Computer Aided Engineering Design

80 COMPUTER AIDED ENGINEERING DESIGN

ρ

κκ

=

=^1 =^1

3

2

2

3

3

d

du

d

du

d

du

d

du

ds

du

r

rr

r

× ⎛

⎝

⎞

⎠

(3.30c)

In other words, the scalar κis the inverse of the radius of curvature, ρ for which reason κ is referred
to as curvature. Note that if the curve lies on a plane, so does the osculating circle and hence B is
invariant, that is, dB/ds = 0. Otherwise, dB/ds can be computed in the following manner. Noting that
B · T= 0,

T · (dB/ds) + B ·(dT/ds) = 0

From Eq. (3.26), (dT/ds) = κNand since B is orthogonal to N,T · (dB/ds) = 0, implying that dB/ds
is perpendicular to T. Moreover, since B is a unit vector, B ·(dB/ds) = 0 and thus dB/ds is parallel
toB×T or N. Define dB/ds as

dB/ds = −τN (3.31)

whereτ is termed as the torsionof the curve. Now, since T,N and B are mutually orthogonal, using
N=B×Tand differentiating, we get

dN/ds= (dB/ds)×T + B× (dT/ds)

= (dB/ds)×T+κ B×N

=−τN×T+κ B×N

=τB−κT (3.32)

Eqs. (3.23), (3.26), (3.31) and (3.32) are collectively termed as the Frenet-Serret formulae summarized
as follows:

dr/ds=T

dT/ds=κN

dB/ds=−τN

dN/ds=τB−κT (3.33)

Most often, it is easier to work with parameter u as opposed to the natural or arc length parameter s
for which the Frenet-Serret formulae can be modified accordingly using Eq. (3.20). Such conditions
provide useful information on the slope and curvature of the segments which is very helpful when
implementing the continuity requirements at the common data points (or junction points) of piecewise
composite curves.

Example 3.4. Consider the helix r(t) = a cos ti +a sin tj+btkin parameter t. Determine the tangent,
normal, bi-normal, radius of curvature, curvature and torsion at a point on the helix.
The unit tangent vector Tis given by

rijk ̇( ) = – sin + cos + tatatb

T

r ̇

r ̇

ijk ijk

=

()

| ( ) |

=

• sin + cos +
  (– sin ) + ( cos ) +

=

• sin + cos +

(^22222) +
t
t
ata tb
at a t b
ata tb
ab