Graphics Reference
In-Depth Information
mations is not as computationally expensive. The analysis of different algorithms in
[ElLK97] showed that the one described in [TilH84] performed best on piecewise quad-
ratic curves. The question of which curves have rational offsets has also been studied
in various papers, because otherwise one is basically only left with approximations. For
more information about this subject see [Faro92], [Pott95], [Lü95], or [FarS95].
14.9.2
Offset Surfaces
We have already talked about offset-type surfaces when we considered bump map-
pings in Section 9.8. Here we give a few definitions and state some properties of the
basic offset surfaces.
Suppose that p(u,v) is a regular parameterization for a surface
S
. If p(u,v) is dif-
ferentiable, then
∂
∂
p
u
∂
∂
p
v
(
)
=
(
)
¥
(
)
Nuv
,
uv
,
uv
,
(14.38)
is the standard normal vector to the surface. Since N(u,v) does not vanish,
(
)
Nuv
Nuv
,
,
(
)
=
n
uv
,
(14.39)
(
)
is a well-defined unit normal vector to
S
at p(u,v). Let d be any nonzero real number.
Definition.
The
offset surface
p
d
(u,v) to p(u,v), which is a distance |d| from p(u,v) is
defined by
(
)
=
(
)
+
(
)
puv puv d uv
,
,
n
,
.
(14.40)
d
Offset surfaces are called parallel surfaces in differential geometry. Just as in the
case of offset curves, the reader is assumed to have familiarity with some basic dif-
ferential geometry. The relevant material for this section can be found in Section 9.14
in [AgoM05] along with proofs of many of the mathematical assertions made here.
There are formulas that express the basic intrinsic geometric properties of an
offset surface in terms of the corresponding properties of the original surface. We list
them here. First of all, the normal vector
∂
∂
p
u
∂
∂
p
v
d
d
(
)
=
(
)
¥
(
)
Nuv
,
uv
,
uv
,
d
to the surface p
d
(u,v) is given by the following formula
(
)
(
(
)
=-
2
)
Nuv
,
12
HdKdNuv
+
,
,
(14.41)
d
where K and H are the Gauss and mean curvature of
S
, respectively. Let
