Developable Ngons

The goal of the project is to fabricate an efficient and optimized Parabolic cyclide with the help of Developable Ngons.

To have a better approximation to the surface to be developed, we will use non-planar polygons, in which the vertices will be projected in the base geometry, to begin with a minimum deviation from the original surface. 

Ways of Developing Foldable Ngons

If the polygons being unfolded have a sum of central angles greater than 2π, at the time of manufacture, this could represent a problem, since when converting the polygon into a flat surface, in order to preserve its propions it will overlap, what would force to cut the polygon in 2 so that each one has the necessary surface for its development. for this reason one of the main restrictions is to work with polygons that the sum of central angles is 2π

Implementation

A | Conversion of surface to a quad mesh with well placed singularities

B | Diagonilization of the quad mesh.

C | Triangulation of diagonalized mesh.

D | Dual graph obtaing hexagons, heptagons and octagons.

E | Final Mesh.

F | Optimization of each polygon to obtain an unfolding angle of 2π.

The best optimal fabrication method can be achieved by analyzing three main metrics:

• Angle defect:The sum of the interior angles of the polygon that must add the expected amount of 2π.i.e angle dflection must be kept zero.
• Fairnesss: The minimum deviation of the mesh having as its point of origin the original surface.
• Apect ratio: Describes the relationship of an object's width to it's height. It is the proportions of a shape, we are interested that most of the pieces are similar.

Four meshes were created each one with an approximation to the different design, in order to evaluate them and compare them select the best

Unfolding Method

Since the interior angles of the polygons were kept at 360°, they could be cut without splitting the polygons, the pieces of the mesh were projected onto the xy plane and then numbered according to their edges and faces. Flaps were then fabricated so that each polygons could be joined or stapled to one another.

To optimise the printing and fabrication process the pieces were then run in K2 simulations so that they could be collided and nested such a way that there would be least wastage of material as shown in the image on the left hand side.

This project was created in the framework of the Master Parametric Design in Architecture - Barcelona 2019

Team: Anupam Gupta - Muzafar Mahmood - Camila Muñoz