Page 208 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 18: Theory of Space Curves

3. If x is equivalent to y and if y is equivalent to z, then x is equivalent to z Notes
A relation that satisfies these properties is called an equivalence relation. Precisely speaking, a
curve is considered be an equivalence class of parametrized curves.

18.2 Curvature and Fenchel’s Theorem

If x is an immersed curve, with x’(t)  0 for all t in the domain, then we may define the unit
x'(t)
tangent vector T(t) to be . If the parameter is arclength, then the unit tangent vector T(s) is
|x'(t)|
given simply by x’(s). The line through x(t ) in the direction of T(t ) is called the tangent line at
0
0
x(t ). We can write this line as y(u) = x(t ) + uT(t ), where u is a parameter that can take on all real
0
0
0
values.
Since T(t) · T(t) = 1 for all t, we can differentiate both sides of this expression, and we obtain
2T’(t)· T(t) = 0. Therefore T’(t) is orthogonal to T(t). The curvature of the space curve x(t) is defined
T'(t)
by the condition (t) = , so = (t)s’(t) = |T’(t)|. If the parameter is arclength, then
|x'(t)|
x’(s) = T(s) and (s) = |T’(s)| = |x”(s)|.
Proposition 1. If (t) = 0 for all t, then the curve lies along a straight line.

Proof. Since (t) = 0, we have T’(t) = 0 and T(t) = a, a constant unit vector. Then x’(t) = s’(t)T(t)
= s’(t)a, so by integrating both sides of the equation, we obtain x(t) = s(t)a + b for some constant
b. Thus, x(t) lies on the line through b in the direction of a.

Curvature is one of the simplest and at the same time one of the most important properties of a
curve. We may obtain insight into curvature by considering the second derivative vector x”(t),
often called the acceleration vector when we think of x(t) as representing the path of a particle at
time t. If the curve is parametrized by arclength, then x’(s)·x’(s) = 1 so x”(s)·x’(t) = 0 and
(s) = |x”(s)|. For a general parameter t, we have x’(t) = s’(t)T(t) so x”(t) = s”(t)T(t) + s’(t)T’(t). If
we take the cross product of both sides with x’(t) then the first term on the right is zero since x’(t)
is parallel to T(t). Moreover x’(t) is perpendicular to T’(t) so
|T’(t) × x’(t)| = |T’(t)||x’(t)| = s’(t) (t) .
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Thus,
x”(t) × x’(t) = s’(t)T’(t) × x’(t)

and
|x”(t) × x’(t)| = s’(t) (t) .
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This gives a convenient way of finding the curvature when the curve is defined with respect to
an arbitrary parameter. We can write this simply as
|x"(t) × x'(t)|
 (t) = .
|x'(t)x'(t)| 3/2

Notes The curvature (t) of a space curve is non-negative for all t. The curvature can
be zero, for example at every point of a curve lying along a straight line, or at an isolated
point like t = 0 for the curve x(t) = (t, t , 0). A curve for which (t) > 0 for all t is called
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non-inflectional.

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