Surface Components Basics

It is not possible to cover a double curvature surface with planar quads. Here is one method that overcomes quad tiling on double curvature by pulling one vertex of the quads to the plane defined by the other three. This method was used in architecture on several occasions such as the exterior facade of The Yas Hotel, designed by Asymptote Architecture in 2009. The same approach is also evident in the roof design covering the central courtyard of the Smithsonian…

This is the basic form of a surface division, based on curvature. As each point on the surface has a curvature value, this might be used to dispatch those values and see the points at flat and curved parts of the surface. I used my favourite surface equation definition (here) as implemented equation of cos(x)+cos(y) in the animation below.

This is another popular “math surface” being rediscovered by designers nowadays, in 2012. Saddle surfaces, seen above as mentioned earlier (here) have a special type named “Monkey Saddle Surface”. This surface was a dramatic example of how Grasshopper can control equations and instantly show graphical results. The mathematical equations start with Z=… this makes it very easy for us to transform any x-y grid centers (a 2d data tree of 3d points) and add z values to them according to…

This experiment is based on a traditional surface-component definition. However, the variation of components is associated with Gaussian curvature. We just control the subdivision and a multiplier value. Results are interesting in as an educational tool to explain NURBS surface curvature and its utilization for Design Geometry. Different surface shapes generate exciting results. Of course, this could be much improved by recognizing positive and negative curvature values, (probably only accepting positive ones maybe).

This is an update on my 2012 exercise on rhombille tiling. This reminded me of the old-school subdivisions of surfaces again after 11 years. It was while reading about rhombic dodecahedron (the stackable solid), that I came across this cute tiling. It’s quite simple, just a hexagonal grid with a special reconstruction. I decided to recreate it by using well-known native Grasshopper components. I animated the result with a variation of the Breaststroke surface function (described here). Then I reconstructed…