All edges are parameterized and one can visualize the parameterization using the s and s1 variables (2D and 3D). In general the edge parameter will go from 0 to 1, and if it does not it can be normalized to do so. Nothing similar is immediately possible for surfaces in 3D, because although a parameterization exists (s1 and s2), it is a trimmed parameterization. The 3D polynomial regularization thus relies on computing an untrimmed parameterization using a PDE, and this is only possible for triangular and quadrilateral boundaries.

Here n is the order and dmax is the maximum displacement. The equation shown here is a Bernstein polynomial for an edge in 2D. In contrast to Lagrange polynomials, Bernstein polynomials respect the maximum displacement constraint strictly. Similar to the PDE-based regularization, terms can be excluded to prevent deformation of the boundary near fixed boundaries, and continuity of the normal near planar symmetry boundaries can also be controlled by restricting the design freedom, at least in 2D. Finally, it is possible to preserve continuity of the normal between boundaries in 2D, but this is not possible in 3D and therefore the PDE-based regularization is preferred for most cases of 3D shape optimization.