Chapter 2: Space Curves
Section 2.7: Frenet-Serret Formalism
Table 2.7.1 lists the equations and definitions that constitute the Frenet-Serret formalism for the analysis of a curve in two or three dimensions. The left-hand column of the table lists the results of previous sections. The three differential equations on the right complete the formalism. The first two of these equations are simply a rewrite of definitions appearing on the left. The third equation is new, and completes the formalism.
B′= −τ N
N′= −κ T+τ B
Table 2.7.1 The Frenet-Serret formalism; s denotes arc length and prime, dds
Table 2.7.2 provides a partial answer to the question "To what extent does the curvature of a curve determine the curve?" Here, the question is answered for a plane curve where necessarily, the torsion is zero.
Let s denote arc length; and the prime, dds. If κs is prescribed as the curvature for a plane curve, then a position-vector description of that curve is given by
R=a+∫cscosθu ⅆu i+b+∫cssinθu ⅆu j
where θu=∫cuκp ⅆp and Rc passes through the initial point a,b.
Table 2.7.2 Prescription for a plane curve with a given curvature
Theorem 2.7.1, not proven at this level, states that two curves with identical curvature and torsion, are congruent, that is, can be made to coincide. Thus, Theorem 2.7.1 actually says that a curve is completely determined by its curvature and torsion. Therein lies the importance of the Frenet formalism.
If s is arc length, and the curves defined by R1s and R2s both have the same curvature κs and the same torsion τs, then the two curves are congruent.
Theorem 2.7.2 states that given suitably well-behaved curvature and torsion, there exists a curve whose curvature and torsion are those given. This is an existence theorem, and does not provide a recipe for finding the curve from the curvature and torsion. Just the first few terms of a series representation of the curve are given in the third conclusion of the theorem.
1. s is arc length
2. κs and τs are positive and analytic (have a convergent Maclaurin expansion)
1. There exists some curve defined by Rs having κs and τs as its curvature and torsion, respectively.
2. Rs has a convergent Maclaurin expansion ∑n=0∞Rn0n!sn.
The Darboux vector, given in Definition 2.7.1, is related to the angular velocity vector encountered in the study of dynamics.
Definition 2.7.1 - The Darboux Vector
d=τ T+κ B
Remarkable simplification and symmetry enters the Frenet formalism when expressed in terms of the Darboux vector, as seen in Table 2.7.3.
Table 2.7.3 Frenet formalism in terms of the Darboux vector; prime denotes differentiation with respect to arc length s
Relevant Maple Commands
Table 2.7.4 summarizes the Maple commands that are relevant to the Frenet-Serret formalism.
Computes κ, the curvature of a curve R.
Computes 1/κ, the reciprocal of the curvature of a curve R.
With "output = plot", returns a graph of R and the circle of curvature.
Computes τ, the torsion of a curve R.
Computes R′, a vector tangent to a curve R.
With the option normalized, returns T, the unit tangent vector.
With "output = plot", returns a graph of R and representative (unit) tangent vectors.
With "output = animation", returns a graph of R and a representative T traversing R.
For a curve R, computes a vector along N, the principal normal vector.
With the option normalized, returns N, the unit principal normal.
With "output = plot", returns a graph of R and representative (unit) principal-normal vectors.
With "output = animation", returns a graph of R and a representative N traversing R.
For a curve R, computes a vector along B, the binormal vector.
With the option normalized, returns B, the unit binormal.
With "output = plot", returns a graph of R and representative (unit) binormal vectors.
With "output = animation", returns a graph of R and a representative B traversing R.
Returns a sequence of T, N, and B, the (unit) tangent, principal normal, and binormal vectors for a curve R.
With "output = plot", returns a graph of R and representative triples of T, N and B vectors.
With "output = animation", returns a graph of R and a representative triple of T, N and B vectors traversing R.
Table 2.7.4 Commands relevant to the Frenet-Serret formalism
The Space Curve tutor implements the graphical aspects of the commands in Table 2.7.4. The computational aspects are captured in the Context Panel when the Student VectorCalculus package is installed.
If s is arc length, establish the Frenet equation T′s=κ N.
If s is arc length, establish the Frenet equation B′s= −τ N.
If s is arc length, establish the Frenet equation N′s= −κ T+τ B.
If s is arc length and the prime denotes dds, prove that T′·B′= −κ τ.
If s is arc length and the prime denotes dds, show that R″R‴R⁗=κ5ddsτκ.
With c=a=b=0 in the prescription in Table 2.7.2, obtain and graph the plane curve whose curvature is κs=1/1+s2.
With c=1 and a,b=3,−2 in the prescription in Table 2.7.2, obtain and graph the plane curve whose curvature is κs=1/s.
With c=a=b=0 in the prescription in Table 2.7.2, obtain and graph the plane curve whose curvature is κs=coss. Hint: The integrals defining R can only be obtained numerically.
Verify the prescription in Table 2.7.2; that is, show the resulting position vector R defines a curve for which T=R′s satisfies T=1 and T′s=κs.
Prove Theorem 2.7.1.
Obtain the expansion in the third conclusion of Theorem 2.7.2.
If C is the helix Rp=cosp i+sinp j+p k in Example 2.6.1,
Obtain its Darboux vector d.
Verify the three identities in Table 2.7.3.
Show that T′×T″=κ2d, where primes denote differentiation with respect to arc length s.
If C is the curve given by Rp=p i+3 p2 j+p3 k in Example 2.6.2,
If C is the curve given by Rp=lncosp i+lnsinp j+2p k in Example 2.6.3,
Obtain its Darboux vector d.
If C is the curve given by Rp=3 p−p3 i+3 p2 j+3 p+p3 k in Example 2.6.4,
Prove that T′×T″=κ2d.
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