In section 3.3 we defined the tangent and principal normal vectors,
T(t) and P(t), and let T'(t) = κ(t)s'(t)P(t),
where κ(t) is the curvature of the curve at t. We now wish
to write P'(t) and B'(t) in terms of the Frenet frame as well.
Observe that since P(t) and B(t) are unit vectors, P'(t) is perpendicular
to P(t) and B'(t) is perpendicular to B(t). Therefore,
To solve for a(t), we can use the fact that P(t)·T(t) = 0.
Then,
0 = (P(t)·T(t))' = P'(t)·T(t) + P(t)·T'(t)
and
P'(t)·T(t) = -T'(t)·P(t) = -κ(t)s'(t)
So, a(t) = -κ(t)s'(t). For b(t), we define the binormal
component of P'(t) as P'(t)·B(t) = s'(t)τ(t),
where τ(t) is called the torsion of the curve at t.
Using the equations for T'(t) and P'(t) it is fairly straightforward
to show that c(t) = 0 and d(t) = s'(t)τ(t). This gives us three equations
knowns as the Frenet-Serret Equations.
T'(t) = κ(t)s'(t)P(t)
P'(t) = -κ(t)s'(t)T(t) + τ(t)s'(t)B(t)
B'(t) = -τ(t)s'(t)P(t)
These equations can also be expressed in matrix form:
(
T'(t)
)
=
s'(t)
(
0
κ(t)
0
)
(
T(t)
)
P'(t)
-κ(t)
0
τ(t)
P(t)
B'(t)
0
-τ(t)
0
B(t)
Since T-P-B is an orthonormal frame, the matrix which relates them to their derivatives is an antisymmetric matrix.
Although, we used the motion of the Frenet frame to define the curvature and
torsion functions of a curve, it is helpful to consider their corresponding
qualitative definitions as well. We can think of curvature as being
"deviation from straightness" and torsion as being "deviation from flatness".
From the Frenet-Serret equations, we can see that if κ(t)=0, the curve
has a constant tangent vector, and is, therefore, a straight line. Similarly,
if τ=0, then B(t) is constant and the curve lies in a plane with B(t) as
its normal vector.
This demonstration draws a parameterized curve, X(t), and shows the Frenet
frame at the point X(t0). You can travel along the curve by using the t0
tapedeck in the control panel. In two separate windows, we graph the curvature
and torsion functions, κ(t) and τ(t). Notice that if X(t) is a
planar curve, then τ = 0 and if X(t) is a straight line, then
κ(t) = 0.
1. See what happens when the space curve is actually a plane curve.
2. Take a look at the space cardioid family given by X(t) =((1+cos(t))*cos(t), (1+cos(t))* sin(t),c*sin(t)).
3. Derive the Frenet-Serret equation for B'(t). That is, show that
B'(t) = τ(t)s'(t)P(t)
4. Show that the torsion function can be written as
τ(t) = [X'''(t)·(X'(t)×X''(t))]/|X'(t)×X''(t)|2
5. Show that if κ(t) = 0 then X(t) lies along a line.