Space curves

VECTOR CALCULUS
z
n̂
C
P
t̂
b̂
r(u)
O
y
x
Figure 10.3 The unit tangent t̂, normal n̂ and binormal b̂ to the space curve C
at a particular point P .
since the first term is zero by (10.10), and the second is zero because it is the vector product
of two parallel (in this case identical) vectors. Integrating, we obtain the required result
r×
dr
= c,
dt
(10.11)
where c is a constant vector.
As a further point of interest we may note that in an infinitesimal time dt the change
in the position vector of the small mass is dr and the element of area swept out by the
position vector of the particle is simply dA = 12 |r × dr|. Dividing both sides of this equation
by dt, we conclude that
dr |c|
dA
1
= r × =
,
dt
2
dt
2
and that the physical interpretation of the above result (10.11) is that the position vector r
of the small mass sweeps out equal areas in equal times. This result is in fact valid for
motion under any force that acts along the line joining the two particles. 10.3 Space curves
In the previous section we mentioned that the velocity vector of a particle is a
tangent to the curve in space along which the particle moves. We now give a more
complete discussion of curves in space and also a discussion of the geometrical
interpretation of the vector derivative.
A curve C in space can be described by the vector r(u) joining the origin O of
a coordinate system to a point on the curve (see figure 10.3). As the parameter u
varies, the end-point of the vector moves along the curve. In Cartesian coordinates,
r(u) = x(u)i + y(u)j + z(u)k,
where x = x(u), y = y(u) and z = z(u) are the parametric equations of the curve.
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10.3 SPACE CURVES
This parametric representation can be very useful, particularly in mechanics when
the parameter may be the time t. We can, however, also represent a space curve
by y = f(x), z = g(x), which can be easily converted into the above parametric
form by setting u = x, so that
r(u) = ui + f(u)j + g(u)k.
Alternatively, a space curve can be represented in the form F(x, y, z) = 0,
G(x, y, z) = 0, where each equation represents a surface and the curve is the
intersection of the two surfaces.
A curve may sometimes be described in parametric form by the vector r(s),
where the parameter s is the arc length along the curve measured from a fixed
point. Even when the curve is expressed in terms of some other parameter, it is
straightforward to find the arc length between any two points on the curve. For
the curve described by r(u), let us consider an infinitesimal vector displacement
dr = dx i + dy j + dz k
along the curve. The square of the infinitesimal distance moved is then given by
(ds)2 = dr · dr = (dx)2 + (dy)2 + (dz)2 ,
from which it can be shown that
ds
du
2
=
dr dr
· .
du du
Therefore, the arc length between two points on the curve r(u), given by u = u1
and u = u2 , is
u2
dr dr
·
du.
(10.12)
s=
du du
u1
A curve lying in the xy-plane is given by y = y(x), z = 0. Using (10.12),
show that the
b
arc length along the curve between x = a and x = b is given by s = a 1 + y 2 dx, where
y = dy/dx.
Let us first represent the curve in parametric form by setting u = x, so that
r(u) = ui + y(u)j.
Differentiating with respect to u, we find
dr
dy
=i+
j,
du
du
from which we obtain
dr dr
·
=1+
du du
341
dy
du
2
.
VECTOR CALCULUS
Therefore, remembering that u = x, from (10.12) the arc length between x = a and x = b
is given by
2
b
b
dy
dr dr
s=
1+
dx.
·
du =
du du
dx
a
a
This result was derived using more elementary methods in chapter 2. If a curve C is described by r(u) then, by considering figures 10.1 and 10.3, we
see that, at any given point on the curve, dr/du is a vector tangent to C at that
point, in the direction of increasing u. In the special case where the parameter u
is the arc length s along the curve then dr/ds is a unit tangent vector to C and is
denoted by t̂.
The rate at which the unit tangent t̂ changes with respect to s is given by
d t̂/ds, and its magnitude is defined as the curvature κ of the curve C at a given
point,
2 d t̂ d r̂ κ = = 2 .
ds
ds
We can also define the quantity ρ = 1/κ, which is called the radius of curvature.
Since t̂ is of constant (unit) magnitude, it follows from (10.8) that it is perpendicular to d t̂/ds. The unit vector in the direction perpendicular to t̂ is denoted
by n̂ and is called the principal normal at the point. We therefore have
d t̂
= κ n̂.
ds
(10.13)
The unit vector b̂ = t̂ × n̂, which is perpendicular to the plane containing t̂
and n̂, is called the binormal to C. The vectors t̂, n̂ and b̂ form a right-handed
rectangular cooordinate system (or triad) at any given point on C (see figure 10.3).
As s changes so that the point of interest moves along C, the triad of vectors also
changes.
The rate at which b̂ changes with respect to s is given by d b̂/ds and is a
measure of the torsion τ of the curve at any given point. Since b̂ is of constant
magnitude, from (10.8) it is perpendicular to d b̂/ds. We may further show that
d b̂/ds is also perpendicular to t̂, as follows. By definition b̂ · t̂ = 0, which on
differentiating yields
d b̂
d t̂
d · t̂ + b̂ ·
b̂ · t̂ =
0=
ds
ds
ds
d b̂
· t̂ + b̂ · κ n̂
=
ds
d b̂
· t̂,
=
ds
where we have used the fact that b̂ · n̂ = 0. Hence, since d b̂/ds is perpendicular
to both b̂ and t̂, we must have d b̂/ds ∝ n̂. The constant of proportionality is −τ,
342
10.3 SPACE CURVES
so we finally obtain
d b̂
= −τ n̂.
(10.14)
ds
Taking the dot product of each side with n̂, we see that the torsion of a curve is
given by
d b̂
τ = − n̂ ·
.
ds
We may also define the quantity σ = 1/τ, which is called the radius of torsion.
Finally, we consider the derivative d n̂/ds. Since n̂ = b̂ × t̂ we have
d n̂
d b̂
d t̂
=
× t̂ + b̂ ×
ds
ds
ds
= −τ n̂ × t̂ + b̂ × κ n̂
= τ b̂ − κ t̂.
(10.15)
In summary, t̂, n̂ and b̂ and their derivatives with respect to s are related to one
another by the relations (10.13), (10.14) and (10.15), the Frenet–Serret formulae,
d t̂
= κ n̂,
ds
d n̂
= τ b̂ − κ t̂,
ds
d b̂
= −τ n̂.
ds
(10.16)
Show that the acceleration of a particle travelling along a trajectory r(t) is given by
a(t) =
v2
dv
t̂ + n̂,
dt
ρ
where v is the speed of the particle, t̂ is the unit tangent to the trajectory, n̂ is its principal
normal and ρ is its radius of curvature.
The velocity of the particle is given by
v(t) =
dr
dr ds
ds
t̂,
=
=
dt
ds dt
dt
where ds/dt is the speed of the particle, which we denote by v, and t̂ is the unit vector
tangent to the trajectory. Writing the velocity as v = v t̂, and differentiating once more
with respect to time t, we obtain
a(t) =
dv
d t̂
dv
t̂ + v ;
=
dt
dt
dt
but we note that
ds d t̂
v
d t̂
=
= vκ n̂ = n̂.
dt
dt ds
ρ
Therefore, we have
a(t) =
v2
dv
t̂ + n̂.
dt
ρ
This shows that in addition to an acceleration dv/dt along the tangent to the particle’s
trajectory, there is also an acceleration v 2 /ρ in the direction of the principal normal. The
latter is often called the centripetal acceleration. 343