Begin2.DVI

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.