Table 2.7.1   The Frenet-Serret formalism; $s$ denotes arc length and prime, $\frac{d}{\mathrm{ds}}$

Let $s$ denote arc length; and the prime, $\frac{d}{\mathrm{ds}}$. If $\mathrm{\κ}\left(s\right)$ is prescribed as the curvature for a plane curve, then a position-vector description of that curve is given by

$\mathbf{R}\=\left(a\+{\int }_c^s\cos \left(\mathrm{\θ}\left(u\right)\right) \mathit{\ⅆ}u\right) \mathbf{i}\+\left(b\+{\int }_c^s\sin \left(\mathrm{\θ}\left(u\right)\right) \mathit{\ⅆ}u\right) \mathbf{j}$ 

where $\mathrm{\θ}\left(u\right)\={\int }_c^u\mathrm{\κ}\left(p\right) \mathit{\ⅆ}p$ and $\mathbf{R}\left(c\right)$ passes through the initial point $\left(a\,b\right)$.

Table 2.7.2   Prescription for a plane curve with a given curvature

Theorem 2.7.1

If $s$ is arc length, and the curves defined by ${\mathbf{R}}_1\left(s\right)$ and ${\mathbf{R}}_2\left(s\right)$ both have the same curvature $\mathrm{\κ}\left(s\right)$ and the same torsion $\mathrm{\τ}\left(s\right)$, then the two curves are congruent.

Theorem 2.7.2

1. $s$ is arc length

2. $\mathrm{\κ}\left(s\right)$ and $\mathrm{\τ}\left(s\right)$ are positive and analytic (have a convergent Maclaurin expansion)

⇒

1. There exists some curve defined by $\mathbf{R}\left(s\right)$ having $\mathrm{\κ}\left(s\right)$ and $\mathrm{\τ}\left(s\right)$ as its curvature and torsion, respectively.

2. $\mathbf{R}\left(s\right)$ has a convergent Maclaurin expansion $\sum _{n\=0}^{\infty }\frac{{\mathbf{R}}^{\left(n\right)}\left(0\right)}{n\!}{s}^n$.

3. $\mathbf{R}\left(s\right)\=\mathbf{R}\left(0\right)\+s\mathbf{T}\left(0\right)\+\frac{s^2}{2\!}\mathrm{\κ}\left(0\right)\mathbf{N}\left(0\right)\+\frac{s^3}{3\!}\left(-{\mathrm{\κ}}^2\left(0\right)\mathbf{T}\left(0\right)\+\mathrm{\κ}\prime \left(0\right)\mathbf{N}\left(0\right)\+\mathrm{\κ}\left(0\right)\mathrm{\τ}\left(0\right)\mathbf{B}\left(0\right)\right)\+\cdots$

Definition 2.7.1 - The Darboux Vector

$\mathbf{d}\=\mathrm{\τ} \mathbf{T}\+\mathrm{\κ} \mathbf{B}$

Table 2.7.3   Frenet formalism in terms of the Darboux vector; prime denotes differentiation with respect to arc length $s$

Command

Comments

Curvature

RadiusOfCurvature

Torsion

TangentVector

PrincipalNormal

Binormal

TNBFrame

Table 2.7.4   Commands relevant to the Frenet-Serret formalism

Example 2.7.1

If $s$ is arc length, establish the Frenet equation $\mathbf{T}\prime \left(s\right)\=\mathrm{\κ} \mathbf{N}$.

Example 2.7.2

If $s$ is arc length, establish the Frenet equation $\mathbf{B}\prime \left(s\right)\= -\mathrm{\τ} \mathbf{N}$.

Example 2.7.3

If $s$ is arc length, establish the Frenet equation $\mathbf{N}\prime \left(s\right)\= -\mathrm{\κ} \mathbf{T}\+\mathrm{\τ} \mathbf{B}$.

Example 2.7.4

If $s$ is arc length and the prime denotes $\frac{d}{\mathrm{ds}}$, prove that $\mathbf{T}\prime ·\mathbf{B}\prime \= -\mathrm{\κ} \mathrm{\τ}$.

Example 2.7.5

If $s$ is arc length and the prime denotes $\frac{d}{\mathrm{ds}}$, show that $\left[\mathbf{R}″\mathbf{R}‴\mathbf{R}⁗\right]\={\mathrm{\κ}}^5\frac{d}{\mathrm{ds}}\left(\frac{\mathrm{\τ}}{\mathrm{\κ}}\right)$.

Example 2.7.6

With $c\=a\=b\=0$ in the prescription in Table 2.7.2, obtain and graph the plane curve whose curvature is $\mathrm{\κ}\left(s\right)\=1\/\left(1\+s^2\right)$.

Example 2.7.7

With $c\=1$ and $\left(a\,b\right)\=\left(3\,-2\right)$ in the prescription in Table 2.7.2, obtain and graph the plane curve whose curvature is $\mathrm{\κ}\left(s\right)\=1\/s$.

Example 2.7.8

With $c\=a\=b\=0$ in the prescription in Table 2.7.2, obtain and graph the plane curve whose curvature is $\mathrm{\κ}\left(s\right)\=\cos \left(s\right)$. Hint: The integrals defining R can only be obtained numerically.

Example 2.7.9

Verify the prescription in Table 2.7.2; that is, show the resulting position vector R defines a curve for which $\mathbf{T}\=\mathbf{R}\prime \left(s\right)$ satisfies $∥\mathbf{T}∥\=1$ and $∥\mathbf{T}\prime \left(s\right)∥\=\mathrm{\κ}\left(s\right)$.

Example 2.7.10

Prove Theorem 2.7.1.

Example 2.7.11

Obtain the expansion in the third conclusion of Theorem 2.7.2.

Example 2.7.12

If $C$ is the helix $\mathbf{R}\left(p\right)\=\cos \left(p\right) \mathbf{i}\+\sin \left(p\right) \mathbf{j}\+p \mathbf{k}$ in Example 2.6.1,

Example 2.7.13

If $C$ is the curve given by $\mathbf{R}\left(p\right)\=p \mathbf{i}\+3 p^2 \mathbf{j}\+p^3 \mathbf{k}$ in Example 2.6.2,

Example 2.7.14

If $C$ is the curve given by $\mathbf{R}\left(p\right)\=\ln \left(\cos \left(p\right)\right) \mathbf{i}\+\ln \left(\sin \left(p\right)\right) \mathbf{j}\+\sqrt{2}p \mathbf{k}$ in Example 2.6.3,

Example 2.7.15

If $C$ is the curve given by $\mathbf{R}\left(p\right)\=\left(3 p-p^3\right) \mathbf{i}\+3 p^2 \mathbf{j}\+\left(3 p\+p^3\right) \mathbf{k}$ in Example 2.6.4,

Example 2.7.16

Prove that $\mathbf{T}\prime \times \mathbf{T}″\={\mathrm{\κ}}^2\mathbf{d}$.