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

1.10. The Frenet formulae. The torsion 53
therefore the mixed product of the first three derivatives of r reads
� r ′ (t), r ′′ (t), r ′′′ (t) � = � ρ ′ (λ(t)) · λ ′ (t), ρ ′′ (λ(t)) · λ ′2 (t) + ρ ′ (λ(t)) · λ ′′ (t),
ρ ′′′ (λ(t)) · λ ′3 (t) + 3ρ ′′ (λ(t)) · λ ′ (t) · λ ′′ (t) + ρ ′ (λ(t)) · λ ′′′ (t) � =
= λ ′6 (t) � ρ ′ (λ(t)), ρ ′′ (λ(t)), ρ ′′′ (λ(t)) � ,
all the other mixed products from the right hand side vanishing, because two of the
factors are colinear. Hence
� ρ ′ (λ(t)), ρ ′′ (λ(t)), ρ ′′′ (λ(t)) � = 1
λ ′6 (t)
� r ′ (t), r ′′ (t), r ′′′ (t) � .
But, since ρ is naturally parameterized and the two curves are positively equivalent, we
have λ ′ = �r ′ �, therefore the previous formula becomes
� ′ ′′ ′′′
ρ , ρ , ρ � = (r′ , r ′′ , r ′′′ )
�r ′ �6 .
Moreover (see (1.7.2)), the curvature can be expressed with respect to the derivatives of
r through the formula
k = �r′ × r ′′ �
�r ′ �3 ,
hence, from the expression of the torsion (1.10.4) and the previous relation, we get
χ(t) = (r′ , r ′′ , r ′′′ )
�r ′ × r ′′ . (1.10.5)
�2 Exercise 1.10.1. Let ω = χτ + kβ. Show that Frenet’s formulae can be written as
⎧
τ
⎪⎨
⎪⎩
′ = ω × τ
ν ′ = ω × ν
β ′ (1.10.6)
= ω × β
The vector ω is called the Darboux vector.
1.10.1 The geometrical meaning of the torsion
The torsion is, in a way, an analogue of the curvature (this is the reason why in the oldfashioned
books the torsion is called the second curvature). What we mean is that the
torsion can be also interpreted as being the speed of rotation of a straight line, this time
the binormal. In other words, we have