Prenet's Formulas 31the principal unit normal vector at P is
p =-«'(«); (1.3.15)the unit binomial vector at P is
b(s) = u(s) x p{s).EXERCISE 1.3.9.C Parameterizing the circle by the arc length, verify that
the curvature of the circle of radius R is \/R.
To define the torsion, we derive the relation between b'(s) and the
vectors u,p,b. Using Lemma 1.1 once again, we conclude that b'(s) is
orthogonal to b(s). Next, we differentiate the relation b(s) • u(s) = 0 with
respect to s and use the product rule (1.3.5) to find b'(s)-u(s)+b(s)-u'(s) —
- By construction, the unit vectors u,p,b are mutually orthogonal, and
then the definition (1.3.15) of the vector p implies that b(s) • u'(s) = 0.
As a result, b'(s) • u(s) = 0. Being orthogonal to both u(s) and b(s), the
vector b'(s) must then be parallel to p(s). We therefore define the torsion
of the curve C at point P as the number r = T(S) so that
b'(s) = -T(s)p(s); (1.3.16)the choice of the negative sign ensures that the torsion is positive for the
right-handed circular helix (1.3.13).
Note that the above definitions use the canonical parametrization of the
curve by the arc length s; the corresponding formulas can be written for an
arbitrary parametrization as well; see Problem 1.11 on page 412.
Relations (1.3.15) and (1.3.16) are two of the Prenet formulas. To derive
the third formula, note that p(s) = b(s) x u(s). Differentiation with respect
to s yields p' = bxu' + b'xu = bxKp — rpxu, and
p'{s) = -Ku{s) + Tb(s). (1.3.17)Different sources refer to relations (1.3.15) - (1.3.17) as either the
Frenet or the Frenet-Serret formulas. In 1847, the French mathemati-
cian JEAN FREDERIC FRENET (1816-1900) derived two of these formulas in
his doctoral dissertation. Another French mathematician, JOSEPH ALFRED
SERRET (1819-1885), gave an independent derivation of all three formulas,
but we could not find the exact time of his work. Of course, neither Frenet
nor Serret used the modern vector notations in their derivations.