Page 214 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 18: Theory of Space Curves

Since the origins of the two frames coincide at the value t0, we have Notes
x(t) – x(t) = x(t ) – x(t ) = 0
0
0
for all t.
This completes the proof that two families of frames satisfying the same set of differential
equations differ at most by a single affine motion.

Exercise 13: Prove that the equations E (t) = q (t)E(t) can be written E (t) = d(t) × E (t), where
'
'
i
i
i
j
ij
d(t) = q (t)E (t) + q (t)E (t) + q (t)E (t). This vector is called the instantaneous axis of rotation.
12
3
2
1
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Exercise 14: Under a rotation about the x -axis, a point describes a circle x(t) = (a cos(t), a sin(t), b).
3
Show that its velocity vector satisfies x’(t) = d × x(t) where d = (0, 0, 1). (Compare with the
previous exercise.).
.
Exercise 15: Prove that (v v)(w·w)”(v·w)2 = 0 if and only if the vectors v and w are linearly
dependent.
18.5 Curves at a Non-inflexional Point and the Frenet Formulas
A curve x is called non-inflectional if the curvature (t) is never zero. By our earlier calculations,
this condition is equivalent to the requirement that x’(t) and x”(t) are linearly independent at
every point x(t), i.e. x’(t) × x”(t)  0 for all t. For such a non-inflectional curve x, we may define a
pair of natural unit normal vector fields along x.
x'(t)× x"(t)
Let b(t) = , called the binormal vector to the curve x(t). Since b(t) is always
|x'(t)× x"(t)|
perpendicular to T(t), this gives a unit normal vector field along x.
We may then take the cross product of the vector fields b(t) and T(t) to obtain another unit
normal vector field N(t) = b(t) × T(t), called the principal normal vector. The vector N(t) is a unit
vector perpendicular to T(t) and lying in the plane determined by x’(t) and x”(t). Moreover,
x”(t) · N(t) = k(t)s’(t) , a positive quantity.
2
Note that if the parameter is arclength, then x’(s) = T(s) and x”(s) is already perpendicular to T(s).
x"(s)
It follows that x”(s) = k(s)N(s) so we may define N(s) = and then define b(s) = T(s) × N(s).
k(s)
This is the standard procedure when it happens that the parametrization is by arclength. The
method above works for an arbitrary parametrization.
We then have defined an orthonormal frame x(t)T(t)N(t)b(t) called the Frenet frame of the
non-inflectional curve x.
By the previous section, the derivatives of the vectors in the frame can be expressed in terms of
the frame itself, with coefficients that form an antisymmetric matrix. We already have
x’(t) = s’(t)T(t), so
p (t) = s’(t), p (t) = 0 = p (t) .
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3
2
Also T’(t) = k(t)s’(t)N(t), so
q (t) = k(t)s’(t) and q (t) = 0 .
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13
We know that
b’(t) = q (t)T(t) + q (t)N(t), and q (t) = –q (t) = 0 .
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