Any boundary defined by a mathematical function results in a smooth and continuous surface.

A technical overview of the most common 3d data structures in architectural design, from the various surface representations to logical models of geometric relationships.

Any boundary defined by a mathematical function results in a smooth and continuous surface.

Many surfaces may be described by well-known equations, and are often controlled by the relevant geometric properties. Planes, cylinders, cones, spheres are often defined in this way in CAD packages, which express them as unique objects rather than general surfaces, and offer direct control over properties such as their radius, axis, or dimensions.

So-called free-form surfaces are generally described by some sort of interpolation function through a set of control points. There are many such functions, but the most prevalent by far in architectural applications are NURBS, which are discussed in detail below.

Solid modeling may be seen as an extension of one of the above surface definition schemas, but is distinguished from surface modeling by its emphasis on the maintenance of physical fidelity, including the property of water-tightness. Solid modeling is often used in engineering, manufacturing and fabrication applications, as it is appropriate for the creation, exchange, visualization, animation, interrogation, and annotation of digital models of physical objects.

Voxels are not often employed in architectural design, and are more often used in medical imaging, the simulation of volumetric dynamics (such as wind or water movements), structural analysis and the representation of terrain in games and simulations.

Given their increasing prominence in architectural design, the "logical" model for defining form is worth a mention here. Although these are more rightly defined as a variety of approaches to the interface and data-structure of CAD (rather than as descriptions of form) we should be familiar with the following models of CAD software.

Parametric modeling is an activity focused on the relationships between geometries rather than their description. Grasshopper is a popular plugin for Rhino that facilitates parametric modeling.