Gross - 9 
4 Beyond Geometry: Physically-Based Models 
The third and last paradigm to be addressed here is the deformable model, initially pro 
posed by [12]. This model incorporates some of the physics of the object and as a result 
it is well shaped for any type of nonrigid object behavior under the presence of external 
forces. When using it for 3D shape design, the computation is figured out most elegantly 
using finite element methods (FEM). Since the deformable model has been set up for 
curves, the first step is to extend it to surfaces. Due to the restrictions of the tensor product 
approach, the more flexible barycentric extensions should be preferred [13]. One difficulty, 
however, coming along with the introduction of FEM-methods is the choice of the shape 
function. Since we have to guarantee at least (^-continuity when dealing with shape mod 
eling, some computational expensive functions have to be selected. 
Fig. 6 illustrates the shape of a 3 patch triangular surface element using different refine 
ment steps for visualization. Note, that the (^-continuity is preserved at all surface points. 
Fig. 6 Illustration of the shape functions interpolating a 3 patch surface with a displaced mid 
point. The surface is displayed via polygons at different subdivision steps, (from [2]) 
a) Initial triangulation of the 3 patch surface element. 
b) Shape approximated with 4 subdivisions. 
c) Shape approximated with 8 subdivisions. 
d) Shape approximated with 16 subdivisions.