Openmirrors. Abstracting a circle

A circle is unambiguously defined by only three points. Fortunately, there is a mathematical model for precisely describing curved surfaces. Figure 4. But modelling a free-form surface means more than just tweaking control points; in order to come up with something buildable in the end, it means understanding the mathematical concepts behind those surfaces and relating them to the material world. ALA Arkitekter AS, Kilden Performing Arts Center, Kristiansand, Norway, due for completion 2012

The facade towards the waterfront is clad by straight oak boards, only twisted around their longitudinal axis. For the intended prefabrication concept all generatrices had to be aligned with the building axes, a demand that could not be met with the default ‘loft’ method found in standard CAD packages (left), but needed a custom NURBS – definition (right). The structure is composed of six layers of double-curved girders that were precisely pre-cut on a computer-controlled machine. ALA Arkitekter AS, Kilden Performing Arts Center, Kristiansand, Norway, due for completion 2012

The facade’s shape is defined by a ruled surface with a straight upper and a curved lower edge. Shigeru Ban, Centre Pompidou, Metz, France, 2010

The roof during erection. For curved surfaces (right), this approach fails because no projection plane would preserve the metrics. Additionally, the geometric definition of a circle lets us now identify the exact location of infinitely many more points than the 30 we started with. After discovering the shape behind those 30 points we can throw away 27 of them (90 per cent of the data) and still have the same figure defined in the drawing. NURBS allow the precise definition of complex shapes through control points. Meshes are an easy way to define complex shapes, but they have a severe disadvantage: the planar facets of a mesh can only approximate a curved shape, which is usually acceptable for rendering an image, but certainly is not for digital fabrication as the approximation errors quickly exceed the machine precision (typically some 1/10 millimetre for large-scale fabrication equipment) and are duly and visibly reproduced. When used properly, significantly fewer control points are needed for a NURBS surface than vertices for a similar mesh, while at the same time NURBS allow the precise calculation of all in-between points on the

Figure 2. Developed in the 1950s and 1960s, the computational complexity of NURBS meant it took almost 50 years until they started their impressive career in architecture. So it is more efficient to work with NURBS from the beginning, which is exactly what modern 3-D modelling software offers.