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

1.11. The local behaviour of a parameterized curve around a biregular point 63
We expand r in a Taylor series around the origin. Up to the third order, the Taylor
expansion is
r(s) = r(0) + sr ′ (0) + 1
2 s2r ′′ (0) + 1
6 s3r ′′′ (0) + o(s 3 ). (1.11.1)
We want to express the derivatives of r as functions of the Frenet vectors τ0, ν0, β0. We
have, obviously, since r is naturally parameterized,
r ′ (0) = τ0. (1.11.2)
On the other hand, from the definition of the curvature vector, we have
Moreover, if we differentiate the relation
we get
r ′′ (0) = k(0) = k(0) · ν0. (1.11.3)
r ′′ (s) = k(s) = k(s) · ν (1.11.4)
r ′′′ (s) = k ′ (s)ν + k(s)ν ′ = k ′ (s)ν + k(s)(−k(s)τ + χ(s)β) =
= −k 2 (s)τ + k ′ (s)ν + k(s)χ(s)β,
where we have made use of the second Frenet equation.
Substituting s = 0 in (1.11.5), we get
Thus, the relation (1.11.1) becomes
or
(1.11.5)
r ′′′ (0) = −k 2 (0)τ0 + k ′ (0)ν0 + k(0)χ(0)β 0. (1.11.6)
r(s) − r(0) = sτ0 + 1
2 s2k(0)ν0 + 1
6 s3 � −k 2 (0)τ0 + k ′ �
3
(0)ν0 + k(0)χ(0)β0 + o(s ) (1.11.7)
r(s) − r(0) =
+
�
s − 1
6 k2 (0)s 3 + o(s 3 �
)
� 1
6 k(0)χ(0)s3 + o(s 3 )
�
1
τ0 +
2 k(0)s2 + 1
6 k′ (0)s 3 + o(s 3 �
) ν0+
�
β0. (1.11.8)
We consider now a coordinate frame with the origin at M0 and having as axes the Frenet
axes. The position vector of a point of the curve with respect to this frame is exactly the