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ˆ
()
T
t
ˆ
()
N
t
()
1
κt
()
P
t
Figure 11.20. The osculating circle lies in the plane determined by the tangent direction
()
ˆ
ˆ
()
T
t
and the normal direction
N
t
. The radius of the osculating circle is the reciprocal of
()
the curvature
κ t .
d
()
a
=
v t
T
dt
vt
[
()
]
()
2
a
=
(11.108)
N
ρ t
are called the tangential and centrifugal components of the acceleration, respec-
tively. The centrifugal component agrees with the acceleration corresponding to
the centrifugal force given by Equation (14.8).
()
We can complete a three-dimensional orthonormal basis at a point
P
t
by
ˆ
()
defining the unit binormal
B
t
as
ˆ
ˆ
ˆ
()
()
()
B
t
TN .
t
t
(11.109)
ˆ B is called the Frenet
frame . The derivatives of the axes with respect to the distance s along a path can
be written in terms of the axes themselves. For the tangent direction
ˆ
ˆ
()
()
()
The coordinate system having the axes
T
t
,
N
t
, and
ˆ
()
T
t
, we
have
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