As in the 2D morphing system of [2], the animator
identifies two corresponding features in 
and 
, by
defining a pair of elements 
. These features
should be transformed to one another during the morph. Such a
transformation requires that the feature of 
be moved,
turned, and stretched to match respectively the position, orientation,
and size of the corresponding feature of 
. Consequently, for
each frame of the morph, our warp should generate a volume 
from 
with the following property: the feature of 
should possess an intermediate position, orientation and size in
. This is achieved by computing the warp in two steps:
and scaling factors of elements 
and
to produce an interpolated element 
. This
element encodes the spatial configuration of the feature in
.
Figure 3: Single element warp. In order to find the point 
in
volume 
that corresponds to 
in 
, we
first find the coordinates 
of 
in the scaled
local system of element 
; 
is then the point with
coordinates 
in the scaled local system of element
. To simplify the figure, we have assumed unity scaling
factors for all elements.
of 
,
we find the corresponding point 
in 
in two simple
steps (see figure 3): (i) We find the coordinates of
in the scaled local system of element 
by
(ii) 
is the point with coordinates 
, 
and 
in
the scaled local system of element 
, i.e. the point
.