A curve is called an oriented curve if
(i) The curve is piecewise smooth.
(ii) The position vector r =r (t), when expressed in terms of a parameter t, determines
the direction of the tangent vector to each point on the curve.
(iii) The direction of the tangent vector is said to determine the orientation of the
curve.
(iv) A plane curve which is a simple closed curve which does not cross itself is said
to have either a clockwise or counterclockwise orientation which depends upon
the directions of the tangent vector at each point on the closed curve.
Tangents to Space Curve
In three-dimensions the derivative vector dr
dt
=x′(t)ˆe 1 +y′(t)ˆe 2 +z′(t)ˆe 3 is tangent
to the point (x(t), y (t), z (t)) on the curve r =r (t) = x(t)ˆe 1 +y(t)ˆe 2 +z(t)ˆe 3 for any fixed
value of the parameter t. The tangent line to the curve r =r (t)at the point where
the parameter has the value t=t∗is given by
R =R(λ) = r (t∗) + λdr
dt t=t∗
−∞ < λ < ∞
where λ is a parameter. The tangent line defined by the vector R can also be
expressed in the expanded form
R =R(λ) = (x(t∗) + λx ′(t∗)) ˆe 1 + (y(t∗) + λy ′(t∗)) ˆe 2 + (z(t∗) + λz ′(t∗)) ˆe 3
where t∗ represents some fixed value of the parameter t. The element of arc length
ds along the curve r =r (t)is obtained from the relation
ds^2 =dr ·dr = (dx )^2 + (dy)^2 + (dz)^2 =
[(
dx
dt
) 2
+
(
dy
dt
) 2
+
(
dz
dt
) 2 ]
(dt)^2
and ds =
√(
dx
dt
) 2
+
(
dy
dt
) 2
+
(
dz
dt
) 2
dt
so that one can write ds
dt
=
∣∣
∣∣dr
dt
∣∣
∣∣=
√(
dx
dt
) 2
+
(
dy
dt
) 2
+
(
dz
dt
) 2
(7 .1)
The total arc length for the curve r =r (t)for t 0 ≤t≤t 1 is given by
arc length of curve =
∫t 1
t 0
ds =
∫t 1
t 0
√(
dx
dt
) 2
+
(
dy
dt
) 2
+
(
dz
dt
) 2
dt
The unit tangent vector to the curve can therefore be expressed by
ˆet=∣∣dr^1
dt
∣∣dr
dt
=ds^1
dt
dr
dt
=dr
ds
(7 .2)
which shows that the derivative of the position vector with respect to arc length s
produces a unit tangent vector to the curve.
