**Property c)** In terms of height functions, this condition
says that there is a unit vector **V** near **N**(**P**) such
that _{V} has two
distinct critical points **Q**_{1} and **Q**_{2} near
**P** with
_{V}(**Q**_{1}) =
_{V}(**Q**_{2}).
For a function graph
**X**(*x*, *y*) = (*x*, *y*, *f*(*x*, *y*)),
this condition translates to **N**(**Q**_{1}) =
(*a*, *b*) = **N**(**Q**_{2}) and
_{(a, b)}(**Q**_{1}) =
_{(a, b)}(**Q**_{2}), i.e.

*f*_{x}(*x*_{1}, *y*_{1}) =*a* = *f*_{x}(*x*_{2}, *y*_{2})

*f*_{y}(*x*_{1}, *y*_{1}) = *b* = *f*_{y}(*x*_{2}, *y*_{2})

*f*(*x*_{1}, *y*_{1}) + *ax*_{1} + *by*_{1} = *f*(*x*_{2}, *y*_{2}) + *ax*_{2} + *by*_{2}

so (

For Menn's surface we use the symmetry about the plane of the
second two coordinates. If
*x*_{1} = -*x*_{2} and
*y*_{1} = *y*_{2}, then
*f*(*x*_{1}, *y*_{1}) = *f*(*x*_{2}, *y*_{2}),
*f*_{x}(*x*_{1}, *y*_{1}) = -*f*_{x}(*x*_{2}, *y*_{2}), and
*f*_{y}(*x*_{1}, *y*_{1}) = *f*_{y}(*x*_{2}, *y*_{2})
Thus if
*f*_{x}(*x*, *y*) = 0,
*x* 0, then
**Q**_{1} = (*x*, *y*) and
**Q**_{2} = (-*x*, *y*) will be a
pair of distinct points with the same tangent plane. Now
0 = *f*_{x}(*x*, *y*) = 4*x*^{3} + 2*x**y* = 2*x*(2x^{2} + *y*) if and only if
*x* = 0 or y = -2*x*^{2}. Since this last curve
contains **P** = (0, 0), (c) holds for Menn's
surface. (Note that if = -1/4,
this curve coincides with the parabolic curve, since the Gaussian
image of the parabolic curve is a single point. The same thing happens
at the top of a torus of revolution.) Theorem 3.1 (c) will be proved
in chapter 5.

**Property d)** Suppose that **P** in *U* is not an
umbilic point of **X**, i.e. **X** has two distinct principal
curvatures *k*_{1} and *k*_{2} at
**P**. Let **V**_{1} and **V**_{2} be unit
vectors in the corresponding principal directions. Then **P** is a
ridge point of **X** , with associated principal curvature
*k*_{1}, if the directional derivative of
*k*_{1} in the direction **V**_{1} is zero at
**P** [Por1].

Suppose that the plane *g* in **R**^{3} is a plane
of symmetry of the surface **X** . The intersection of *g*
with the image of **X** is a ridge curve of **X**. More
precisely if **X**(**P**) is in *g*, then **P** is
either a ridge point or an umbilic point. For let
: (-*a*, *a*) -> *U* be a curve
with (0) = **P**, and let
= **X** ° . If we choose
so that r((*t*)) = (-t), where r
is reflection in *g*, and *t* is in
(-*a*, *a*), then '(0) will be in a principal
direction. ('(0) is an eigenvector of the Jacobian of the
Gauss map of **X**, by symmetry.) Suppose that **P** is not an
umbilic point, and let *k* be the principal curvature in the
direction of '(0). Along the curve , we have
*K*(*t*) = *K*(-*t*), so
*K*'(0) = 0, i.e. the directional derivative of
*k* in the direction of '(0) is zero, so **P** is a
ridge point of **X**. Note that for condition (d) to be satisfied,
we must also have that *K*(0) = 0.

The shoe surface has the plane *y* = 0 as a plane
of symmetry, so (*t*, 0) is a curve of ridge points of
**X**. The parabolic curve is (0, *t*), so this ridge
crosses the parabolic curve at **P** = (0, 0). We
can take (*t*) = (0, *t*), the parabolic
curve, so
(*t*) = (0, *t*, -1/2*t*^{2}),
and *K*(0) is the curvature of at zero, since the plane
of is normal to **X** at **P**. So
*K*(0) 0, and (d) does not hold at **P**.

Similarly, the plane *x* = 0 is a plane of symmetry
of Menn's surface, so (0, *t*) is a ridge curve, which
crosses the parabolic curve (*t*,-(6
+ 1)*t*^{2}) at **P**
= (0, 0). If we let
(*t*) = (*t*, 0), then
(*t*) = (*t*, 0,
*t*^{4}) and
(0) = 0, so (d) does hold at **P**.

