**Previous: **Chapter 2: Gauss mappings of surfaces

**Up: **Table of Contents

**Next: **Chapter 2: Examples 4 and 5

### Example 3a. The perturbed monkey saddle:

The modified Gauss
mapping is then

and the parabolic set is given by
0 = = 4 (*x*^{2} + *y*^{2} -
^{2}). Since
grad = 4 (2 *x*, 2 *y*),
the map **Ñ** is good if and only if 0. If
= 0, then the parabolic set is just the origin. The
modified Gauss mapping then has a two-fold ramification point at the
origin, as can be seen by expressing **Ñ** in polar
coordinates:

If = 0, we have

which takes a circle of radius *r* into a circle of radius
*r* doubly covered, with the opposite orientation. If
= 0, the parabolic set is the circle
*x*^{2} + *y*^{2} =
^{2}, parametrized by *x* = |
| cos*t*, *y* = |
| sin *t*. The restriction of **Ñ** to this
circle gives

This
curve is singular when

If > 0, this yields the solutions
*t* = 0, 2/3, 4/3, and if < 0 the
solutions are *t* = , /3, 5/3. In either case,
it is easy to check that **Ñ**'(*t*) = 0
implies **Ñ**''(*t*) 0. For
0, the curve **Ñ**(*t*) is a
hypocycloid of three cusps. Thus the Gauss map of the perturbed
monkey saddle is stable for all 0. This unfolding
of the Gauss map of the monkey saddle is identical with the unfolding
of the complex squaring map described by Arnold [A1, pp. 4-5] and
Callahan [C1, pp. 233-234]. (The focal set of the monkey saddle has
an elliptic umbilic singularity at infinity - cf. Section 6 below.)

#### Figure 2.6

The monkey saddle and its spherical image ( = 0).

#### Figure 2.7

The perturbed monkey saddle and its spherical image ( = -1/2).

#### Figure 2.8

The perturbed monkey saddle and its spherical image ( = 1/2).
### Example 3b. The handkerchief surface:

The modified Gauss mapping is

and the parabolic set is given by
0 = = -4(*x*^{2} - y^{2} - ^{2}).
Since grad = -4(2*x*, -2*y*), the
map **N** is good if and only if 0. If
= 0, the parabolic set is the union of the two lines
*y* = *x* and *y* = -*x*, and
the modified Gauss mapping is a ``quarter folded handkerchief''

which maps each of the four quadrants
*A* = {(x, y) | *x* y, *x* -*y*},
*B* = {(*x*, *y*) | *x* *y*, *x* -*y*},
*C* = {(*x*, *y*) | *x* *y*, *x* -*y*},
*D* = {(*x*, *y*) | *x* *y*, *x* -*y*} homeomorphically
onto the quadrant *C*. For 0 the parabolic curve is the
hyperbola *x*^{2} - *y*^{2} = ^{2}. To find the cusps of Ñ, we
parametrize the parabolic curve by *x* = ± cosh *t*, *y* = sinh *t*. The restriction of Ñ to the parabolic curve is then

This curve is singular when

For 0 this
yields the unique solution *t* = 0,
*x* > 0, and Ñ'(t) = 0 implies
Ñ"(t) 0. So the Gauss
map of the handkerchief surface is stable for all 0. This family
of Gauss maps is the same as the unfolding of the quarter folded
handerchief described by Arnold [A1] and
Callahan [18]. (The focal set of the
handkerchief surface for = 0 has a
hyperbolic umbilic singularity at infinity.)

#### Figure 2.9

The handkerchief surface and its spherical image ( = 0).

#### Figure 2.10

The handkerchief surface and its spherical image
( = 1/2).

#### Figure 2.11

The handkerchief surface and its spherical image ( = -1/2).

**Previous: **Chapter 2: Gauss mappings of surfaces

**Up: **Table of Contents

**Next: **Chapter 2: Examples 4 and 5