Lecture Notes for 120 - UCLA Department of Mathematics

CHAPTER 3
Space Curves
3.1. The Fundamental Equations
The theory of space curves dates back to Clairaut in 1731. He considered
them as the intersection of two surfaces given by equations. In that work Clairaut
showed that space curves have two curvatures, but they did not corresponds exactly
to the curvature and torsion we introduce below. The subject was later taken
up by Euler who was the first to think of curves as parametrized by arclength.
Lancret in 1806 introduced the concepts of unit tangent, principal normal and binormal
and with those curvature and torsion as we now understand them. It is,
however, possible that Monge had some inklings of what torsion was, but he never
presented an explicit formula. Cauchy in 1826 considerably modernized the subject
and formulated some of the relations that later became part of the Serret and Frenet
equations that we shall introduce below.
In order to create a set of equations for space curves q (t) :[a, b] ! R 3 we
need to not only assume that the curve is regular but also that its velocity and
acceleration are always linearly independent. This is equivalent to assuming that
q is regular and that the unit tangent T also defines a regular curve, i.e., that the
curvature never vanishes. In this case it is possible to define a suitable positively
oriented orthonormal frame T, N, andB by declaring
T = v
|v|
a (a · T) T
N =
|a (a · T) T|
B = T ⇥ N
The new normal vector B is called the bi-normal. We define the curvature and
torsion by
apple = N · dT
ds
⌧ = B · dN
ds
We should check that these definitions for N and apple are consistent with our
earlier definitions where we used ✓ as arclength parameter for T. We already saw
that the earlier definitions implied the formulas we just gave. So it is a question of
checking that our new definitions conversely imply the old ones. We’ll do this after
having established the next theorem using only the definitions just given.
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