Metamorphosis is the blending of one shape into another. Work on 3D shape blending includes [8, 16]. The blending of 2D shapes has widespread application in illustration, animation, etc., and simple (e.g., linear) interpolation techniques usually produce unsatisfactory results [29]. Shinagawa and Kunii [31] propose an method which interpolates differential properties of the 2D shape using the elastic surfaces of [37, 36]. Motivated by their approach, we propose a new technique which exploits the properties of D-NURBS surfaces. D-NURBS provide minimal-energy blends which are more general than linear interpolants and which may be controlled through various additional constraints specific to the NURBS geometry. For example, since NURBS can represent conics, we can exploit their ability to generate helical surfaces in order to represent rotational components of shape metamorphoses.
Our technique interpolates a D-NURBS generalized cylinder between two or more planar shapes with known correspondence. The interpolant is a constrained skinned surface between the two end curves. We interpret the parametric coordinate along the length of the surface, say u, as the (temporal) shape blending parameter. The u coordinates of the control points are fixed, while the v coordinates are subject to the D-NURBS deformation energy and additional constraints. We obtain intermediate shapes by evaluating cylinder cross sections at arbitrary values of u.
Some examples will help to explain our technique in more detail.
Fig. 7 shows minimal-energy D-NURBS surfaces with
control points (3 control points along u) interpolating
between two closed elliptical curves. Fig. 7(b)
shows a linear generalized cylinder obtained with high surface tension
in the u direction: 
and
. Note that the morphing ellipse shrinks
as it rotates, a typical artifact of linear interpolation
[29]. The rotational component can be preserved,
however, by imposing a geometric constraint on the D-NURBS which
creates a helical surface in the u direction of the cylinder, as
shown in Fig. 7(c). Here the only nonzero deformation
energy parameter is the rigidity 
. Note that the
interpolating surface now bulges outside the convex hull between the
two ellipses. As a consequence the interpolated ellipses rotate
instead of shrinking (Fig. 7(d)). In general, we can
obtain a family of blending surfaces between these two extremes by
using intermediate values of tension 
and rigidity
parameters. Fig. 8 illustrates the
morphing between two planar polygonal shapes. The D-NURBS interpolant
is a 
surface. The parts of this figure are similar to
those of the previous one.
Figure 7: Metamorphosis between two planar elliptical curves using
D-NURBS interpolating surface. (a) Control points and patch outline of
cylindrical surface terminated by the two planar curves. (b) Linear
interpolating surface. (c) Constrained nonlinear interpolating surface
combines rigid rotation with nonrigid deformation. (d) An intermediate
morphed curve obtained as cross section of surface in (c).
Figure 8: Metamorphosis between two planar polygonal curves using
D-NURBS interpolating surface. (a) Control points and patch outlines
of cylindrical surface terminated by the two planar curves. (b) Linear
interpolating surface. (c) Constrained nonlinear interpolating surface
combines rigid rotation with nonrigid deformation. (d) An intermediate
morphed curve obtained as cross section of surface in (c).
