The components of 
may take arbitrary finite values in
, but this is not the case for the weights 
. Negative
components of may cause the denominator to vanish at some
evaluation points, causing the matrices to diverge. Although not
forbidden, negative weights are not useful. We include constraints in
our D-NURBS which enforce positivity of weight values. Such a
constraint is easily implemented by establishing a positive lower
bound on the weight values and enforcing it in the numerical solution
using a projection method.
Another potential difficulty is that smaller values of tend
to flatten the surface in the vicinity of the control points, which
lowers the deformation energy. Consequently, the will tend
to move toward zero. To counteract this tendency, we can associate
with the potential energy the penalty term
in which 
are desired weights and c is scaling factor.
We have implemented both techniques. Experiments indicate that the
projection scheme works very well. Consequently, we do not make use
of the penalty scheme in our current modeling system. It may be
useful, however, if the modeler wants to constrain the weights to
assume values near certain target values .