Graphics Reference
In-Depth Information
z
z
H
1
L
H
2
L
y
y
(0,b,0)
B
Q
(-c,0,0)
(-a,0,0)
A
(c,0,0)
P
B
A
(a,0,0)
x
(0,-b,0)
P
x
E
(a)
(b)
Figure 9.32.
A string construction for a cyclide.
wrap it around
E
. The cyclide is the locus of points swept out by the end point
B
of
the string as we keep it taut and ensure that there are no lateral forces at the point
P
where it touches
E
. The line
L
through
P
and
B
will intersects
H
.
9.13.2. Theorem.
Cyclides are the only surfaces all of whose lines of curvature are
circles.
Proof.
See [HilC99].
It is useful to divide the cyclides into different types. Recall that the spines of
cyclides are anticonics.
Definition.
A
central
,
parabolic
, or
revolute cyclide
is a cyclide whose spines are an
ellipse/hyperbola, parabola/parabola, or circle/straight line pair, respectively. A
degen-
erate cyclide
is a cyclide at least one of whose spines is a degenerate conic.
For a finer subdivision of each cyclide type into “horned,” “ring,” and “spindle”
cyclides see [ChDH89].
Now, in general, the evolute of a surface
S
consists of two sheets of surface. One
can show that the normals to
S
are tangent to the evolute at the foci. Therefore the
normals to
S
are the common tangents to the two sheets of the evolute. One can turn
this around.
Definition.
Given two arbitrary surfaces
S
1
and
S
2
, a surface
S
whose normals define
a family of lines that are tangent to both of the given surfaces is called the
involute
of the surfaces.
The involute
S
has the property that the two surfaces
S
1
and
S
2
are the evolute
of it. One can show that a necessary and sufficient condition for this to happen is
that the tangent planes at the points where a normal of
S
touches
S
1
and
S
2
must
be orthogonal. A pair of confocal quadrics of unlike type is an example of this. See
[HilC99].
