A Survey of the Representation and Design of Surfaces

The surface appropriate for a given problem depends on the application; there is no universal surface form. The numerous applications of surface methods include modeling physical phenomena (e.g., combustion) and designing objects such as airplanes and cars. In addition to these 3D surfaces, there are interesting 4D "surfaces" such as temperature as a function of the three spatial variables. Because the geometric information for these problems can be located arbitrarily in 3D or 4-Dimensional space, the schemes must be able to handle arbitrarily located data. The standard (and easier) approach to surfacesusing tensor products of curve methods-restricts the surface method's applicability to rectangularly "gridded" data. Two broad classes of methods suitable for solving these problems (i.e., problems for which simplifying geometric assumptions cannot be made) are (1) surface interpolants defined over triangles or tetrahedra and (2) distanceweighted interpolants. Users ordinarily want smoother surfaces than their data imply directly, so additional information must usually be created. (A notable feature of the methods shown here is that the smoothness of the surface is always greater than or equal to the smoothness of the defining data. The author covers Surface form selection (Interpolation versus approximation; Representation versus design; Smoothness; Shape fidelity; Local versus global methods; and Rendering), and Interpolation surfaces. It is noted that triangular interpolants and distance-weighted interpolants excel as surface methods because of their smooth interpolation of arbitrarily located data.