Page 216 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 18: Theory of Space Curves

and Notes
x”’(t) · x’(t) × x”(t) = k (t)s (t) w(t) .
2
6
’
Thus, we obtain the formula
x"'(t).x'(t)xx"(t)
w(t) = 2 ,
x'(t) x"(t)

valid for any parametrization of x.
Notice that although the curvature k(t) is never negative, the torsion w(t) can have either algebraic
p
sign. For the circular helix x(t) = (r cos(t), r sin(t), pt) for example, we find w(t) = , so the
2
r + p 2
torsion has the same algebraic sign as p. In this way, the torsion can distinguish between a right-
handed and a left-handed screw.
Changing the orientation of the curve from s to –s changes T to –T, and choosing the opposite
sign for k(s) changes N to –N. With different choices, then, we can obtain four different right-
handed orthonormal frames, xTNb, x(–T)N(–b), xT(–N)(–b), and x(–T)(–N)b. Under all these
changes of the Frenet frame, the value of the torsion w(t) remains unchanged.
A circular helix has the property that its curvature and its torsion are both constant. Furthermore,
the unit tangent vector T(t) makes a constant angle with the vertical axis. Although the circular
helices are the only curves with constant curvature and torsion, there are other curves that have
the second property. We characterize such curves, as an application of the Frenet frame.
Proposition 7. The unit tangent vector T(t) of a non-inflectional space curve x makes a constant
w(t)
angle with a fixed unit vector a if and only if the ratio is constant.
k(t)
Proof. If T(t) · a = constant for all t, then differentiating both sides, we
obtain
.
.
T’(t) a = 0 = k(t)s’(t)N(t) a,
so a lies in the plane of T(t) and b(t). Thus, we may write a = cos()T(t) + sin()b(t) for some
angle . Differentiating this equation, we obtain 0 = cos()T’(t) + sin()b’(t) = cos()k(t)s’(t)N(t) –
w(t) sin( )

sin()w(t)s’(t)N(t), so   tan( ). This proves the first part of the proposition and

k(t) cos( )

identifies the constant ratio of the torsion and the curvature.
w(t)
Conversely, if = constant = tan() for some , then, by the same calculations, the expression
k(t)
cos()T(t) + sin()b(t) has derivative 0 so it equals a constant unit vector. The angle between T(t)
and this unit vector is the constant angle .
Curves with the property that the unit tangent vector makes a fixed angle with a particular unit
vector are called generalized helices. Just as a circular helix lies on a circular cylinder, a generalized
helix will lie on a general cylinder, consisting of a collection of lines through the curve parallel
to a fixed unit vector. On this generalized cylinder, the unit tangent vectors make a fixed angle
with these lines, and if we roll the cylinder out onto a plane, then the generalized helix is rolled
out into a straight line on the plane.

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