Math 497C
Oct 1, 2004
1
Curves and Surfaces
Fall 2004, PSU
Lecture Notes 7
1.17
The FrenetSerret Frame and Torsion
Recall that if
α
:
I
→
R
n
is a unit speed curve, then the unit tangent vector
is defined as
T
(
t
) :=
α
(
t
)
.
Further, if
κ
(
t
) =
T
(
t
)
= 0, we may define the principal normal as
N
(
t
) :=
T
(
t
)
κ
(
t
)
.
As we saw earlier, in
R
2
,
{
T, N
}
form a moving frame whose derivatives
may be expressed in terms of
{
T, N
}
itself. In
R
3
, however, we need a third
vector to form a frame. This is achieved by defining the
binormal
as
B
(
t
) :=
T
(
t
)
×
N
(
t
)
.
Then similar to the computations we did in finding the derivatives of
{
T, N
}
,
it is easily shown that
T
(
t
)
N
(
t
)
B
(
t
)
=
0
κ
(
t
)
0

κ
(
t
)
0
τ
(
t
)
0

τ
(
t
)
0
T
(
t
)
N
(
t
)
B
(
t
)
,
where
τ
is the
torsion
which is defined as
τ
(
t
) :=

B , N .
Note that torsion is well defined only when
κ
= 0, so that
N
is defined.
Torsion is a measure of how much a space curve deviates from lying in a
plane:
1
Last revised: September 27, 2007
1
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Exercise 1.
Show that if the torsion of a curve
α
:
I
→
R
3
is zero everywhere
then it lies in a plane. (
Hint
: We need to check that there exist a point
p
and a (fixed) vector
v
in
R
3
such that
α
(
t
)

p, v
= 0. Let
v
=
B
, and
p
be any point of the curve.)
Exercise 2.
Computer the curvature and torsion of the circular helix
(
r
cos
t, r
sin
t, ht
)
where
r
and
h
are constants. How does changing the values of
r
and
h
effect
the curvature and torsion.
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 Spring '08
 Staff
 Math, Geometry, Torsion, nonvanishing curvature

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