Gauss' original draft of his 1827 paper on curved surfaces was written in 1825 [Gau, p. 117]. We follow the outline of the 1825 draft and begin our study with the case of a smooth curve in the plane.
Let X(t),
a < t < b, be a regular smooth
plane curve, so that the velocity vector X'(t) is
non-zero at each value of t. We define the unit tangent
vector T(t) by the condition
X'(t) = |X'(t)| T(t)
and we let N(t) denote the unit normal vector
obtained by rotating T(t) ninety degrees in a
counter-clockwise direction. The curve N is called the
circular image of the curve X, or the Gauss
mapping of X. Since
N(t) . N(t) = 1, we
have
N'(t) . N(t) = 0,
so we may define the curvature k(t) by the condition
N'(t) = -k(t) X'(t). The
singularities of the Gauss mapping (those points where it is not
one-to-one) are the points where k(t) = 0. A
curve X is general if
k'(t) 
0 whenever
k(t) = 0, which implies that the curvature
changes sign as it passes through a singular value. For such a curve,
the singularities of the Gauss map are all folds, where the
circular image doubles back on itself (Figure 1.1). This situation is
stable: folds of the Gauss map of X persist under small
perturbations of the curve X.
If X(t) = (t, f(t))
for a smooth function f(t), then
X'(t) = (1, f'(t)) so
T(t) = (1 / [1+[f'(t)]2]1/2) (1, f'(t))
and
N(t) = 1/[1+[f'(t)]2]1/2(-f'(t), 1).
The curvature is given by
k(t) = f"(t)/{1+[f'(t)]2}3/2,
so the singularities of the Gauss map N occur at the inflection
points of f. The condition that X is general is that
f'''(t) 0 whenever
f"(t) = 0.
The singularities of the Gauss map of a plane curve can be characterized geometrically in several ways:
0. (Contact
of order n is also called (n+1)-point contact.) Thus a
line L has order of contact greater than 1 with a regular curve
X at t if and only if L is the tangent line to
X at t. If X is general, then no line has
third-order contact with X, and L has second-order
contact with X at t if and only if X has an
inflection at t, i.e. t is a singular point of the Gauss
map.
V(t) = X(t)
. V
For a regular curve X, t is a critical point of the
function V(t) if
and only if V = ± N(t). (If
X(t) = (t, f(t)) and
V = (0, 1), then V(t) = f(t)). If
X is general, and t is not a singular point of the Gauss
map, then V(t) has
a local minimum [maximum] at t if
V = (sign k)N [V = -(sign k)N].
If t is a singular point of the Gauss map, and
V = ±N(t), then V(t) has a degenerate critical point
at t. If V is moved slightly in one direction, this
critical point bifurcates into a local maximum and a local minimum.
If V is moved slightly in the other direction, the critical
point disappears.
All three of the characterizations (a), (b), (c) can be interpreted
using the fact that the Gauss map is the catastrophe map of the family
V(t) of real-valued
functions, parametrized by V on the unit circle. We now turn
to characterizations which bear the same relation to the family of
distance functions from points in the plane.
E(t) = X(t) + 1/k(t) N(t)
The singularities of the evolute occur when
k'(t) = 0, i.e. t is a vertex of
X. If we suppose that k"(t) 0 when
k'(t) = 0, then no circle has contact greater
than 3 with X, and C has third order contact with
X at t if and only if C is tangent to X at
t, the center of C is the center of curvature of
X, and t is a vertex of X. Furthermore, the
singularities of the evolute are cusps. As t approaches an
inflection point of X, the evolute curve goes to infinity, so
the singular points of the Gauss mapping are the infinite points of
the evolute.
DA(t) = |A - X(t)|2
For a regular curve X, t is a critical point of
DA if and only if A lies on the normal line
to X at t. This critical point is degenerate if and only
if A is the center of curvature of X at t. If
t is not a vertex of X, this critical point bifurcates
into a maximum and a minimum as A moves to one side of the
evolute curve, and the critical point disappears as A moves to
the other side of the evolute curve. If
k'(t) = 0 and
k"(t) 0, and A is the center of
curvature of X at t, then the critical point of
DA at t bifurcates into a maximum and two
minima as A moves inside the cusp (Figure 1.4). In the
terminology of catastrophe theory, the image of the evolute curve is
the bifurcation set of the family of real-valued functions
DA (cf. [T2] [Gu]).
The curve X(t) = (t, t2) and its evolute E, with normals to points A inside the cusp of E and B outside the cusp of E.
To relate the family DA to the Gauss map of X, we consider for each r > 0 the curve of points in the plane at distance r from X, the parallel curve at distance r:
Xr (t) = X(t) + r N(t)
We find the singularities of Xr(t)
by computing
0 = Xr'(t) = X'(t) + r N'(t) = (1-r k(t)) X'(t).
Since X'(t) 0 for all t (X
regular), we get a singularity at t0 when
k(t0) = 1/r, i.e. precisely
when the radius of curvature of X is r. This is exactly
when the parallel curve intersects the evolute:
Xr(t0) = E(t0).
In fact the parallel curve Xr has a
cusp in general at such a point t0, for if
k(t) - 1/r changes sign at
t0 then
As we let the distance r approach infinity, the values of t for which k(t) = 1/r will approach the values for which k(t) = 0, the singularities of the Gauss mapping. We may interpret this phenomenon by stating: The Gauss mapping is the parallel curve at infinity. We may express this in another way by considering the linear interpolation between the curve X and its Gauss map N, i.e. the family of curves Iu(t), 0 < u < 1 defined by
Iu(t) = (1-u)X(t) + uN(t)
= (1-u) (X(t) + u/(1 - u) N(t))
In this preliminary section we have seen how the singularities of the Gauss mapping of a plane curve may be characterized from several geometric viewpoints. The real power of these methods comes forth when we apply the same approach to the study of surfaces.
