where x, y, z define some parametric representation of the curve C. The element of
arc length along the curve, when squared, is given by
ds^2 =dr ·dr =dx^2 +dy^2 +dz^2.
An integration (summation) produces the following formulas for the arc length s.
1. If y=y(x)and z=z(x) are known in terms of the parameter x, the arc length
between two points P 0 (x 0 , y 0 , z 0 )and P 1 (x 1 , y 1 , z 1 )on the curve can be represented
in the form
s=
∫x 1
x 0
√
1 +
(
dy
dx
) 2
+
(
dz
dx
) 2
dx. (6 .92)
2. If the parametric equations of the curve are given by x=x(t), y =y(t)and z=z(t),
the arc length between two points P 0 and P 1 on the curve is given by
s=
∫t 1
t 0
√(
dx
dt
) 2
+
(
dy
dt
) 2
+
(
dz
dt
) 2
dt, (6 .93)
where the parametric values t=t 0 and t=t 1 correspond to the points P 0 and P 1
and
x(t 0 ) = x 0 , y(t 0 ) = y 0 , z (t 0 ) = z 0
x(t 1 ) = x 1 , y(t 1 ) = y 1 , z (t 1 ) = z 1.
Figure 6-18. Curve C partitioned into n−segments between P 0 and P 1.
The above formulas result indirectly from the following limiting process. On
that part of the curve between the given points P 0 (x 0 , y 0 , z 0 )and P 1 (x 1 , y 1 , z 1 ),the arc
length along the curve is divided into nsegments by a set of numbers
