Figure 6-15.
Tangent line and normal line to plane curve change with position.
Recall from our earlier study of calculus that the arc length smeasured along a
curve from some fixed point (x 0 , f (x 0 )) is given by
s=
∫x
x 0
√
1 + [f′(x)]^2 dx (6 .65)
and the derivative of this arc length with respect to the parameter xis
ds
dx
=
√
1 + [f′(x)]^2. (6 .66)
Using chain rule differentiation one finds
dr
ds
ds
dx =
dr
dx =
dr
ds
√
1 + [f′(x)]^2
or
ˆet=dr
ds
=√^1
1 + [f′(x)]^2
dr
dx
which shows the unit tangent vector to the curve is the derivative of the position vec-
tor with respect to arc length. The choice of the sign on the square root determines
the direction of the unit tangent vector.
At each point on the plane curve the unit tangent vector ˆet makes an angle θ
with the constant unit vector ˆe 1. The absolute value of the rate of change of this
angle with respect to arc length is called the curvature and is denoted by the Greek
letter κ. The curvature is thus represented by
κ=
∣∣
∣∣dθ
ds
∣∣
∣∣.
