Graphics Reference
In-Depth Information
Figure 15.15.
Blending with density functions.
2
2
2
.
(
)
(
)
(
)
r
=-
x
x
+-
y
y
+-
z
z
1
1
1
Given a collection of n atoms, we sum the densities for the collection to get a density
function of the form
n
Â
1
(
)
=
-
a
ii
Dxyz
,,
be
,
(15.11)
i
i
=
where r
i
is the distance of (x,y,z) to the center of the ith atom. Actually, for efficiency
reasons, r
i
2
was used in the exponents of equation (15.11) rather than simply r
i
. One
then used a cutoff value c and only displayed those points for which D(x,y,z) > c. See
Figure 15.15. By changing the constants b
i
and a
i
one could achieve different effects.
One problem with trying to apply this type of blending in a CAGD program is that
the objects would get bigger or smaller because the density function would modify
any original sharp boundaries.
Next, [Wood87] attributes an early example of
volume bounded blends
to M.A.
Sabin. Given two surfaces defined by F = 0 and G = 0, define
(
)
+
(
)
2
2
2
2
HF u
=-
1
Gu cu
+
1
-
u
,
(15.12)
where
P
PQ
u
=
(15.13)
+
for some auxiliary surfaces defined by P = 0 and Q = 0. The surfaces defined by P and
Q define what are usually called the
contact curves
,
link curves
, or
trim (ming) curves
on the surfaces defined by F and G, respectively. These curves define the boundaries
of the blending surface. See Figure 15.16. The equation H = 0 then defines the blended
surface. The term cu
2
(1 - u
2
) in equation (15.12) was needed to prevent the blend
from passing through the intersection of F = 0 and G = 0 and creating a “bump” at
that point.
A related approach originated by Liming ([Limi44]) depends on projective
properties of conics. The problem Liming was concerned with was designing airplane
fuselages with conic cross-sections. First, note that if we have two conics defined
by F = 0 and G = 0, then
(
)
1
-
tF tG
+=
0
(15.14)
