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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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