Computer Aided Engineering Design

186 COMPUTER AIDED ENGINEERING DESIGN

MK LN

GG

H

LG NG

GG

= 0 = = , 2 = ( + ) =

+

12

11 22

12

22 11

11 22

⇒ κκ κ κ (6.45)

Create a surface S* parallel to S by shifting each point P on S through a distance a along the unit

normaln

rr

rr

=

| |

u

u

×

×

v

v

on S at P. From Figure 6.17, the point P* on the parallel surface is given by

r(u,v), where r(u,v) = r(u,v) + an. The tangents on the parallel surface S* are given by

rr nr r nuu u**= + , = + aavvv (6.46)

Figure 6.17 Parallel surfaces

O

r(u,v)

ru

rv S

P

an

r*(u,v)

P*

rv* S*

n* = n

OP=r(u,v)

PP*=an

OP*=r*(u,v)

n* =n

ru*

To find the Gaussian and mean curvatures, KandH (of the parallel surface S), one needs the
coefficients GGG 11 , , 12 22 and L,M,N* of the first and second fundamental forms in terms of G 11 ,
G 12 ,G 22 and L,M,N of surface S.

It can be shown that nu and nv are normal to n and therefore, lie in the tangent plane at point P
on surface S

n·n = 1 ⇒nu·n + n·nu= 0⇒nu·n = 0 ⇒nu⊥n. Similarly, nv⊥n.

Since,ru and rv are orthogonal vectors through P and lie on the tangent plane, they can be used as
orthogonal basis for nu and nv. Thus, nu and nv can be expressed as linear combinations of ru and rv

nu = a 1 ru + b 1 rv⇒nu · ru = a 1 ru·ru + b 1 rv·ru⇒–L = a 1 G 11 + b 1 G 12 ⇒a 1 = – L
G 11

(QG 12 = 0)