Graphics Reference
In-Depth Information
(2) Parts of complex geometric objects, such as rounded corners, may have a more
natural description with triangular patches.
Disadvantages of triangular domains:
(1) Triangular domains are not convenient for everything, so that it would not be
desirable to restrict all parameterizations to have such domains.
(2) Triangular patches involve a specialized data structure which would have to
be integrated into an existing modeling system. This could be expensive.
For a discussion of how one can define surface patches over domains that are arbi-
trary convex polygons see [LooD89].
Finally, recursive subdivision surfaces have an advantage over regular B-spline
surfaces when one can define a polygonal structure that fits the surface better. On the
other hand, one pays a price because a recursive structure is usually much harder to
work with than simple formulas.
12.19
A Little Bit of History
We would like to end our discussion of curves and surfaces with a brief history of
their development as far as it relates to design and manufacturing. A good reference
is [Sabi90]. See also [NowR83], [Hoch83], [Elsa83], and [Fari83].
The earliest significant instance involving a systematic design of surfaces was in
the building of boats. For hundreds of years craftsmen designed boat hulls by using
a sequence of frames that defined the lateral shape of the hull. Planking was then
applied to these frames to produce the hull. If the frames were defined well, then the
planks produced a smooth shape. It took great skill to end up with a shape that had
the desired property. Eventually, pencil and paper drawings were used to help in this
process. Designing a boat involved roughly two stages:
The Design Stage. Some initial specifications for the hull led to a preliminary line
plan that consisted of a collection of planar curves representing an orthogonal set of
cross-sections of the targeted shape. Figure 12.31(b) shows some sample views of the
boat in Figure 12.31(a). The goal of this stage was to resolve any potential problems
caused by perhaps conflicting specifications.
The Fairing Stage. The goal of this stage was to adjust the line plan obtained in the
first stage to obtain a smoother looking surface while still meeting the basic initial
requirements. The process involved checking the shape of the surface along some
other control sections. The two strategies used here were to either modify some lines
after the fact or to have defined a parameterized set of lines to begin with and to tweak
these parameters.
Basically, one would draw a sequence of frames on a drawing board to full scale
and horizontal and vertical sections were determined. This required a large drawing
board that was situated on the floor of the loft above the room where the ship was
being built, hence the term “lofting.” When ships became too big for full-scale
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