14.3. Arc Length and Curvature of Space Curves
The length of a space curve is defined in the same way as a
length of a planar curve.
Given a space curve
r
(
t
) =
h
f
(
t
)
, g
(
t
)
, h
(
t
)
i
,
a
≤
t
≤
b
or equivalently with parametric equations
x
=
f
(
t
)
,
y
=
g
(
t
)
,
z
=
g
(
t
)
,
a
≤
t
≤
b.
If
f
0
, g
0
, h
0
are continuous on [
a, b
] and the curve is traversed exactly
once if
t
varies from
a
to
b
, then its length
L
is
L
=
Z
b
a
q
[
f
0
(
t
)]
2
+ [
g
0
(
t
)]
2
+ [
h
0
(
t
)]
2
dt
and this can be put in more compact form:
L
=
Z
b
a

r
0
(
t
)

dt
r
0
(
t
) =
h
f
0
(
t
)
, g
0
(
t
)
, h
0
(
t
)
i
.
Example 1.
Find the length of the arc (the part) of the circular he
lix with vector equation
r
(
t
) = cos(
t
)
i
+ sin(
t
)
j
+
t
k
from the point
(0
,
1
, π/
2) to the point (1
,
0
,
2
π
).
Different parametrization of a curve.
Note that one curve can
have many parametric representations.
However,
the formula for
the length of a curve does not depend on the parametric rep
resentation of the curve.
Example 1.
The vector functions
r
1
(
t
) =
h
t, t
2
, t
3
i
, t
∈
[1
,
2];
r
2
(
t
) =
h
e
v
, t, e
2
v
, e
3
v
i
, v
∈
[0
,
ln(2)]
are two parametric representations of the twisted cubic
t
=
e
v
, v
= ln(
t
).
Parametrization of a curve with respect to its arc length.
Define the arc length function
s
(
t
) as the length of a curve
r
(
t
) =
h
f
(
t
)
, g
(
t
)
, h
(
t
)
i
,
a
≤
t
≤
b
from the initial value of the parameter
a
1