Computer Aided Engineering Design

78 COMPUTER AIDED ENGINEERING DESIGN

The vector QP×QW can be computed as

QP×QW =[r(u + Δu)−r(u)]× [r(u + Δu)−r(u−Δu)] (3.28)
Using the first order Taylor series expansion and ignoring higher order terms in Δu, Eq. (3.28) can
be rewritten as

QP QW

r r rr rr

= +^1

2

2 + 2 =

2

2

2

2

2

2 2

2

× ⎡^3

⎣

⎢

⎤

⎦

⎥×

⎡

⎣

⎢

⎤

⎦

⎥ ×

⎡

⎣

⎢

⎤

⎦

⎥

d

du

u d

du

u

d

du

u

d

du

u

d

du

d

du

ΔΔ ΔΔ Δu (3.29)

AsΔs→ 0,

Δ

Δ

TT

r

s

d

ds

= ( )→ ′′s (3.25)

To determine the direction of T′(s) or r′′(s), consider r′(s) · r′ (s) = 1 differentiating which with
respect to s yields

r′ · r′′ + r′′ · r′ = 0 ⇒ 2 r′ · r′′ = 0 ⇒r′ · r′′ = 0

Thus,r′andr′′ are orthogonal to each other implying that r′′ is perpendicular to T. We may, therefore,
defineN (a unit normal vector) such that

κN

T

=

d

ds

(3.26)

whereκ =

d

ds

T

is the scaling factor to ensure that Nis a unit vector. Also note that

Nr r = /| | =

(/)

=

/(/)

=

(/)

′′ ′′

dds

d

ds

ddududs

d

du

du

ds

ddu

d

du

T

T

T

T

T

T

The binormal B can then be defined as

B=T×N (3.27)

The plane containing T and B is termed the rectifying plane while that containing T and N is referred
to as the osculating plane. To interpret the scalar κphysically, let P,Q and W be three points on
the curve in close vicinity with values r(u),r(u + Δu) and r(u−Δu), respectively, as shown in
Figure 3.9.

Figure 3.9 Determination of κκκκκ

W

P

r(u) Q r(u + Δu)

r(u – Δu)