The perturbed monkey saddle has three planes of symmetry:
*y* = 0,
*y* = 3^{1/2} *x*,
*y* = -3^{1/2} *x*. So we have three
ridge curves (*t*, 0),
(*t*,3^{1/2} *t*),
(*t*, -3^{1/2} *t*). Recall that if
0 the parabolic curve is the circle
*x*^{2} + *y*^{2} =
^{2}, which crosses the ridge curves at
the six points *r* = ||,
= 0, /3, 2/3, , 4/3, 5/3 in polar
coordinates. For example consider the ridge curve (*t*, 0)
which crosses the parabolic curve at
**P**_{1} = (, 0)
and **P**_{2} = (-
, 0). Let
_{1}(*t*) = (
, *t*), so
_{1}(*t*) = (
, *t*,4
^{3}/3). The curvature of
_{1} is zero at *t* = 0
(_{1} is a straight line) so (d) is satisfied at
**P**_{1}.
Let _{2}(*t*) = (-
, *t*), so
_{2}(*t*) = (-
, *t*, 2^{3}/3 + 2*t*^{2}), and the curvature of
_{2} is zero at *t* = 0, so (d) is not
satisfied at **P**_{2}. Since **X** has 3-fold symmetry
about the *z*-axis, we conclude that if
> 0 then (d) is satisfied at
*r* = ,
= 0, 2/3, 4/3, and if
< 0 then (d) is satisfied at
*r* = -,
= /3, , 5/3.

For surfaces of revolution every plane containing the *z*-axis
is a plane of symmetry, so all the meridians are ridges. Along a
circle of latitude corresponding to an inflection of the profile
curve, the principal curvature associated to the meridian ridges is
constant, nonzero. Along a latitude corresponding to an extremum of
the profile curve the principal curvature associated to the meridian
ridges is identically zero.

Finally, the warped torus has *n* vertical planes of symmetry,
each of which intersects the parabolic curve in four points, and (d)
holds at each of these points.

Theorem 3.1(d) will be proved in chapter 6.

**Property e)** For example, the parallel surface of a canal
surface of a space curve is a parallel tube of . A
parallel tube of radius *D* has swallowtail singularities at
those points in space where the osculating sphere of has radius
*D*. (Cf. chapter 6 below.) Thus (e) says that the osculating
sphere of has infinite radius, which occurs at the torsion
zeros of . Theorem 3.1(e) will be proved in chapter 6.

**Property f)** If we assume that the singular locus of the
pedal surface of **X** from the point **A** is a cuspidal edge
with isolated swallowtail points, and a degenerate singularity at
**A** itself, then it is easy to check that the swallowtail points
of the pedal surface correspond to the cusps of the Gauss map. (This
assumption is valid if the Gauss map is stable, as we shall show in
chapter 6). For simplicity we assume that **A** is the origin of
**R**^{3}. The pedal map **W** of **X** from the
origin is defined by

**W** = (**X** ^{.} **N**)**N**,

**W**_{x} = (**X**_{x} ^{.} **N** + **X** ^{.} **N**_{x})**N** + (**X** ^{.} N)N_{x},

**W**_{y} = (**X**_{y} ^{.} **N** + **X** ^{.} **N**_{y})**N** + (**X** ^{.} N)N_{y},

**W**_{x} x **W**_{y}
=
(**X**_{x} ^{.} **N** + **X** ^{.} **N**_{x})
(**X** ^{.} **N**)
**N** x **N**_{y}

+
(**X**_{y} ^{.} **N** + **X** ^{.} **N**_{y})
(**X** ^{.} **N**)
**N**_{x} x **N**

+
(**X** ^{.} **N**)^{2} **N**_{x} x **N**_{y}

**W**(*t*) = (**X**(*t*) ^{.} **N**(*t*))**N**(*t*),

**W**'(*t*) = (**X**'(*t*) ^{.} **N**(*t*) +
**X**(*t*) ^{.} **N**'(*t*))**N**(*t*) +
(**X**(*t*) ^{.} **N**(*t*))**N**'(*t*)

If
**X**(*t*) ^{.} **N**(*t*) 0,
then **W**'(*t*) = 0 only if **N**'(*t*) is
parallel to **N**(*t*), which can happen only if
**N**'(*t*) = 0. Thus the cusps of the cuspidal
edge **W**(*t*), which are the swallowtail points of **W**,
correspond to the cusps of the parabolic image **N**(*t*),
which are the cusps of the Gauss mapping **N**.

Theorem 3.1(f) will be proved in chapter 6.

**Property g)** The space curve (*t*) has
*n*^{th} order contact with the surface
*G*(*x*, *y*, *z*) = 0 at the
point (*t*_{0}) if and only if the function
*G*(*t*) = *G*((*t*)) vanishes to
order *n* at *t*_{0}:

(cf. [St, p. 24]). Thus a line is tangent to a surface if and only if it has order of contact at least one with the surface. For the shoe surface, we have

The origin (0, 0, 0) is a parabolic point, and the
tangent plane at the origin is horizontal. For the tangent line
(*t*) = (*t* cos *s*, *t* sin *s*, 0),
0 < *s* < ,
*G*^{(2)}(0) = -sin^{2} *s*,
*G*^{(3)}(0) = 2cos^{3} *s*,
*G*^{(4)}(0) = 0, so has first order
contact with the shoe if *s* 0 and second order
contact if *s* = 0. Next consider Menn's surface

Again (0, 0, 0) is a parabolic point with horizontal tangent plane, and for the line above we have

The perturbed monkey saddle contains the three lines

which are tangent to the parabolic curve at the three points

respectively. These three points are cusps of the Gauss map.

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