three-dimensional graphic statics
KW - Constraint manipulation of polyhedral diagrams
KW - Geometric degrees of freedom
KW - Polyhedral reciprocal diagrams
KW - Static degrees of indeterminacies
KW - Tension and compression combined polyhedra
UR - http://www.scopus.com/inward/record.url?scp=85056894749&partnerID=8YFLogxK
U2 - 10.1016/j.cad.2018.08.003
DO - 10.1016/j.cad.2018.08.003
M3 - Journal article
AN - SCOPUS:85056894749
SN - 0010-4485
VL - 108
SP - 30
EP - 41
JO - CAD Computer Aided Design
JF - CAD Computer Aided Design
ER -