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Example 3a. The perturbed monkey saddle:

The modified Gauss mapping is then

and the parabolic set is given by 0 =  = 4 (x2 + y2 -  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 x2 + y2 =  2, parametrized by x = | | costy = | | 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(x2 - y2 - 2). Since grad  = -4(2x, -2y), 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 = {(xy) | x  yx  -y}, C = {(xy) | x  yx  -y}, D = {(xy) | x  yx  -y} homeomorphically onto the quadrant C. For   0 the parabolic curve is the hyperbola x2 - y2 = 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).

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