Graphics Reference
In-Depth Information
S
2
S
2
S
3
S
3
S
S
1
S
1
S
(a)
(b)
Figure 9.31.
A cyclide as an envelope of spheres.
Dupin's definition
([Dupi22])
:
A
cyclide
is the envelope of spheres that touch three
fixed spheres.
The example in Figure 9.31 captures Dupin's basic idea. Figure 9.31(a) shows four
spheres
S
1
,
S
2
,
S
3
, and
S
with centers in the x-y plane. The sphere
S
touches the other
three. Imagine raising the center of the sphere
S
in the z-direction while at the same
time letting its radius expand so that we maintain contact with the other three spheres.
We keep raising the center in this way until the radius becomes “infinity.” At that point,
we wrap around to “minus infinity” and let the radius decrease so that the z-coordi-
nate of the center now moves from minus infinity to zero. Note that in the first stage
the spheres
S
i
were outside of
S
but now they are inside. See Figure 9.31(b). The enve-
lope of the moving sphere
S
is our cyclide. See [ChDH89] for a more thorough dis-
cussion of this process and why Maxwell's and Dupin's definition of a cyclide are
equivalent.
Definition.
A pair of conics is said to be an
anticonic pair
or
anticonics
if they lie in
orthogonal planes and the vertices of one are the foci of the other and vice versa.
9.13.1. Theorem.
The two spines of a cyclide are conics. In fact, they are an anti-
conic pair.
Proof.
See [ChDH89].
There is a nice geometric construction using a string on which Maxwell's defini-
tion is based. Figure 9.32(a) shows the construction in the special case of a torus. We
tie a string of fixed length to the center
A
of a planar circle and then wrap the string
around the circle. The torus will consist of all points swept out by the endpoint
B
of
the string assuming that we keep it taut and there are no lateral forces at the point
P
where the string touches the circle. It is easy to see that the normal lines to the surface
through
P
and
B
will either intersect the vertical axis
L
or be parallel to it. Figure
9.32(b) shows the construction for a general cyclide. Consider an ellipse
E
in the x-y
plane with foci at (±c,0,0) and vertices at (±a,0,0) and (0,±b,0). Let
H
1
and
H
2
be
branches of the hyperbola
H
in the x-z plane whose foci and vertices are the vertices
and foci of the ellipse
E
, respectively. Tie a string to the focus
A
= (-c,0,0) of
E
and
