Computer Aided Engineering Design

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