|
|
| |
|
|
Subdivision Surfaces for Architectural Modelling |
|
|
Subdivision SurfacesSubdivision Surfaces are a technique for representing curved forms (surfaces or volumes) in three dimensions. They are defined using a very course triangular control mesh and an algorithm is then applied which recursively splits each triangle into smaller children. The position of each new triangle is carefully calculated to result in a very smooth meshed surface. Theoretically this process can be continued forever, producing infinitely many, infinitely small child triangles which converge to lie on what is known as the "Limit Surface". The limit surface of such a process can have guaranteed smoothness and predictable properties. For example it is not necessary to carry out an infinite subdivision process to find its position or normal. There are also schemes such as "Catmull-Clark" which can operate on quadrelaterals. |
 |
 |
| There are two types of subdivision scheme, Interpolating and Approximating. Interpolating schemes such as "Butterfly" calculate new positions only for the child-vertices, whereas approximating schemes also calculate new positions for the parent verices as well. |
 |
 |
Constrained SubdivisionFor architectural purposes however, subdivision needs to be supported by a system of constraints allowing, for example, boundaries to be locked into place, or creases to be formed. Otherwise the resulting surfaces could pull away from a required boundary defined by the project brief. For example the picture on the left shows the mesh being pulled away from the corners and also shrinks into the central hole. Once nodes can be constrained to a point, edges to a curve, this means that all vertices inserted into the edge during subdivision will be constrained to the same curve as shown in the picture on the right. This allows subdivision surfaces to be used to define surfaces for building geometries and is especially suitable to doubly curved envelopes and grid-shells. |
 |
 |
Attributes may be attached to each topological element, but most importantly to nodes. An important benefit of subdivision is that numerical attributes can be subdivided in the same manner as the spatial coordinates, yielding a smooth structure, for example of texture (u,v) coordinates.
There is an open issue about the representation of surfaces of revolution, which are not uncommon architecturally. No stationary (using same averaging rule at every level) subdivision process will produce one. Constraints may be a solution, or alternatively there are non-stationary schemes that will create them. |
 |
 |
|
