Graphics Reference
In-Depth Information
9.14
Parallel Surfaces
The main reason for this section on parallel surfaces is that, like parallel curves, the
topic is important for CAGD.
Let p(u,v) be a regular parameterization for a surface
S
in
R
3
. Let
∂
∂
p
u
∂
∂
p
v
(
)
=
(
)
¥
(
)
Nuv
,
uv
,
uv
,
.
(9.75)
Since we have a regular parameterization, N(u,v) is nonzero and we can normalize it
to be of unit length. If
(
)
Nuv
Nuv
,
,
(
)
=
n
uv
,
,
(9.76)
(
)
then
n
(u,v) is a unit normal vector to
S
at p(u,v).
Definition.
A
parallel surface
to p(u,v) is a surface
S
d
with a parameterization p
d
(u,v)
of the form
(
)
=
(
)
+
(
)
puv puv d uv
d
,
,
n
,
(9.77)
where d is any nonzero real number. In CAGD a parallel surface is called an
offset
surface
.
It follows from equation (9.77) that
()
=+
()
=+
p
p
d
n
and
p
p
d
n
.
(9.78)
d
u
u
u
d
v
v
v
Assumption:
In the rest of this section we shall assume that p
u
and p
v
are principal
directions!
There is no loss of generality with this assumption because the quantities we shall
want to compute are independent of parameterizations and by Theorem 9.9.22 such
parameterizations always exist. The assumption will greatly simplify our computa-
tions because F = M = 0 in this case by Theorem 9.9.23.
Substituting
n
u
and
n
v
from the Weingarten equations with F = M = 0 into equa-
tions (9.78) and using Theorem 9.9.25 gives
d
L
E
Ê
Ë
ˆ
¯
()
=
(
)
p
p
1
-
=
pd
1
-
k
(9.79a)
d
u
u
u
1
