76 COMPUTER AIDED ENGINEERING DESIGN
AsQ approaches P, i.e. in the limit Δu→0, the length Δs becomes the differential arc length dsof the
curve, that is
ds
d
du
= = | | du d u = du
r
rrr ̇ ̇ ̇⋅ (3.20)
For a reference value u 0 , the arc length s(u) at parameter value umay be computed from Eq. (3.20) as
su du x y z du
u
u
u
u
( ) = = + +
00
222
∫∫
rr ̇ ̇⋅ ̇ ̇ ̇ (3.21)
The parametric velocity v may be defined as
v
r
= = ( )
d
du
r ̇u (3.22)
A unit tangent Tat point Pis along the direction of the parametric velocity, that is, T = v/| v |, where
|v | = |dr/du| = ds/du from Eq. (3.20). Thus
T
̇
= ̇
()
| ( ) |
=
()
= ( )
r
r
r
r
u
u
ds
ds
′ s (3.23)
Therefore, ̇
rr
rrT = = = ( ) =
d
du
d
ds
ds
du
′ svv
wherevis the parametric speed equal to | v |. The unit tangent Tis expressed above as a function
of the arc length. On a parametric curve r=r(u),P is said to be a regular point if | | 0. ̇r ≠ If P
is not regular, it is termed singular. The curve can be represented either in the form r≡r(u), or
r≡ r(s); the first is dependent on the parameter u and thus on the co-ordinate axes chosen while
the second is independent of the co-ordinate axes and is a function of the natural parameter or the
arc length s.
Figure 3.7 Cylindrical helix
O
•
•
z
bu
r(u)
y x
a sin u a cos u