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

1.4. Analytical representations of curves 27
entire curve. Thus, any point a of the curve has an open neighbourhood which is the
support of the parameterized curve
⎧
⎪⎨ x = x(t)
⎪⎩ y = y(t)
. (1.4.1)
The equations (1.4.1) are called the parametric equations of the curve in the neighbourhood
of the point a. Usually, unless the curve is simple, we cannot use the same set of
equations to describe the points of an entire curve.
Explicit representation. Let f : I → R be a smooth function, defined on an open
interval from the real axis. Then its graph
is a simple curve, which has the global parameterization
The equation
C = {(x, f (x)) | x ∈ I}, (1.4.2)
⎧
⎪⎨ x = t
⎪⎩ y = f (t)
. (1.4.3)
y = f (x) (1.4.4)
is called the explicit equation of the curve (1.4.2).
In the literature for the explicit representation of a curve it is, also, used the term nonparametric
form, which we do not find particularly appealing since, in fact, an explicit
representation can be thought of as a particular case of parameter representation (the
parameter being the coordinate x).
Implicit representation. Let F : D → R be a smooth function, defined on a domain
D ⊂ R 2 , and let
C = {(x, y) ∈ D | F(x, y) = 0} (1.4.5)
be the 0-level set of the function F. Generally speaking, C is not a regular curve (we can
only say that it is a closed subset of the plane). Nevertheless, if at the point (x0, y0) ∈ C
the vector grad F = {∂xF, ∂yF} is non vanishing, for instance ∂yF(x0, y0) � 0, then, by
the implicit functions theorem there exist:
• an open neighborhood U of the point (x0, y0) in R 2 ;