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Complex Analysis and Differential Geometry

Notes Thus b’(t) is a multiple of N(t), and we define the torsion w(t) of the curve by the condition
b’(t) = –w(t)s’(t)N(t),

so q (t) = –w(t)s’(t) for the Frenet frame. From the general computations about moving frames,
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it then follows that
N’(t) = q (t)T(t) + q (t)b(t) = –k(t)s’(t)T(t) + w(t)s’(t)b(t) .
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The formulas for T’(t), N’(t), and b’(t) are called the Frenet formulas for the curve x.
If the curve x is parametrized with respect to arclength, then the Frenet formulas take on a
particularly simple form:

x’(s) = T(s)
T’(s) = k(s)N(s)
N’(s) = –k(s)T(s) + w(s)b(s)
b’(s) = –w(s)b(s) .

The torsion function w(t) that appears in the derivative of the binormal vector determines
important properties of the curve. Just as the curvature measures deviation of the curve from
lying along a straight line, the torsion measures deviation of the curve from lying in a plane.
Analogous to the result for curvature, we have:
Proposition 6. If w(t) = 0 for all points of a non-inflectional curve x, then the curve is contained
in a plane.
Proof. We have b’(t) = –w(t)s’(t)N(t) = 0 for all t so b(t) = a, a constant unit vector. Then, T(t)a = 0
.
.
.
.
.
for all t so (x(t) a)’ = x’(t) a = 0 and x(t) a = x(a) a, a constant. Therefore, (x(t) – x(a)) a = 0
and x lies in the plane through x(a) perpendicular to a.
If x is a non-inflectional curve parametrized by arclength, then
.
w(s) = b(s) N’(s) = [T(s),N(s),N’(s)] .
x"(s)
Since N(s) = , we have,
k(s)
x"'(s)  k'(s)
N’(s) =  x"(s) ,
k(s) k(s) 2
so


w(s) = x'(s), x"(s) x'''(s)  x"(s)  k'(s)  [x'(s),x"(s),x"'(s)] .
,
2 

 k(s) k(s) k(s)  2
We can obtain a very similar formula for the torsion in terms of an arbitrary parametrization of
the curve x. Recall that
x”(t) = s”(t)T(t) + k(t)s’(t)T’(t) = s”(t)T(t) + k(t)s’(t) N(t),
2
so
x”’(t) = s”’(t)T(t)+s”(t)s’(t)k(t)N(t) + [k(t)s’(t) ]’N(t) + k(t)s’(t) N’(t) .
2
2
Therefore,
x”’(t)b(t) = k(t)s’(t) N’(t)b(t) = k(t)s’(t)2w(t)s’(t),
2

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