Lecture
In this section we consider critical points of smooth maps R → R², i.e. singularities of plane curves given parametrically:
x=φ(t), y=ψ(t), (6.1)
where φ, ψ are smooth functions. A point of the curve (6.1), corresponding to the value t of the parameter, is called critical (other terms: singular,
non-regular), if
φ'(t) =ψ'(t) =0. (6.2)
In a neighborhood of any noncritical (regular) point the curve
(6.1) is the graph of some smooth function y = f(x) or
x = g(y), and in suitable local coordinates it becomes a straight
line. Near a critical point, however, the curve can look
quite different. For example, consider the curves
x=t², y=tⁿ, n>2, (6.3)
with a critical point at zero. For n = 2 this is a semicubical parabola;
at zero it has a cusp — a semicubical cusp point, which is also
called a cusp (English: cusp); see Fig. 6.1 (center, framed).
Cusps are the simplest and most frequently encountered singularities of plane
curves given parametrically.
For n = 3 the curve (6.3) outwardly resembles an ordinary parabola and
can be represented in the form of a graph y = f(x); however, the function f
is not C∞-smooth at zero, but belongs only to the class C¹. We
invite the reader to work out independently what the curves
(6.3) look like for all other n.
62
PROBLEM 6.1. Using the functions constructed in Problem 5.1, express the graph y = |x| in the form (6.1) with smooth φ and ψ.
Fig. 6.1 The semicubical parabola (center) and its perturbations (6.4): for
ε > 0 (left) and for ε < 0 (right)
Critical points of curves (6.1) are unstable, i.e. they can be destroyed by arbitrarily small perturbations of the curves. This follows from the
obvious observation: after perturbation the critical value of the parameter t0 at which (6.2) holds will turn into two values
t1 and t2, for which φ'(t1) = 0, ψ'(t1) ≠ 0 and φ'(t2) ≠0, ψ'(t2) = 0,
and there will be no critical point. Nevertheless, their study makes
sense, because critical points occur in a stable way in families of curves depending on parameters, similarly to
the way degenerate critical points occur for families of functions (see the discussion in Section 4.1), as well as in certain special
cases, considered in Section 6.2.
EXAMPLE 6.1. The semicubical parabola x = t², y = t³ is an
element of the family of curves
x=t², y=t³+εt, 0≤ε<1, (6.4)
The curves corresponding to any ε ≠ 0 have no critical points. They
are shown in Fig. 6.1. Thus the cusp is destroyed by an arbitrarily
small perturbation of the original curve x = t², y = t³. However,
in the family (6.4) the cusp is stable. Indeed, consider a slightly
perturbed family
x=t²+f(t,ε), y=t³+εt+g(t,ε), 0≤ε<1,
where |f(0,ε)| < c and |g(0,ε)| < c for all 0 ≤ ε < 1. The equality
f'(0,ε) + ε = 0 holds for at least one value of ε from the given interval. From the second condition it follows that the corresponding
critical point is a cusp.
6
Comments