Calculating the Fold Angles of Any Vertex Roof Using a Spherical Image Technique

Abstract

The shape of a vertex roof is defined by the geometry of its constituent flat facets and the relative angle of folding across them. The spherical image of the roof, originally from Gauss, expresses these properties simultaneously. We present a method for calculating the image properties and thence the shape of any vertex roof based on subdividing the image into an array of suitable spherical triangles. In particular, we introduce a truss representation of the vertex for choosing viable subdivisions of the image, which allow full calculations to be made. Additionally, this allows us to construct generalized closed-form expressions for the fold angles of vertex roofs, presented here for roofs with up to six facets.