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

1.3. The definition of the curve 25
general, the support of an arbitrary parameterized curve is not a regular curve. Take, for
instance, the lemniscate of Bernoulli (R, r(t) = (x(t), y(t), z(t))), where
⎧
⎪⎨
x(t) = t(1+t2 )
1+t4 y(t) = t(1−t2 )
1+t4 ⎪⎩ z = 0
r is continuous, even bijective, but the inverse is not continuous. In fact, the support has
a self intersection, because limt→−∞ r = limt→∞ r = r(0) (see the figure 1.3). However,
we can always restrict the domain of definition of a parameterized curve such that the
support of the restriction is a regular curve.
Figure 1.3: The Bernoulli’s lemniscate
Theorem 1.3.2. Let (I, r = r(t)) a regular parameterized curve. Then each point t0 ∈ I
has a neighbourhood W ⊂ I such that r(W) is a simple regular curve.
Proof. The regularity of r at each point means, in particular, that r ′ (t0) � 0. Without
restricting the generality, we may assume that x ′ (t0) � 0. Let us consider the mapping
ψ : I × R 2 → R 3 , given by
ψ(t, u, v) = r(t) + (0, u, v),
where (u, v) ∈ R 2 . ψ is, clearly, smooth and its Jacobi matrix at the point (t0, 0, 0) is
.