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:

- Interpolation:
- We interpolate the local coordinate
systems 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.

- Inverse mapping:
- For every point in 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 .

Last update: 11 May 1995 by Apostolos "Toli" Leriostolis@cs.stanford.edu