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Math 251-copyright Joe Kahlig, 15A
Section 11.7: Arc Length and Curvature
In Cal II, the arc length of a two-dimensional smooth curve that is only traversed once on an interval
I was given by
L=
Z
ds or L =
I
Z q
[x′ (t)]2 + [y ′ (t)]2 dt
I
This can be extended to a space curve. If r(t) = hf (t), g(t), h(t)i on the interval a ≤ t ≤ b, then the
length of the curve is given by
L=
Zb q
[f ′ (t)]2
+ [g′ (t)]2
+
[h′ (t)]2 dt
or
L=
a
Zb
a
′ r (t) dt.
Note: A curve given by a vector function, r(t), on an interval I is called smooth if r′ is continous and
r′ (t) 6= 0 (except possibly at any endpoints of I).
Example: Find the length of the arc for r(t) = h3t, 2 sin(t), 2 cos(t)i from the point (0, 0, 2) to (6π, 0, 2).
Definition: The arc length function, s, is s(t) =
Zt
a
The arc length s is called the arc length parameter.
′
r (u) du.
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Math 251-copyright Joe Kahlig, 15A
Example: Find the arc length function for r(t) = et , et sin(t), et cos(t) from the point (1, 0, 1) in the
direction of increasing t.
Example: Reparametrize the curve r(t) = h1 + 2t, 3 + t, −5ti with respect to arc length measured
from the point where t = 0 in the direction of increasing t.
Math 251-copyright Joe Kahlig, 15A
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Curvature
Definition: The curvature, κ, of a curve is defined to be the magnitude
of the rate of change of the
dT unit tangent vector with respect to the arc length is given by κ = ds Theorem The curvature of the curve given by the vector function r is
dT |T′ (t)|
|r′ (t) × r′′ (t)|
=
= |r′′ (s)|
3
′
ds |r′ (t)|
|r (t)|
κ = E
D
√
Example: Find the curvature of r(t) = = − 2 sin t, cos t, cos t .
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Math 251-copyright Joe Kahlig, 15A
D
Example: Find the curvature of r(t) = 1 + t, 1 − t, 3t2
Note: The unit normal vector is defined as N (t) =
B(t) = T(t) × N(t).
E
T′ (t)
and the binormal vector is defined as
|T′ (t)|