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d
d
ˆ
ˆ
ˆ
ˆ
()
()
()
() ()
T
t
=
T
t
N
t κ t
=
N
t
.
(11.110)
ds
ds
The derivative of the binormal can be written as
d
d
d
ˆ
ˆ
ˆ
ˆ
ˆ
()
()
()
()
()
B
t
=
T
t
×
N
t
+
T
t
×
N
t
.
(11.111)
ds
ds
ds
Since the derivative of the tangent direction is parallel to the normal direction,
the first cross product is zero. The derivative of the normal direction must be
perpendicular to the normal direction itself (because it has constant length) and
can therefore be expressed as a linear combination of the tangent and binormal
directions. Thus, using the functions
()
()
α t and
τ t , we simplify Equation
(11.111) as follows.
d
ˆ
ˆ
ˆ
ˆ
()
()
() ()
() ()
BT
t
t α t
T
t τ t
+
B
t
ds
ˆ
() ()
=−
τ t
N
t
(11.112)
Finally, the derivative of the normal direction yields
d
d
ˆ
ˆ
ˆ
()
()
()
N
t
=
B
t
×
T
t
ds
ds
d
d
ˆ
ˆ
ˆ
ˆ
()
()
()
()
=
BT B
t
×
t
+
t
×
T
t
ds
ds
ˆ
ˆ
ˆ
ˆ
() ()
()
()
() ()
τ t
NT B
t
×+×
t
t κ t
N
t
ˆ
ˆ
() ()
() ()
=
τ t
B
t κ t
T
t
.
(11.113)
()
()
(This shows that the value of
α t in Equation (11.112) is
κ t
.) Taken together,
the three relations
d
ˆ
ˆ
()
() ()
T
t κ t
=
N
t
ds
d
ˆ
ˆ
ˆ
()
() ()
() ()
N
t τ t
=
B
t κ t
T
t
ds
d
ˆ
ˆ
()
() ()
B
t τ t
=−
N
t
(11.114)
ds
are called the Frenet formulas .
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