**Property h)** A curve
(*x*(*t*), *y*(*t*)) in *U* is
*asymptotic* at *t* if **X**'(*t*) is perpendicular
to **N**'(*t*). The vector **X**'(*t*) is an
*asymptotic vector*; its direction is an *asymptotic
direction*. The curve is an *asymptotic curve* if it is
asymptotic for all *t*. If
(*x*(*t*), *y*(*t*)) is an asymptotic curve,
the point
(*x*(*t*_{0}), *y*(*t*_{0}))
is an *asymptotic inflection point* if the curve
**X**(*t*) has curvature zero at *t*_{0}.

For any curve (*x*(*t*), *y*(*t*)),
**X**'(*t*) ^{.} **N**(*t*) = 0 for
all *t*, so the curve is asymptotic at *t* if
0 = **X**'(*t*) ^{.} **N**'(*t*) = -**X**"(t) ^{.} **N**(t). In
the case of a function graph,

**X**' = (*x*', *y*', *f _{x}x*' +

*f _{xx}*(

(*x*', *y*', *f _{x}x*' +

So there are 0, 1, or 2 asymptotic directions, according as the
discriminant
(*f*_{xy})^{2} - *f*_{xx}*f*_{yy}
is negative, zero, or positive. The discriminant and the Gaussian
curvature have opposite signs, so there are no asymptotic directions
at an elliptic point, one at a parabolic point, and two at a
hyperbolic point. The asymptotic direction at a parabolic point is the
zero principal curvature direction, which is tangent to the parabolic
curve at a Gaussian cusp. (Cf. (a) above).

For a generic immersion **X** , there are three topologically
distinct configurations of asymptotic lines near a cusp of the Gauss
map (see figures 3.1, 3.2, 3.3). One of these (3.1) occurs at a
hyperbolic cusp, and the other two (3.2, 3.3) occur at an elliptic
cusp. (Thanks to J. Callahan for pointing out an error in our original
figure 3.2). This classification follows from the classification of
singularities of generic multiform differential equations. (See [Lak],
[A5], and also [T3], [Bro], [KeT].)

The general form for asymptotic vectors for a canal surface
(example 5) is difficult to express, but it is still possible to
obtain the asymptotic vectors along the parabolic curve. For a curve
given by **X**(*x*(*t*), *y*(*t*)), we have
(assuming d*x*/d*s* = 1)

The condition for an asymptotic direction is then

or

If we are on the parabolic curve where cos

The parabolic curve itself is given by

and the tangent vector to this curve is given by

Thus the tangent to the parabolic curve will be an asymptotic vector if and only if the torsion of the center curve of the canal surface is zero (cf. the warped torus).

If the curve (*x*(*t*), *y*(*t*)) in
*U* has the property that **X**(*t*) is a straight line,
then not only is (*x*(*t*), *y*(*t*)) an
asymptotic curve, but also each point of the curve is an asymptotic
inflection point. For example, there are three such straight lines in
the perturbed monkey saddle, and these lines meet the parabolic curve
precisely at the cusps of the Gauss map (
0). Menn's surface for
= 0 also contains a straight line,
which meets the parabolic curve at the cusp of the Gauss map.

For the case of a function graph, we can explicitly calculate the inflections of asymptotic curves. The following observation simplifies computations considerably:

If
**X**(*x*, *y*) = (*x*, *y*, *f*(*x*, *y*)),
with grad *f*(0, 0) = 0, and
(*t*) = (*t*, *y*(*t*)) is a
curve in the parameter domain *U*, with
(0) = 0, then the curvature of the space curve
**X**(*t*) is zero at 0 if and only if the curvature of
(*t*) is zero at 0 and **X**'(0) is an asymptotic
vector.

To prove this observation, recall that the curvature of
**X**(*t*) is zero at 0 if and only if the vectors
**X**'(0) and **X**"(0) are linearly dependent. We have

**X**'(0) = (1, *y*'(0), grad *f*(0, 0) ^{.} '(0))

**X**"(0) = (0, *y*"(0), grad *f*(0, 0) ^{.} "(0) + *D*^{2}*f*(0, 0)('(0), '(0)))

Now we apply this observation to compute the asymptotic inflection points of Menn's surface. The asymptotic curves are the solution curves of the differential equation

The asymptotic inflection points are characterized, by the above observation, as those points at which d

On the other hand, the parabolic curve is given by

and so if -1/4, the asymptotic inflection curve and the parabolic curve meet only at the origin, where the Gauss map has a cusp.

Theorem 3.1(h) will be proved in chapter 7.

