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# 1 Prelude: Gauss mappings of plane curves

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(tX'(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) = (tf(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.

### Example 1.

If f(t) = 1/3 t3 then k(t) = 2 t/[1 + t4]3/2 so the curvature changes sign at exactly one point, where t = 0. The curve has an inflection point at the origin, and the Gauss map has a fold at this point (Figure 1.1). This situation is stable in that any nearby curve obtained by adding lower order terms to the function f(t) = 1/3 t3 will also have a single inflection point.

#### Figure 1.1

The curve X(t) = (t, 1/3 t3), -1 < t < 1, with its normal vectors N(t) and circular image.

### Example 2.

If f(t) =ف/4 t4 then k(t) = 3t2 / (1+t6)3/2 so again the only place where the curvature can vanish is at t = 0. However the curvature does not change sign at this point and the Gauss mapping does not have a fold (Figure 1.2). The situation is not stable since the nearby curves f(t) =ف/4 t4+a t2 will have inflection points at the solutions of 0 = f''(t) = 3t2 + 3a, so there will be two inflections if a < 0 and none if a > 0.

#### Figure 1.2

The curve X(t) = (t,ف/4 t4), -1 < t < 1, with its normal vectors 1/2 N(t) and circular image.

### Example 3.

If f(t) = 1/5 t5 then k(t) = 4t3 / (1+t8)3/2 so the curvature changes sign when t= 0. However k'(0)= 0 so this curve is not general. The nearby curves determined by the functions f(t) = 1/5 t5 + a t3 will have inflections where 0 = f"(t) = 4t + 6a t so there will be one inflection if a > 0 and three inflections if a < 0.

The singularities of the Gauss map of a plane curve can be characterized geometrically in several ways:

### a) Contact with lines

(cf. [St, p. 24]). The curve X(t) = (tf(t)) has contact of order n with the x-axis at the origin if and only if f(k)(t) = 0 for 0 < k <n and f(n+1)(t 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.

### b) Projection to lines.

For a unit vector V in the plane, consider the composition of X with orthogonal projection to the line spanned by V:

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.

### c) The pedal curve.

The pedal curve W of a regular curve X is defined by W(t) = (X(t. N(t)) N(t)
The point W(t) is the foot of the perpendicular from the origin to the tangent line of X at t. For a general curve X, which does not pass through the origin, an easy computation shows that the singularities of the pedal curve W are isolated cusps which occur precisely when t is an inflection point of X (Figure 1.3).

#### Figure 1.3

The curve X(t) = (t + 3/4, 1/3 t3 + 3/2) and its pedal curve W.

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.

### d) Contact with circles.

A circle C has order of contact greater than or equal to 1 with a regular curve X at t if and only if C is tangent to X at t. The circle C has order of contact greater than or equal to 2 with X at t if and only if C is tangent to X at t and the center A of C coincides with the center of curvature of X at t. The locus of centers of curvature of X is the evolute curve:

E(t) = X(t) + 1/k(tN(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.

### e) Distance from points.

Dual to contact with circles is the analysis of critical points of the family D of distance functions from points A in the plane:

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]).

#### Figure 1.4

The curve X(t) = (tt2) 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 - uN(t))

Each intermediate curve Iu(t) is a homothetic image of the parallel curve at distance r = u/(1-u), and the limit as u -> 1 of Iu(t) = N(t). In terms of singularities, we may state the basic observation: The folds of the Gauss mapping are the limits of the cusps of the parallel curves.

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.

#### Figure 1.5

The curve X(t) = (t,t2), its evolute E, and its parallel curve X1 at the distance 1.

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