The rounding of solids is a common operation for the design of mechanical parts. A goal of this operation is to construct a fillet surface that smooths by interpolating between two or more surfaces. In geometric modeling, this is usually done by enforcing parametric or geometric continuity requirements on the fillet.
D-NURBS provide a natural solution to the solid rounding problem. In contrast to the geometric approach, the D-NURBS can produce a smooth fillet with the proper continuity requirements by minimizing its internal deformation energy. Additional position and normal constraints may be imposed across the boundary of the surface. The dynamic simulation automatically produces the desired final shape.
Fig. 2 demonstrates edge rounding using D-NURBS
surfaces. In Fig. 2(a1), we round an edge at the
intersection of two planar faces. The faces are formed using quadratic
D-NURBS patches with 
control points. Multiple control
points are used to produce the sharp corner. We free the control
points near the corner and fix the remaining control points at the far
boundaries to impose position and surface normal constraints. After
initiating the physical simulation, the D-NURBS rounds the corner as
it achieves the minimal energy equilibrium state shown in
Fig. 2(a2).
Figure 2: Solid rounding: (a) rounding an edge between polyhedral
faces; (b) rounding a trihedral vertex. (a1) Initial configuration of
control points and patches. (a2) Rounded D-NURBS surface in static
equilibrium. (b1) Initial configuration of control points and
patches. (b2) Rounded D-NURBS surface. In both examples, the control
points along edges have multiplicity 2.
Fig. 2(b1) illustrates the rounding of a trihedral
corner of a cube. The corner is represented using a quadratic D-NURBS
surface with 
control points. The corner is rounded with
position and normal constraints along the far boundaries of the faces
(Fig. 2(b2)).
The above rounding technique is easily extensible to any number of surfaces meeting at arbitrary angles. To round a complete solid, we can apply the technique to all of its edges, corners, etc.
