The boundary method of computing surfaces starts with a boundary and some initial approximation of a spanning surface. The boundary is a connected set of (straight or curved) linear elements. In the simplest case, the boundary is a simple loop, and the initial surface is a simply-connected patch spanning the loop. More complicated structures can also be used, however. The final surface approximation is arrived at by repeatedly refining the spanning surface and using some process to reposition the vertices based on their relationship to adjacent vertices, and other possible factors such as forces describing pressure, gravity, etc. The boundary need not be fixed, but can include elements that are merely constrained in some way -- for example, to lie on some plane. The ^ Brakke Surface Evolver is a software package which uses the boundary method to compute surface approximations, and has considerable sophistication in how the boundary can be specified.
The Plateau method is a special kind of boundary method in which the spanning surface is constrained to be minimal.