Computer Aided Engineering Design

DIFFERENTIAL GEOMETRY OF SURFACES 193

6.10 Curve of Intersection between Two Surfaces

In engineering design, one has to deal with situations where two surfaces are made to intersect.
Examples can be that of design of the outer shell of an automobile where surfaces such as the glass
window, the roof top and the bonnet are made to
intersect along various curves of intersection. In
case of air-conditioning ducts as well, various
cylindrical and spherical surfaces intersect in
curves. These curves of intersection are important
from the manufacturing viewpoint as they define
the boundaries of various surfaces to be assembled
or conformed. In many cases, it is often difficult
to determine the curve of intersection in explicit
form. It is useful, therefore, to know the properties
of such a curve, like torsion and curvature, using
which we can numerically integrate to determine
the curve of intersection.
Letf (x,y,z) = 0 and g (x,y,z) = 0 be two
surfaces intersecting in a curve of intersection
whose equation cannot be determined in a simple
form. The curvature and torsion of the curve may
still be determined in the following manner. Let
the curve of intersection be given by r = r(s). The
unit tangent vector t at any point on this curve is
orthogonal to the surface normals at that point on each of the surfaces. The surface normals are given
by

∇

∂

∂

∂

∂

∂

∂

∇

∂

∂

∂

∂

∂

∂

f

f

x

f

y

f

z

g

g

x

g

y

g

z

= ijk + + , = ijk + + (6.53)

Thus,t will be proportional to p = ∇f×∇g. For a scalar λ which is a function of s

λt = ∇f×∇g = p⇒ (λt)· (λt) = (∇f×∇g)· (∇f×∇g)⇒λ^2 = (∇f×∇g)^2 (6.54)

λλt λ Δ

r

= = Define oparator (^) 12 3 + +
d
ds
fg d
ds
h
x
h
y
h
x
∇×∇ ⎯⎯⎯⎯⎯→ ≡
∂
∂
∂
∂
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟≡ (6.55)
where, h
dx
ds
h
dy
ds
h
dz
(^123) ds
= λλλ, = , =.
Using the operator Δ on λt results in
λ
λ
λλ
λ
Δλκλ
d λ
ds
d
ds
d
ds
d
ds
()
=^22 + = = +
tt
tP n t (6.56)
Whereκ is the curvature and t the unit normal to the curve of intersection.
Taking the cross product with λt = p
λλ λ
λ
t Δκ
t
tpp t
t
×^2 + = Here = , = 0bt t
⎛
⎝
⎜
⎞
⎠
⎟ ×⋅ × ×
d
ds
d
ds
d
ds
with b being the unit binormal.
f = 0
g = 0
Figure 6.23 Curve of intersection between
two surfaces
∇f
T = ∇f×∇g
∇g