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

62 Chapter 1. Space curves
Corollary 1.10.1. The circular cylindrical helices, the curves of constant torsion (in
particular the plane curves), and the curves of constant curvature are Bertrand curves.
Remark. For a plane curve, the principal normals are, in fact, the normals, therefore,
as one can see easily, any plane curve has an infinity of Bertrand mates, all of them
congruent (they are, all of them, parallel to the given one).
Corollary 1.10.2. If a Bertrand curve has more than one mate, then it has an infinity
and it is a circular cylindrical helix.
Proof. As we saw from the proof of the theorem of Bertrand, the constant a from the
relation (1.10.19) is the one that identifies the Bertrand mate of our curve. Thus, if a
curve r has more then one Bertrand mate, this means that there are (at least) two distinct
pairs of real numbers (a, b), (a1, b1) such that
a · k + b · χ = 1,
a1 · k + b1 · χ = 1.
Subtracting side by side the two relations, we obtain
(a − a1) · k = (b1 − b) · χ.
At least one of the coefficients (a − a1) and (b1 − b) is different from zero, therefore
the ratio between the curvature and torsion is a constant. This means, via the Lancret’s
theorem, that the curve is a general helix. However, as, say, χ = c · k, with c – constant,
from (1.10.19) follows that
(a + b · c) · k = 1,
which means that k (and, hence, also χ) is a constant, therefore the curve is a circular
cylindrical helix. Clearly, for the case of a cylindrical helix, when both the curvature
and the torsion are constants, we can get an infinity of pairs of real numbers that verify
(1.10.19), so our curve has an infinity of Bertrand mates. They lie on circular cylinders
which have the same axis of symmetry with the cylinder associated to the given
curve. �
1.11 The local behaviour of a parameterized curve around a
biregular point
Let (I, r) be a naturally parameterized curve. We shall assume that 0 ∈ I, as an interior
point and M0 ≡ r(0) is a biregular point of the curve. We shall make use of the following
notations: τ0 = τ(0), ν0 = ν(0), β 0 = β(0).