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

1.4. Analytical representations of curves 29
the neighborhood of that point, but we cannot claim the opposite, either. Consider, as a
trivial example, the equation
Then we have
and if we denote
F(x, y) ≡ (x − y) 2 = 0.
grad F(x, y) = 2{x − y, −(x − y)}
C = {(x, y) ∈ R 2 | F(x, y) = 0},
then grad F = 0 at all the points of C. But, clearly, C is a curve (it is easy to see that it is
the first bissector of the coordinate axes, i.e. a straight line).
1.4.2 Space curves
Parametric representation. As in the case of plane curves, with a local parameterization
⎧⎪⎨⎪⎩
x = x(t)
y = y(t)
z = z(t)
(1.4.6)
we can represent either the entire curve, or only a neighbourhood of one of its points.
Explicit representation. If f, g : I → R are two smooth functions, defined on an open
interval from the real axis, then the set
C = {(x, f (x), g(x)) ∈ R 3 | x ∈ I} (1.4.7)
is a simple curve, with a global parameterization given by
⎧
x = t
⎪⎨
y = f (t)
⎪⎩ z = g(t)
. (1.4.8)
The equations ⎧
⎪⎨ y = f (x)
⎪⎩ z = g(x)
(1.4.9)
are called the explicit equations of the curve. Let us note that, in fact, each equation of
the system (1.4.9) is the equation of a cylindrical surface, with the generators parallel
to one of the coordinate axis. Therefore, representing explicitly a curve actually means
representing it as an intersection of two cylindrical surfaces, with the two families of
generators having orthogonal directions.