Graphics Reference
In-Depth Information
Figure 15.19.
Blending with potential surfaces.
The curve f(s,t) = 0 defines a parabola (see Figure 15.19(b)) and satisfies the hypothe-
ses in Theorem 15.6.1, so that V(F) will be the surface we want, where
(
)
=
4
2
2
2
4
3
2
2
F x y z
,,
x
+
2
x y
-
22
x y
+
y
-
22
y
-
28
x
+
93
y
+
176
y
+
1164
is defined by equation (15.15). See Figure 15.19(a), which shows the horizontal slice
of our surfaces in the xy-plane.
Our construction created a blending surface by a sweeping operation of curves
parameterized by a curve f in a plane. We can think of this as a three-dimensional
construction by thinking of f as defining a conic cylinder in x-y-z space.
Another well-known implicit blending approach is the
rolling ball-type blend
.
As the name suggests, one rolls a ball along the two surfaces one is trying to blend.
The ball will touch the surfaces in a tangential way. Mathematically, the surface
generated by the rolling ball is a canal surface. One problem is that the blending
surface that is obtained in this way is defined by complicated equations even for a
blend between relatively simple surfaces such as cylinders. For that reason one has
sometimes used approximations ([RosR84]). Klass and Kuhn ([KlaK92]) describe a
unified approach to finding a fillet surface based on a rolling ball approach. After
determining the intersection curves that are needed for trimming the original sur-
faces, they end up defining a bicubic Bézier blending surface.
More complicated yet are
variable radius rolling ball blends
. This leads us to
cyclides. The definition and geometry of these surfaces are discussed elsewhere in this
topic (see Section 12.13 in this topic and Section 9.13 in [AgoM05]). The reason for
the renewed interest in them is precisely because of their usefulness for blending.
Figure 15.20 shows how a cyclide can be a blending surface between a cylinder and
a plane. More generally, cyclides work well for blending between the basic quadrics,
namely, planes, spheres, cylinders, and cones. Allen and Dutta ([AllD97a] and
[AllD97b]) define the problem carefully and show that cyclides can achieve singular-
ity-free variable radius rolling ball blends. They give necessary and sufficient condi-
tions for the existence of certain blends and describe constructions for them. They
also indicate limitations with single cyclide blends. The paper [AllD97c] extends their
results to supercyclides. The supercyclides allow more freedom in the shape of blends
