In Rhino a [...]]]> In Architecture as well some other disciplines rationalization is the process of modifying a design in order to build it economically. Actually we prefer to talk about optimization for fabrication. Some of the current student projects in our forum are about ruled surfaces, in particular the hyperbolic paraboloid. So let’s have a closer look at some famous examples.

In Rhino a hyperbolic paraboloid can be modeled in many ways. For example you can make one by simply lofting (or _EdgeSrf) two skewed lines. The result will be a doubly ruled surface. With the _Curvature command you can evaluate a surface and create the minimum and maximum principial curvature circles at the picked points.

The surface curvature is calculated by 1/RadiusOfCurvature while the value on the backside of the surface is negative. Minimum and maximum curvature are always orthogonal to each other. The Gaussian Curvature is defined as (minimum curvature)*(maximum curvature). Therefore the Gaussian curvature of a hyperbolic paraboloid is non-positive everywhere on the surface. Surfaces with just negative curvature are called anticlastic while a surface with positive curvature (like for example a sphere) is called synclastic.

With the _CurvatureAnalysis command you can display the curvature indicated by false colors on the surface.

A ruled surface can be physically modelled if a ruler is moved along a pair of edges. This method was used for Corbusier’s Philips Pavilion which Kurt Brosnan is looking into. Pretty “low-tech” but at least no change of the desired shape.

Triangulation

For St. Mary’s Cathedral, Pier Luigi Nervi used the triangulation approach which actually can be used for any shape. The interior structure of the cupola is made up of 1680 pre-cast triangular coffers of 128 different sizes.

The challenge on more complex shapes is to get a nicely harmonized triangulation. But for these simple ruled surfaces any regular grid or mesh just have to be applied for Rhino to space it correctly.

Since hyperbolic surfaces have non-positive curvature you cannot make a simple casing from developable material. The mould shape has to be patched from smaller pieces which allow the deformation with the required compound curvature. Still, all the pieces have to be fitted exactly.

St. Mary’s Cathedral by Kenzo Tange:

Surface Analysis

Developable Architecture

With developable material, curved surfaces can be made relatively cost-efficient . However, it is not possible to make shapes with compound curvature (doubly curved). Therefore freeform-shapes have to be changed i.e. rationalized. The Gaussian curvature of developable surfaces is zero at any point on the surface.

Variations:

Quadrilaterals, also called Planar Quads are also very economic. Especially for glazing, this is the preferred topology due to simplicity of the connections, low number of parts and clipping waste.