Figure 11.8:
Definitions of unit tangent and normal vectors (adapted from [254])
In this entire chapter we employ the convention (b) (see Fig.
3.7 (b) and Table 3.2) such
that the curvature
of a curve at point
is positive when
the center of curvature
is on the opposite direction of the unit
normal vector
as illustrated in
Fig. 11.8. Following this convention, the Frenet-Serret
formulae for a planar curve
,
with arbitrary speed (2.57) reduce to
(11.3)
where
is the parametric speed. The second equation of
(11.3) can be rewritten
as follows:
(11.4)
A planar offset curve
with signed offset distance
to the
progenitor planar curve
is defined by
(11.5)
The unit tangent and normal vectors and the curvature of the
offset curve are given by [102]
(11.6)
(11.7)
(11.8)
where (11.3) (11.4) are
used for the derivation.
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December 2009