For a parametric space curve
, a radial curve is
defined as
(8.12)
where
is a fixed reference point in space,
is the unit
principal normal vector and
is a nonzero curvature of the
curve [227]. Here the sign convention (a) (see Fig.
3.7 (a)) is adopted for the curvature.
For sign convention (b) we simply replace the plus sign with a minus
sign. Radial curves are a method for visualizing curvature in a
manner decoupled from the shape of the curve as illustrated in Fig.
8.4. When an inflection point is involved, as in Fig.
8.4, spikes in opposite directions occur.
By viewing the radial curve, a curvature measure is
visualized. However, the user is left without a direct reference as to the
relationship between the curve point and the curvature value. The
radial curve also provides a method for viewing the range of variation of
the curve's normal vector. This range of variation can be obtained
from the angular sector described by all rays emanating from p
and passing through all points on the radial curve. Radial curves are
useful in accessibility and interference analyses.
Figure 8.4:
A radial curve of
with a fixed reference point
. The thin curve is the radial curve of the
thick
curve (
with a curvature plot)