Interpolation-Based Animation - Computer Animation: Algorithms and Techniques - page 139

FIGURE 4.34
Projection lines for star-shaped polygons from objects in Figure 4.33 .
Corresponding slices (corresponding in the sense that they use the same axis parameter to define the
plane of intersection) are taken from each object. All of the slices from one object can be used to recon-
struct an approximation to the original object using one of the contour-lofting techniques (e.g., [ 10 ]
[ 14 ]). The two-dimensional slices can be interpolated pairwise (one from each object) by constructing
rays that emanate from the center point and sample the boundary at regular intervals with respect to the
orientation vector ( Figure 4.34 ) .
The parameterization along the axis and the radial parameterization with respect to the orientation
vector together establish a two-dimensional coordinate system on the surface of the object. Corre-
sponding points on the surface of the object are located in three-space. The denser the sampling,
the more accurate is the approximation to the original object. The corresponding points can then be
interpolated in three-space. Each set of ray-polygon intersection points from the pair of corresponding
slices is used to generate an intermediate slice based on an interpolation parameter ( Figure 4.35 ).
Linear interpolation is often used, although higher-order interpolations are certainly useful. See
Figure 4.36 for an example from Chen [ 9 ].
This approach can also be generalized somewhat to allow for a segmented central axis, consisting of
a linear sequence of adjacent line segments. The approach may be used as long as the star-shaped
restriction of any slice is maintained. The parameterization along the central axis is the same as before,
except this time the central axis consists of multiple line segments.
4.4.4 Map to sphere
Even among genus 0 objects, more complex polyhedra may not be star shaped or allow an internal axis
(single- or multi-segment) to define star-shaped slices. Amore complicated mapping procedure may be
required to establish the two-dimensional parameterization of objects' surfaces. One approach is to
map both objects onto a common surface such as a unit sphere [ 19 ]. The mapping must be such that
the entire object surface maps to the entire sphere with no overlap (i.e., it must be one-to-one and onto).
Once both objects have been mapped onto the sphere, a union of their vertex-edge topologies can be
constructed and then inversely mapped back onto each original object. This results in a new model for