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Figure 5.7: When the handle vertex (hv) is translated with T1, the vertices in the red
circle are influenced. The same thing happens with T2.
Experimentation with other functions is left for future work.
5.3.2 Deformation Field Construction
A deformation field D is now computed which can be applied to any point inside
the voxel grid by interpolating the translations Tvi at each vertex vi. The deformed
position q if a point p is in D is defined:
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q = p + D(p) (5.5)
This deformation can be applied to point based representations such as a polygon
mesh as shown in Figure 5.8a because a set of points p ∈ R 3 are explicitly defined
to represent a surface. On the other hand, implicit surfaces are not defined as a set
of points, but by an arbitrary scalar function f. Since a deformation field D takes
a point p ∈ R 3 as input, D cannot be applied to a scalar function f. To evaluate
an implicit surface, it is necessary to calculate the field values at a large number of
sample points. Instead of deforming f, therefore, the sample points are displaced to
the scalar field by applying an inverse deformation. In Figure 5.8b, when a point q
is evaluated, q is displaced to p with the inverse deformation field D and then f(p)
is returned as the field value of q. Hence, deformation of an implicit surface can be
defined by constructing an inverse deformation field D which approximates D −1 .