**Property i)** For a curve
(*x*(*t*), *y*(*t*)) with images
**X**(*t*) and **N**(*t*), we have
**N**'(0) = 0 if and only if the normal curvature
*K*_{N} of **X**(*t*) in the surface
**X**(*x*, *y*) is zero at t = 0. In other
words, the principal normal of **X**(*t*) is tangent to
**X**(*x*, *y*), i.e. the osculating plane of
**X**(*t*) is the tangent plane of
**X**(*x*, *y*). If **N** is excellent and
(*x*(*t*), *y*(*t*)) is the parabolic curve,
then **N**'(0) = 0 if and only if
**P** = (*x*(0), *y*(0)) is a cusp of
**N**. So **N** has a cusp at **P** if and only if (i) holds,
provided that **N** is excellent and the principal normal of
**X**(*t*) is defined at **P**, i.e. the curvature of the
parabolic curve **X**(*t*) is nonzero at **P**. This proves
all but the last statement of theorem 3.2. The proof of theorem 3.2
will be completed at the end of chapter 7.

For the shoe surface or the bell surface (example 4), the parabolic
curve **X**(*t*) is planar, and its plane is transverse to
**X**(*x*, *y*), so (i) holds nowhere along the
parabolic curve. For the top half of the torus of revolution (examples
4 and 5), the parabolic curve is also planar, and its plane is tangent
to the torus along the entire parabolic curve. This reflects that the
Gauss map is not excellent.

For Menn's surface we have the parabolic curve

with osculating plane tangent to

-1/6. For = -1/6 the curvature of

If the osculating plane of the parabolic curve **X**(*t*)
equals the tangent plane of **X**(*x*, *y*) at
*t* = 0, then **X**(*t*) does not cross its
osculating plane at *t* = 0. It follows that if the
curvature *K* of **X**(*t*) is not zero at
*t* = 0, then the torsion T is zero at
*t* = 0.

For example, the torsion of the parabolic curve **X**(*t*) of Menn's surface is zero if and only if

provided that the curvature of

A torsion zero of the parabolic curve of a generic surface is not necessarily a cusp of the Gauss mapping. For example the perturbed monkey saddle (example 3) has parabolic curve

which has curvature nowhere zero, provided 0. The torsion zeros of

**Property j)** The asymptotic direction map of **X** assigns
to each point **P** of the parabolic curve, the line through the
origin in **R**^{3} in the asymptotic direction at
**P**. Locally we can choose a unit asymptotic vector field
A(**P**) along the parabolic curve, and the asymptotic map takes
**P** to A(**P**) in *S*^{2} . Let
(*x*(*t*), *y*(*t*)) be the parabolic curve of
**X** , with asymptotic image A(*t*) and normal image
**N**(*t*). Recall that the principal directions of **X**
at a point **P** are the eigenspaces of the Weingarten map from the
tangent space of **X** at **P** to itself. The Weingarten map is
the Jacobian of the Gauss map, followed by parallel translation. Along
the parabolic curve the Weingarten map has rank one, so its image is
the nonzero principal direction. Thus if
**N**'(*t*) 0, then **N**'(*t*) is the
nonzero principal direction. Now A(*t*) is in the zero principal
direction, so A(*t*) is orthogonal to **N**'(*t*) for all
*t*, i.e. A(*t*)-**N**'(t) = 0. (Since
A(*t*) ^{.}**N**(*t*) = 0, this is
equivalent to
A'(*t*) ^{.} **N**(*t*) = 0.) This
means that A(*t*) and **N**(*t*) are __dual__ curves
on the sphere: **N**(*t*) is the envelope of the family of
great circles orthogonal to A(*t*), and vice-versa. Thus the
cusps of **N**(*t*) correspond to inflections of A(*t*),
and vice-versa. This implies all but the last statement of theorem
3.3. The proof of theorem 3.3 will be completed in chapter 7.

Let [*x*, *y*, *z*] denote the line through
the origin in **R**^{3} spanned by the nonzero vector
(*x*, *y*, *z*) (homogeneous coordinates for
the projective plane). The asymptotic map of Menn's surface is

[1, *t*, (4 + 1)t^{3}],

[2(4 + 1)t^{3}, -3(4 + 1)t^{2}, 1]

(0, 1, 3(4 + 1)*t*^{2}) ^{.} (2(4 + 1)*t*^{3}, -3(4 + 1)*t*^{2}, 1) = 0

(1, *t*, (4 + 1)*t*^{3}) ^{.} (6(4 + 1)*t*^{2}, -6(4 + 1)*t*, 1) = 0