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### Example 4. Surfaces of revolution:

We obtain

So
**X**_{x} times **X**_{y} = (-*x* *f*'(*x*) cos *y*, -*x* *f*'(*x*) sin *y*, *x*).
If *x* 0 then **X** is regular, and for
*x* > 0 the unif normal is

so again a modified Gauss mapping, equivalent to **N** is given by

The parabolic set occurs when
0 = = -*f*'(*x*) *f*"(*x*),
i.e. at extrema or at inflections of the profile curve. Furthermore,
grad = (-(*f*"(*x*))^{2} - *f*'(*x*) *f*'''(*x*), 0),
so Ñ is good if *f*"(*x*) = 0 implies
*f*'(*x*) 0 and
*f*'''(*x*) 0.

If *x*_{0} is a value for which
*f*''(*x*_{0}) = 0, then the parabolic
curve can be parametrized by
*x*(*t*) = *x*_{0},
*y*(*t*) = *t*, and we obtain

If *f*'(*x*_{0}) 0 and
*f*'''(*x*_{0}) 0, then the Gauss map
has an ordinary fold along the parabolic curve. For example, these
conditions are satisfied by the * bell surface*:

On the other hand, if *x*_{0} is a value for which
*f*'(*x*_{0}) = 0 and
*f*"(*x*_{0}) 0, then the Gauss map
is good, but * not* excellent, because the parabolic curve is
parametrized by *x*(*t*) = *x*_{0},
*y*(*t*) = *t*, and
**N**(t) = (0, 0) for all t. An example is the top
of a * torus of revolution*:

0 < b < *a*,
*a* - *b* < *x* < *a* + *b*,
with *x*_{0} = *a*.

#### Figure 2.12

The bell surface and its spherical image.

#### Figure 2.13

A torus of revolution.

#### Figure 2.14

The sperical image of the torus of revolution.

### Example 5. The canal surface of a space curve.

Let be a regular space curve wit curvature nowhere zero.
Define the * canal surface* about of radius *r* to be

where *P* and *B* are the principal normal and the binormal of the
curve . To find the normal of the surface **X**, we form

where and are the curvatuer and torsion of ,
and *s* is the arclength along . Then

so **X** will be regular if *r* is a sufficiently small positive
number. Moreover,

so the parabolic set occurs when the two vectors

are linearly dependent, i.e. when
cos *y* = 0. This occurs at the curves
*x*(*t*) = *t*,
*y*(*t*) = ±/2, for which we have

A straightforward calculation gives the Gaussian curvature
*K* = cos *y*/(*r* (*r*cos *y* - 1))
and
*K*/*y* = sin *y*/(*r* (*r*cos *y* - 1)^{2}),
so *K* = 0 implies grad *K* 0,
since 0. Therfore the Gauss map is good. Now
taking derivatives with respect to *t*, we obtain

so **N**' = 0 if and only if = 0 and
then **N**" = '*s*'*P*. So the Gauss
map **N** is excellent if ' 0 whenever
= 0, and then **N**(*x*, *y*) has a
cusp at (*t* ,±/2) if and only if
(t) = 0.

For example consider the *warped torus*, a canal surface of
the space curve

(*t*) = (cos *t*, sin *t*, sin(*nt*))

where *n* is an integer, *n* 2. The
curvature of is nowhere zero, since
(*s*')^{3} = |(' *x* ")| 1.
Furthermore

so = 0 if and only if
*t* = /2*n*, 3/2*n*, ...,
(4*n* - 1)/2*n*, provided that
0. Taking derivatives of
^{2}(*s*')^6 and
*n*(1 - *n*^{2})cos *n**t*
shows that *t*au = 0 implies ' 0, so
long as 0. Therfore a canal surface of has
an excellent Gauss map, with 4*n* cusps. For = 0 a
canal surface of is a torus of revolution, and each component
of the parabolic curve is collapsed to a point by the Gauss map.

#### Figure 2.15

The warped torus (n = 2).

#### Figure 2.16

The spherical image of the warped torus
(*n* = 2). (The parabolic image is two anitpodal
curves with 4 cusps each. The left picture is the spherical image of
the hyperbolic region, the right is the spherical image of the
elliptic region.)

#### Figure 2.17

The warped torus (n = 3).

#### Figure 2.18

The spherical image of the warped torus
(*n* = 3). (The parabolic image is two antipodal curves
with 3 cusps each. These curves are doubly covered by the parabolic
curve itself. The left picture is the spherical image of the
hyperbolic region, the right is the spherical image of the elliptic
region.)

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**Next: **Chapter 3: Characterizations of Gaussian cusps