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Figure 11.39.
Curve and its hodograph.
define the inflection points, loops, etc. It is claimed that this new approach is not as
sensitive to degenerate cases as previous solutions. Other aspects of hodographs are
described in [SaWS95] and [Moon99].
11.12
Fairing Curves
Producing curves with a pleasing or “fair” shape is not easy. One is confronted with
two tasks here. First, one has to define what it means for a curve to be fair and provide
tests that one can use to check for this property. The latter step is sometimes called
curve interrogation. Second, one has to describe procedures with which a curve that
is not fair can be turned into one that is. This process is referred to as “fairing” the
curve. A good reference is [HosL93].
There does not seem to be any consensus on a definition of “fair.” Curvature clearly
has something to do with it. A common way to deal with fairing is to analyze a plot
of the curvature. Small differences between curves that may not be visible to the eye
can show up in such plots. See [Fari92a]. We shall define the concept as follows (see
[SuLi89]):
Definition.
A curve p(u) is said to be fair if
(1) it is G 2 continuous,
(2) it has no undesirable inflection points, and
(3) its curvature varies in an even way.
As is pointed out in [SuLi89], conditions (2) and (3) are the most important
usually, with emphasis on condition (3). Plots of the curvature functions are useful.
Basically, one likes curves whose curvature function consists of a few monotone
(preferably linear) regions. The points that separate these regions should be a small
set. Another way to put this is to say that the curvature has few extrema. With this
definition a sine curve is not fair because it has lots of inflection points.
Since curves are often defined by point data, we make the following definition:
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