We now consider the notion of the Darboux vector
defined by [206],
(8.32)
where
and
are the unit tangent and binormal
vectors, respectively. The Darboux vector turns out to be a rotation vector of the Frenet frame while moving along the curve and
therefore, its Euclidean norm
indicates the angular speed
of the moving local frame.
The angular speed
is
sometimes called total
curvature of a curve and defined by
(8.33)
In a planar curve,
reduces to
and the binormal vector becomes the axis of rotation.
We notice that the total curvature
captures the coupled
effect of both intrinsic features of a space curve, and hence,
we may consider it as a criterion function for detecting a significant point
on a space curve [57].
At a nonzero curvature point on a regular space curve, where the
moving frame has its locally highest or lowest angular speed, satisfies
the equation
, i.e.
(8.34)
where
,
,
,
and
are
defined in (8.10), (8.22),
(8.23), (8.19) and (8.30),
respectively. Comparing each function we can roughly see each
contribution of
,
,
and
to (8.34). For a special
example, if
and one of
or
vanish at some
, (8.34) is also satisfied
there. We note
and
are always positive
at a nonzero curvature point. Three points in Fig.
8.8 satisfy (8.34). Two
points, marked by
's, are located close to the points of curvature
extrema, and the other point, marked by
, is located at the midpoint of the
curve where
and
also vanish.
Finally, for the special case where
,
and
consequently
are constant, the curve is a circular helix.
Next: 8.2 Stationary points of
Up: 8.1.5 Fourth-order interrogation methods
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December 2009