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The Frenet-Serret Theorem
Adrian Down
September 07, 2006
1 Review
1.1 Arc length and parameterization
Last time, we defined the arc length s of a curve. We discussed reparameteri- zation, and showed that given any curve, it is always possible to reparameter- ize the curve by arc length. If a curve is parametrized by arc length, |d dsα | = 1, and α is said to be a unit speed curve. Parameterization by arc length is convenient because it simplifies calculations. We will henceforth assume that all curves are parameterized by arc length. It may be difficult to transform to this parameterization in practice, but it is a theoretical simplification.
1.2 Frenet-Serret apparatus
We defined the quantities of the Frenet-Serret apparatus,
T(s) = α′(s) |T′(s)| = κ(s) > 0 N(s) =
T′(s) κ(s) B(s) = T(s) × N(s) τ (s) = − 〈B′(s), N(s)〉
where prime denotes derivative with respect to s. The torsion τ measures the amount by which the curve is coming out of the plane in which it lies. In two dimensions, curvature measures the rate of change of the angle of the tangent vector relative to the horizontal. We showed that the vectors of the Frenet-Serret apparatus form an or- thonormal basis of unit vectors of R^3 ,
|T| = |N| = |B| = 1 〈T, B〉 = 〈T, N〉 = 〈N, B〉 = 0
All of these vectors depend on the parameter s and form a comoving frame that is different at each point on the curve. The way in which these vectors change with the parameter s can give information about the evolution of the curve in space.
2 Frenet-Serret Theorem
2.1 Statement
Theorem (Frenet-Serret). The derivatives of T, N, and B are given by,
T′ N′ B′
κ(s)N(s) −κ(s)T(s) +τ (s)B(s) −τ (s)N(s)
These equations can be conveniently written matrix form as a memory aid, although they are usually not written formally this way, as the compo- nents of the matrices are themselves vectors.
T′
N′
B′
0 κ 0 −κ 0 τ 0 −τ 0
T
N
B
Note. • The definitions of some quantities in the Frenet-Serret apparatus may initially have seemed unmotivated, especially that of the torsion. However, the simplicity of the formulas in the Frenet-Serret theorem provides at least partial justification for these definitions.
- The matrix of coefficients in the Frenet-Serret theorem is skew sym- metric, A = −AT^. This fact is important in higher geometry.
2.2 Proof
This proof uses two techniques that will recur frequently in the theory of curves: expand in an orthonormal basis and “differentiate the heck out of it”.
Proof. 1. The definition of curvature gives the first equation,
T′^ = κN
Combining the results of parts 3 and 4,
N′^ = −κT + τ B
which is the second of the Frenet-Serret equations.
- To obtain the last of the Frenet-Serret equation, expand B′^ in terms of the orthonormal basis of T, N, and B as in part 2 of the proof,
B′^ = 〈B′, T〉 T + 〈B′, N〉 N + 〈B′, B〉 B
Since the magnitude of B is constant, the same argument used in part 2 of the proof shows that 〈B, B′〉 = 0.
- Again, to compute the first inner product in the expansion of B′, dif- ferentiate a related inner product,
0 = 〈B, T〉
⇒ 0 =
d ds
〈B, T〉
= 〈B′, T〉 + 〈B, T′〉
Using the definition of the curvature,
〈B′, T〉 = − 〈B, T′〉 = − 〈B, κN〉 = −κ 〈B, N〉 = 0
since B and N are orthogonal.
- The last inner product in the expansion of B′^ is given by the definition of the torsion, τ = − 〈B′, N〉 Thus,
B′^ = τ N
This is the third of the Frenet-Serret equations.
3 Torsion
3.1 Motivation
With the Frenet-Serret theorem, we have laid out the basic theory of curves. Our next objective will be to interpret these result geometrically. Finally, we will do some applications. These applications fall into two categories: local and global. The interaction between the two is an interesting area of study.
3.2 τ ≡ 0 for plane curves
Torsion is roughly a measure of the amount by which the curve is coming out of the plane. The next theorem clarifies this geometric interpretation.
Theorem. Let α be a unit speed curve with curvature κ(s) > 0. Then these statements are equivalent:
a) The image of α lies in plane
b) B(s) is constant for all s
c) τ (s) = 0 for all s
Proof. To prove all statements equivalent, we proceed in steps, showing each statement implies the other two.
b ⇔ c By the third line of the Frenet-Serret theorem, B′^ = −τ N. Hence,
B′^ ≡ 0 ⇔ τ ≡ 0
a ⇒ b Assume the image of α lies in a plane. Without loss of generality, we may assume that the this plane is the xy plane, in which case the curve may be parameterized as,
α(s) = (x(s), y(s), 0)
Note. Here, we write the curve componentwise to take advantage of the as- sumption that the image of the curve lies in the xy plane. However, in gen- eral it is preferable to use vector notation and avoid dealing with components when possible.
By assumption, B is constant, and so B′^ = 0. The first inner product in the derivative of g can be recast using the Frenet-Serret basis vectors,
g′(s) = 〈α′, B〉 + 〈α(s) − α(s 0 ), 0 〉 = 〈T, B〉 = 0
Hence g(s) = 0 ∀s, and the image of α lies in the plane of which B is the normal.
3.3 Remarks
- The theorem above can be false if κ = 0 at any point. If κ = 0 at some point, the curve could stop at the boundary between two planes and then continue on a different plane.
- The plane spanned by T and N is called the osculating plane. This is the plane in which the curve most nearly lies near the point in question. This is like the two-dimensional analog to the linear approximation. The plane spanned by B and N is called the normal plane. This is the plane to which the curve is perpendicular at the given point. The last plane that can be formed from the Frenet-Serret basis vectors, namely that spanned by B and T, is called the rectifying plane. There is not a strong geometric interpretation of this plane.
- τ measures the rate at which the curve is twisting out of the oscillating plane. It can be positive or negative, unlike κ. If τ > 0, the curve is bending out in the direction of B.
