which implies that the vector d
ˆeb
ds
is also perpendicular to the vector ˆeb. These two
results show that the derivative vector
dˆeb
ds
must be in the direction of the normal
vector ˆen. Hence, there exists a constant K such that
dˆeb
ds
=Kˆen (7 .18)
where K is a constant. By convention, the constant Kis selected as −τ, where τis
called the torsion and the reciprocal σ=
1
τ is called the radius of torsion. Taking the
dot product of both sides of equation (7.18) with the unit vector ˆengives
τ=τ(s) = −ˆen·d
ˆeb
ds
(7 .19)
The torsion is a measure of the twisting of a curve out of a plane and is a measure
of how the osculating plane changes with respect to arc length. The torsion can be
positive or negative and if the torsion is zero, then the curve must be a plane curve.
The three vectors ˆet, ˆeb, ˆenform a right-handed system of unit vectors and so one
can write ˆen=ˆeb׈et. Differentiating this relation with respect to arc length gives
dˆen
ds
=ˆeb×d
ˆet
ds
+d
ˆeb
ds
×eˆt=eˆb×κeˆn−τˆen׈et=−κˆet+τˆeb (7 .20)
These results give the Frenet^1 -Serret^2 formulas
dˆet
ds =κ
ˆen
dˆeb
ds =−τ
ˆen
dˆen
ds =τˆeb−κˆet
(7 .21)
Using matrix notation^3 , the Frenet-Serret formulas can be written as
dˆet
ddsˆe
dsb
dˆen
ds
=
0 0 κ
0 0 −τ
−κ τ 0
eˆt
ˆeb
eˆn
(7 .22)
Recall that if B is a vector which rotates about a line with angular velocity ω,
then d
B
dt
=ω×B. One can use this result to give a physical interpretation to the
Frenet-Serret formulas. One can write
dˆet
dt =
dˆet
ds
ds
dt =κeˆn
ds
dt =κ
ds
dt ˆeb×eˆt=ω׈et where ω=κ
ds
dt ˆeb
(^1) Jean Fr ́ed ́eric Frenet (1816-1900) A French mathematician.
(^2) Joseph Alfred Serret (1819-1885) A French mathematician.
(^3) See chapter 10 for a description of the matrix notation.