So… in this post, I will try to introduce some very useful concepts that are widely used in Architectural Geometry, but that are borrowed from other disciplines like Aeronautical Engineering and, lately, Computer Graphics.

Most of these techniques are extremely useful for architectural purposes, specially in recent years, when the use of 3D Modeling CAD programs has become widespread in Architecture practices.

These new modeling capabilities come with some shortcomings as, most of the time, the initially designed surfaces do not meet the projects requirements, leading to solutions that are not easily built or go way above budget. Hence, the shape of these surfaces usually needs to be modified in meet these requirements without compromising the initial design intention of the architect.

This leads us to the concept of discretization

Discretization
In mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts (Wikipedia)

Ok, now in less technical terms: Discretizing a given surface is a process by which you simplify that surface, making it less smooth while keeping it’s geometric properties as intact as possible; which leads to the concept of polygon meshes.

Polygon meshes
A polygon mesh is a collection of vertices, edges and faces that defines the shape of a polyhedral object in 3D computer graphics and solid modeling. The faces usually consist of triangles (triangle mesh), quadrilaterals, or other simple convex polygons. (Wikipedia)

$\alpha = \beta$