Blaga P. Lectures on the differential geometry of - tiera.ru

152 Chapter 4. General theory of surfaces
Let us assume, now, that the curve ρ from the previous theorem is biregular, which
means, in particular, that is curvature is always strictly positive. From the definition of
the normal curvature it follows immediately that, if ν is the unit principal normal vector
of the curve, then the curve is anasymptotic line if and only if
ν(t) · n(u(t), v(t)) = 0,
where n is the unit normal vector to the surface. This actually means that the principal
normal of the curve lies on the tangent plane of the surface, at each point of the curve.
Thus, we obtain the following characterization of the asymptotic lines:
Theorem 4.3. Let S be an oriented surface and ρ a biregular parameterized curve on
S . Then ρ is an asymptotic line if and only if at each point its osculating plane coincide
with the tangent plane to the surface at that point.
Another immediate result concerning the asymptotic lines is the following:
Proposition 4.14.2. Let S be an oriented surface and (U, r = r(u, v)) – a local parameterization
of S . Then the coordinate lines u = const and v = const are asymptotic lines
on r(U) if and only if D = D ′′ = 0.
Example. Let us consider the helicoid, given by the parameterization
⎧
x = u cos v,
⎪⎨
y = u sin v
⎪⎩ z = b · v
where b is a constant. A straightforward computation leads to D = D ′′ = 0, D ′ (u, v) =
−b/ √ b 2 + u 2 . This means that, in this particular case, we have, at each point of the
surface, two asymptotic lines and these are nothing but the coordinate lines, u = const
and v = const. We notice that the helicoid is a ruled surface (see the last chapter) and at
each point, one of the coordinate lines is a straight line (or a line segment). This is, of
course, the line v = const (see the figure 4.14).
4.15 The classification of points on a surface
The first fundamental form of a surface is positively defined. The second one is not. This
is fortunate, as it allows us to give a classification of the points of the surface, according
to the sign of the second fundamental form or, more specifically, according to the sign
of its discriminant DD ′′ − D ′2 .