Graphics Reference
In-Depth Information
(
)
Nuv
Nuv
,
,
d
d
(
)
=
n
d
uv
,
(
)
and define s by
n
d
=s
n
. Notice that s may equal -1 because the orientation of the
offset surface may not be the same as that of the original surface. The principal normal
curvatures (k
i
)
d
for the offset surface are given by
s
k
s
k
1
2
()
=
()
=
k
and
k
.
(14.42)
1
2
d
d
1
-
d
k
1
-
d
k
1
2
The Gauss curvature K
d
and mean curvature H
d
of the offset surface can be computed by
K
Hd
K
d
=
(14.43)
2
12
-
+
Kd
and
HKd
Hd
-
H
d
=
s
12
2
.
(14.44)
-
+
Kd
Clearly, all the problems that could arise in the context of offset curves are mag-
nified for offset surfaces. There could again be cusps. We could have ridges and, of
course, complicated self-intersections. Regions of high curvature and where the offset
distance is close to the minimum concave radius of curvature cause problems.
Barnhill and Frost ([BarF95]) analyzed three approaches to offset surfaces via approx-
imations based on uniform bicubic Hermite meshes, NURBS surfaces, and uniform
bicubic/biquintic Bézier meshes and then proposed a solution that used triangular
Bernstein-Bézier patches. Other approaches for NURBS can be found in [KuSP02]
and [KuSP03].
Forsyth ([Fors95]) discusses offsetting and the closely related operation of shelling
in a slightly different context. Offsetting is thought of here as an operation on a solid
model that adds or removes a uniform layer to its boundary. Shelling comes in two
forms.
Closed shelling
is where one removes all of the interior of a solid further than
a given distance from its boundary. In
open shelling
one removes all of the solid further
than a given distance from a part of its boundary. Figure 14.42 shows two-
Figure 14.42.
Shelling.
