Computer Aided Engineering Design

DIFFERENTIAL GEOMETRY OF SURFACES 185

6.7 Parallel Surfaces

Creation of parallel surfaces is useful in design and manufacture. Making of dies for forging and
castings require modeling of parallel surfaces. Enhancing or reducing the size of free-from surfaces
requires calculation of curvature and other properties of the new surface, which is parallel to the
original surface.
Let S: r(u,v) define a surface patch with parametric curves

rrr r(, )|uuu uvvvvvv= 00 = (, ), (, )| 0 uu= = ( , ) 0 (6.41)

Parametric curves, r(u,v 0 ),r(u 0 ,v) lie entirely on the surface S and intersect at a point P (r(u 0 ,v 0 )).
For simplicity, let these parametric curves be also the lines of curvature of S. The tangents to these
curves at P are given by

rr

r

rr

r

uuu

u

u

u

u

= ( , ) =

(, )

, = ( , ) =

(, )

0

0

0

v v v^0 v

vv v

∂

∂

∂

∂ (6.42)

These tangents are the two principal direction vectors and hence they are orthogonal. From Eq. (6.14),
this will mean that the angle between the two tangent vectors ru(u,v 0 ),rv(u 0 ,v) is θ = 90°,

cos =^12 = = 0 = 0

11 22

θ 12

G

GG

u G

uu

rr

rr rr

⋅

⋅⋅

v ⇒

vv

(6.43)

From Eq. (6.34), the normal curvature κn satisfies the equations

(M–G 12 κn)du + (N–G 22 κn)dv = 0 ⇒Mdu + (N–G 22 κn)dv = 0

(L–G 11 κn)du + (M–G 12 κn)dv = 0 ⇒ (L–G 11 κn) du + Mdv = 0 (6.44)

These equations are true for any arbitrary values of duanddv, because u and v form an orthogonal
net of lines on the surface. This implies that

( – 22 ) = 0 1 = , ( – ) = 0 =

22

11 2

11

NG N

G

LG L

nnG

κκ⇒⇒κ κ

Figure 6.16 Other ruled but non-developable surfaces

(a) Plucker polar surface (n = 4) (b) Hyperbolic paraboloid