Graphics Reference
In-Depth Information
(4) Polyhedral methods
(5) Other methods
Rolling ball blends have already mentioned. There is a natural spine curve,
namely, the curve swept out by the center of the ball, and the trimlines are just where
the ball meets the base surfaces. See the paper by Vida et al. for a discussion and
references for how the blending surface can be parameterized. For example, Choi and
Ju ([ChoJ89]) use quadratic rational Bézier curves for the cross-sections of the para-
meterizations. Another approach is described in the paper [KlaK92] mentioned
earlier.
In spine-based methods, the spine has to be defined. In the case of rounding edges,
the edge can be used as the spine. More generally, it could be the intersection of two
surfaces, such as offset surfaces from the base surfaces. If sweeping is involved, then
the sweep lines can be so used. Rolling ball blends can be considered as a special case
of spine-based methods. The modeler described by Chiyokura in [Chiy88] basically
does spline-based blending. Trimlines are usually computed automatically. For
example, the trimlines could be defined in terms of the points on the base surfaces
closest to the spine curve. The profile curves are usually required to be curves that lie
in the planes determined by the points on the spine curves and their two assigned
points on the trimlines.
For trimline-based methods there are many ways that those curves could be
specified. One can define trimlines as the intersection of a base surface and another
surface, such as an offset surface for the other base surface. One can also define trim-
lines by specifying curves in the parameter spaces of the base surfaces if they are
parametric surfaces. Once one has the trimlines, the next step is to define profile
curves. This can be done with or without an explicit spine curve. Quite a few
approaches to blending are trimline based. Again see [ViMV94] for more details. An
approach using trimmed tensor product surfaces is described in [ElbC97].
Most parametric blending tends to use rectangular or triangular patches. The
reader interested in blends for n-sided patches should see [HsuT98], where one can
find a number of additional references for this topic.
Blending methods based on polyhedral objects fall into two types. In both cases
one starts with an initial polyhedral object that defines the general shape of the final
object. One can then either use recursive subdivision of the facets or a local round-
ing operation to achieve smoothing. Recursive subdivision surfaces were described
in Section 12.17. Chiyokura and Kimura ([ChiK83]) suggested using local rounding
operations that are built into a modeler. Their idea was that one would first define a
wireframe object by means of Euler operations whose edges would be tagged as needing
rounding or not. The initially straight edges were then replaced by appropriate curved
edges. The last step was to fill in faces with smooth surface patches. The approach
was extended in [Chiy87] and [Chiy88]. A user would first specify fancier appropriate
rounding data for edges and vertices of the original polyhedral model. (Such data
essentially corresponds to defining trimlines.) This would define a curved mesh and
Gregory patches would then be generated for the actual rounding. Another direct
approach is described in [Szil91]. Szilvasi-Nagy uses automatically generated rectan-
gular Ferguson-type bicubic patches for the blending. Cylindrical surfaces are used to
round edges and, at the vertices where they meet, a rectangular patch blends them.
A user can vary the shape of the blend by modifying parameters such as rounding
radii.
