My latest research, co-written with the support of my advisor Professor Alexandra Paio and Professor Adriano Lopes, on Computational Geometry and Parametric Design has just been published with SpringerNature in Nexus Network Journal. Read here: https://rdcu.be/b4APm. The article presents triangulation algorithms, which are based on empirical surveying workflows traditionally used by architects and other building professionals, that may be used to iteratively triangulate polygons. These algorithms may be leveraged to implement an interactive workflow for as-is surveys of room plans or sections that automates the drawing of the polygon.

#### Why is this necessary?

While rooms are tendentially orthogonal, if only one side is not orthogonal to the remaining ones by a slight angular difference, the dimensions of the room sides will not suffice to define the polygon. From these edge lengths it is not possible to infer which side is not orthogonal to the remaining ones. The reason is that all polygons except the triangle are not rigid, so there is an infinite number of polygons for any given sequence of edge lengths. Furthermore, assuming that a room is orthogonal can be misleading. The same edge lengths that define a square room can also define a rhombus. So even if the edge lengths produce a closed orthogonal polygon, there is no definite evidence that the room is in fact orthogonal.

For many applications this precision is not paramount. A contractor may measure the perimeter of the room, multiply by its height and subtract the area of the doors and windows to get the wall area to be painted. He then calculates the quantities of the needed materials and the time involved in getting the work done. To be safe he will likely add a 10 or 20% tolerance to the area. The added geometric precision brings small benefits since most of the work is done onsite. But if there is any amount of pre-fabrication of building components from drawings or site surveys, geometrical and metric precision becomes a requirement. Since the components come to the site pre-cut, the margin for onsite adjustments is smaller, making dimensional errors more costly to fix.

Prefabrication with computer controlled machinery, commonly known as digital fabrication, is one of the enablers of the emerging paradigm of mass customizable construction. Parametric design is another enabler that allows the expression of construction systems in terms of their variables. The architect is required to design computational models that allow users to visualize, simulate and fabricate customized solutions for specific spaces. To do that, users must be able to survey those spaces by themselves.

Since in practice measuring internal angles onsite is difficult, surveyors usually do that indirectly by measuring internal diagonals. When they are back to the CAD software drawing the survey, they can be sure that the plan/section is correct. This article presents 3 triangulation algorithms – we called them Patterns – that can be used to iteratively triangulate any polygon. We have observed these patterns in use by architects when drawing the survey and developed generalized versions. These algorithms when correctly combined will produce an interactive workflow for as-is surveys. I developed and tested several implementations into a plugin, called RoomSurveyor, that will be released shortly on Food4Rhino.