  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.
