Graphics Reference
In-Depth Information
9.12.4. Theorem.
Let k(s) be the curvature function of a space curve g : [a,b] Æ
R
3
.
If K is the Gauss curvature of a canal surface
S
that has constant radius function and
center curve g, then
1
4
S
b
Ú
()
Ú
k sds
=
K dA
.
a
Proof.
See [Spiv75].
9.13
Involutes and Evolutes of Surfaces
The discussion of involutes and evolutes of curves extends to surfaces in interesting
ways. We already defined the evolute (or surface of centers) of a surface in Section
9.9. It also played a role in Theorem 9.12.3. The evolute consists of the set of foci on
the normal lines at the points of the surface. The foci will be the same point if and
only if
p
is an umbilic point. Furthermore, Theorem 9.10.10 implies that one of those
points will be at infinity if and only if the Gauss curvature vanishes at
p
. The common
name, “surface of centers,” for the evolute suggests that it is a surface, but that is not
necessarily the case. For a sphere, the evolute is just the center of the sphere, but one
can show that the sphere is the only surface for which the evolute is a point. Theorem
9.12.3 shows that some surfaces have evolutes that consist of curves, but one can
describe all such surfaces. The only surfaces with this property were discovered by
the French mathematician Dupin in 1822 ([Dupi22]).
Definition.
A
Dupin cyclide
or simply
cyclide
is a surface whose evolute consists of
two curves. The two curves are called the
spines
of the cyclide.
Figure 9.30 shows a “ring” cyclide which is a kind of torus, but rather than rotat-
ing a circle of fixed radius about another circle the radius of the rotating circle varies.
A cyclide is a special case of a canal surface where only
one
sheet of the evolute was
required to be a curve. Cyclides are very interesting surfaces and can be defined in a
number of different ways ranging from very geometric constructions to explicit for-
mulas. We shall discuss a few of their major geometric properties. Good references
for more details are [ChDH89], and [Prat90], [Boeh90], and [HilC99].
First of all, the definition of a cyclide we have given is due to Maxwell ([Maxw68])
and is not Dupin's original definition.
Figure 9.30.
A ring cyclide.
