30 Curves in Space
If the curve is smooth, then ds/dt > 0 and s is a monotone function of t so
that t is a well-defined function of s. Hence, r(t(s)) is a function of s, and
is called the canonical parametrization of the smooth curve by the arc
length. By the rules of differentiation,
dr _ dr dt _ dr 1 r'(t) _ _
ds ~ ~dt ds ~ ~dt ds/dt = \\r'(t)\\ ~ U^''
EXERCISE 1.3.8.C Consider the right-handed circular helix
r(t) = acosti + asintj+tH, a > 0. (1.3.13)
Re-write the equation of this curve using the arc length s as the parameter.
1.3.3 Frenet's Formulas
In certain frames, called inertial, the Second Law of Newton postulates the
following relation between the force F = F(t) acting on the point mass m
and the point's trajectory C, defined by a curve r = r(t):
mfgl=F(t). (1.3.14)
A detailed discussion of inertial frames and Newton's Laws is below on page
- When F(t) is given, the solution of the differential equation (1.3.14)
is the trajectory r(t). However, to get a unique solution of (2.1.1), we
must start at some time to and provide two initial conditions r'(to) and
r(to) to determine a specific path. In other words, r(to) and r'(io) are
reference vectors for the motion. At every time t > to, the vectors r(t) and
r '(t) have a well-defined geometric orientation relative to the initial vectors
r(to), r'(to). The three Frenet formulas provide a complete description of
this orientation. In what follows, we assume that the curve C is smooth,
that is, the unit tangent vector u exists at every point of the curve.
To write the formulas, we need several new notions: curvature, principal
unit normal vector, unit binormal vector, and torsion. We will use the
canonical parametrization of the curve by the arc length s measured from
some reference point Po on the curve.
Let u = u(s) be the unit tangent vector at P, where the parameter s is
the arc length from Po to P. By Lemma 1.1 on page 26, the derivative u'(s)
of u(s) with respect to s is orthogonal to u. By definition, the curvature
K(S) at P is
«(*) = ||tt'(a)||;
