Topic 9.5: Curves & Arc Length
1. Curves: Curves are of major applications of
differential calculus. (Another application is surfaces)
2. Any Curve C in space may occur as a path of a moving
body. That curve may be defined as parametric
representation i.e, function of a parameter t (time).
r (t) = [x (t), y (t), z (t)] = x (t) i + y (t) j + z (t) k
To each value t = to , there
corresponds a point of C
with position vector r(to)
x(to), y(to), z(to)
Types of Curves
Plane Curve: A curve that lies in a plane in space (Circle in
Twisted Curve: A curve that is not plane in space (Circular
Above two types of curves are also called simple curves
(curves without multiple points i.e, without points at which
the curve intersects or touches itself).
Example 1: Circle (Parametric Representation)
Increasing time t is called the positive sense on C
defines direction of travel along C.
Decreasing time t is called the negative sense on C
which defines direction of travel along C in
Example 3: Straight Line (Parametric Representation)
Example 4: Circular Helix (Parametric Representation)
Tangent to a Curve (Tangent Vector)
Tangent: Limiting position of straight line L touching a curve C
through two point P & Q as Q approaches P.
If C is given by r (t), P & Q corresponds to t &t+∆t,
then a vector in the direction of L is
Tangent vector of C at point P is
Unit Tangent Vector is given by
Tangent to a Curve (Tangent Line)
Hence, the Tangent to Curve C at point P is
This is sum of position vector r of P and multiple
of tangent vector r’ of C at P.
w is parameter here just like t.
Compare this with simple line
equation r (t) = a + t b
Length of a Curve
Length L of a curve C will be the limis of the lengths
of broken lines of n chords(n=5 in the fig) with larger
and larger n.
approaches 0 as n∞
Length L is given by
Arc: Portion of a curve between any two point of it.
Arc Length “s” of a curve C is given by
Related Problems from
Problem Set 9.5
Problems: 01 to 10
Problems: 11 to 18 Optional
Problems: 22 to 25
Problems: 26 to 28