Graphics Reference
In-Depth Information
An example of a
range constrained blend
is the superelliptic blend described in
[RocO87], where the cross-sections of the blending surface are superellipses. Another,
which we shall describe briefly, is the so-called
potential method
of Hoffmann and
Hopcroft in [HofH87]. If F, G, ...:
R
3
Æ
R
, let V(F,G, . . .) denote the set of simulta-
neous zeros of the functions F, G, ..., that is,
(
)
=
{
()
=
()
=
}
VFG
,
,...
p
0
=
F
p
G
p
... .
Consider a set V(F). This set partitions space into three sets: The points
p
where
F(
p
) > 0, which we shall call the
outside
of V(F), the points
p
where F(
p
) < 0, which
we shall call the
inside
of V(F), and the set V(F) itself. For any nonzero value s, the
set V(F - s) is either inside or outside of V(F), depending on the sign of s. For example,
if
(
)
=++-
2
2
2
Fxyz
,,
x
y
z
1
,
then V(F - 3) is the sphere of radius 2 that lies entirely outside the unit sphere V(F).
Now
(
)
=
()
«
()
VGH
,
VG
VH
and in general such a set corresponds to the intersection of two surfaces and repre-
sents a curve in space. Consider V(G - s,H - t). This describes a family of space curves
parameterized by s and t. Next, suppose we constrain the parameters s and t to lie on
a curve
C
in the plane defined by an equation
()
= 0
fst
,
.
Define F by
(
)
=
(
(
)
(
)
)
Fxyz
,,
fGxyz Hxyz
,, ,
,, .
(15.15)
Then
U
()
=
(
)
VF
VG s H t
-
,
-
(15.16)
(
)
=
fst
,
0
is a surface. Figure 15.18 shows how V(F) is the union of points
p
that are the inter-
sections V(G - s,H - t) of offsets of V(G) and V(H). If f is chosen appropriately, then
V(F) becomes our blending surface between G and H.
15.6.1 Theorem.
If
C
is tangent to the s-axis at (a,0), then V(F) is tangent to V(H)
along the curve V(G - a,H). Similarly, if
C
is tangent to the t-axis at (0,b), then V(F)
will be tangent to V(G) along V(G,H - b).
Proof.
See [HofH85].
Since we clearly want a blending surface to be tangent to the surfaces between
which it is a blend, the important consequence of Theorem 15.6.1 is that we have
