A fly buzzing around the room, a person riding a roller coaster, and a satellite orbiting the earth all have
something in common. They are moving over some sort of curve in three dimensions.
Denote by R
(t)
the position vector of the point on the curve which occurs at time t. Assume
that R^{′},R^{′′} exist and are continuous. Thus R^{′} = v, the velocity and R^{′′} = a is defined as the
acceleration.
PICT
Lemma 10.1.1Define T
(t)
≡ R^{′}
(t)
∕
|R′(t)|
. Then
|T(t)|
= 1 and if T^{′}
(t)
≠0, then there exists aunit vector N
(t)
perpendicular to T
(t)
and a scalar valued function κ
(t)
, with T^{′}
(t)
= κ
(t)
|v|
N
(t)
.
Proof:It follows from the definition that
|T|
= 1. Therefore, T ⋅ T = 1 and so, upon differentiating
both sides,
T′ ⋅T + T ⋅T′ = 2T ′ ⋅T = 0.
Therefore, T^{′} is perpendicular to T. Let N
(t)
′
|T |
≡ T^{′}. Note that if
′
|T |
= 0, you could let N
(t)
be any
unit vector. Then letting κ
(t)
be defined such that
′
|T |
≡ κ
(t)
|v (t)|
, it follows
′ ′
T (t) = |T (t)|N (t) = κ(t)|v (t)|N (t). ■
Definition 10.1.2The vector T
(t)
is called theunit tangent vector and the vector N
(t)
is calledtheprincipal normal. The function κ
(t)
in the above lemma is called the curvature.The radiusof curvature is defined as ρ = 1∕κ.The plane determined by the two vectors T and N in the casewhere T^{′}≠0 is called theosculating^{1} plane.It identifies a particular plane which is in a sense tangent to this space curve.
The important thing about this is that it is possible to write the acceleration as the sum of
two vectors, one perpendicular to the direction of motion and the other in the direction of
motion.
Theorem 10.1.3For R
(t)
the position vector of a space curve, the acceleration is given by theformula
Finally, it is good to point out that the curvature is a property of the curve itself, and does not depend
on the parametrization of the curve. If the curve is given by two different vector valued functions R
From this, it is possible to give an important formula from physics. Suppose an object orbits a point at
constant speed v. In the above notation,
|v|
= v. What is the centripetal acceleration of this object? You
may know from a physics class that the answer is v^{2}∕r where r is the radius. This follows from the above
quite easily. First, what is the curvature of a circle of radius r? A parameterization of such a curve
is