1. The semi-regular curves in a differentiable plane
In this paragraph we introduce preliminary notions needed in order to state the main
results of Chapter 2.
1.1. Proper transversal immersion. Let M be any smooth surface. A
differentiable mapping a of a (not necessarily compact) interval / of R into M is
said to be a weakly transversal immersion (resp. a transversal immersion) if it
satisfies Conditions (i) and (ii) (resp. Conditions (i), (ii) and (iii)).
(i) (immersibility condition) 6c(t) := &- a(t) * 0 for all t e /.
at
(ii) (source transversality condition) whenever a(a) = a(b) = p for a * b,
the tangent vectors d(a) and d(b) are linearly independent in TpM.
(iii) (target transversality requirement) The mapping a has no triple points, in
other words there exist no a, b, c e / such that a * b * c * a and that a(a)
= a(b) = a(c).
1.1.1. Lemma. Let a be a proper transversal immersion of a not necessarily
compact interval / into a smooth surface M. Then the set of double points of a is a
discrete subset of M.

Suppose from now on that the surface M is diffeomorphic to
R2.
Assume for
convenience that M is oriented and suppose that a b. A crossing point p =
a(a) = a(b) of a will be said to have a positive sign ( sgn(p) = 1 ) when the
basis (d(a\d(b)) has positive orientation and a negative sign ( sgn(p) = -1 )
otherwise.
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