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Reparameterization and the Tools of Frenet-Serret 📂Geometry

Reparameterization and the Tools of Frenet-Serret

Definition

Let’s call β:[a,b]R3\beta : [a,b] \to \mathbb{R}^{3} a regular curve. The arc length reparametrization t=t(s)t = t(s) satisfies s(t)=atβ(t)dts(t) = \int_{a}^{t} \left| \beta^{\prime}(t) \right| dt, and the Frenet-Serre apparatus of the unit speed curve α(s):=β(t(s))\alpha (s) := \beta \left( t (s) \right) {κα(s(t)),τα(s(t)),Tα(s(t)),Nα(s(t)),Bα(s(t))} \left\{ \kappa_{\alpha} \left( s(t) \right), \tau_{\alpha} \left( s(t) \right) , T_{\alpha} \left( s(t) \right) , N_{\alpha} \left( s(t) \right), B_{\alpha} \left( s(t) \right) \right\} is defined as the Frenet-Serre apparatus of β\beta.

Explanation

The definition of the Frenet-Serre apparatus was generalized for regular curves. This is a common method used throughout mathematics, where the unit speed is enforced if not already present. For a bijective arc length reparametrization, one can consider α\alpha, which is not β\beta itself but incorporates its geometry.

Notation

dfds=fanddfdt=f˙ {{ df } \over { ds }} = f^{\prime} \quad \text{and} \quad {{ df } \over { dt }} = \dot{f}

Dot ˙\dot{} and prime ' both indicate differentiation, but in the context of differential geometry, symbols are distinguished as above. Usually, ss represents the parameter of a unit speed curve, and t=t(s)t = t(s) is the parameter of the curve after arc length reparametrization. Additionally, for representing the scalar triple product, the following ternary symbols are used for cross product ×\times and dot product <,>\left< \cdot, \cdot \right>. [u,v,w]:=<u×v,w> \left[ \mathbf{u} , \mathbf{v} , \mathbf{w} \right] := \left< \mathbf{u} \times \mathbf{v} , \mathbf{w} \right>

Theorem 1

If β(t)\beta (t) is a regular curve

  • (a): T=β˙β˙\displaystyle T = {{ \dot{\beta} } \over { \left| \dot{\beta} \right| }}
  • (b): B=β˙×β¨β˙×β¨\displaystyle B = {{ \dot{\beta} \times \ddot{\beta} } \over { \left| \dot{\beta} \times \ddot{\beta} \right| }}
  • (c): N=B×TN = B \times T
  • (d): κ=β˙×β¨β˙3\displaystyle \kappa = {{ \left| \dot{\beta} \times \ddot{\beta} \right| } \over { \left| \dot{\beta} \right|^{3} }}
  • (e): τ=[β˙,β¨,β]β˙×β¨2\displaystyle \tau = {{ \left[ \dot{\beta} , \ddot{\beta}, \overset{\cdot\cdot\cdot}{\beta} \right] } \over { \left| \dot{\beta} \times \ddot{\beta} \right|^{2} }}

Proof

Frenet-Serre Formulas: If α\alpha is a unit speed curve such that T(s)=κ(s)N(s)N(s)=κ(s)T(s)+τ(s)B(s)B(s)=τ(s)N(s) \begin{align*} T^{\prime}(s) =& \kappa (s) N(s) \\ N^{\prime}(s) =& - \kappa (s) T(s) + \tau (s) B(s) \\ B^{\prime}(s) =& - \tau (s) N(s) \end{align*}

Following the Notation introduced, we can easily deduce: dsdt=s˙=β˙=ddtatβ(t)dt {{ ds } \over { dt }} = \dot{s} = \left| \dot{\beta} \right| = {{ d } \over { dt }} \int_{a}^{t} \left| \beta^{\prime}(t) \right| dt Especially, T˙\dot{T} is according to the Frenet-Serre formula T˙=dTdt=dTdsdsst=κNs˙ \dot{T} = {{ dT } \over { dt }} = {{ dT } \over { ds }} {{ ds } \over { st }} = \kappa N \dot{s}

(a)

β˙=αs˙=Ts˙=Tβ˙ \begin{align*} \dot{\beta} =& \alpha^{\prime} \dot{s} \\ =& T \dot{s} \\ =& T \left| \dot{\beta} \right| \end{align*} Going beyond β˙\left| \dot{\beta} \right| T=β˙β˙ T = {{ \dot{\beta} } \over { \left| \dot{\beta} \right| }}

(d)

For convenience, we first prove (d)(d). β˙=s˙T \dot{\beta} = \dot{s} T Differentiating both sides by tt gives β¨=s¨T+s˙T˙=s¨T+s˙2T˙=s¨T+κs˙2N \begin{align*} \ddot{\beta} =& \ddot{s} T + \dot{s} \dot{T} \\ =& \ddot{s} T + \dot{s}^{2} \dot{T}^{\prime} \\ =& \ddot{s} T + \kappa \dot{s}^{2} N \end{align*} Thus β˙×β¨=s˙T×(s¨T+κs˙2N)=κs˙3B=κs˙3 \begin{align*} \left| \dot{\beta} \times \ddot{\beta} \right| =& \left| \dot{s} T \times \left( \ddot{s} T + \kappa \dot{s}^{2} N \right) \right| \\ =& \left| \kappa \dot{s}^{3} B \right| \\ =& \kappa \dot{s}^{3} \end{align*} As s˙=β˙\dot{s} = \left| \dot{\beta} \right|, then κ=β˙×β¨β˙3 \kappa = {{ \left| \dot{\beta} \times \ddot{\beta} \right| } \over { \left| \dot{\beta} \right|^{3} }}

(b)

From the proof of (d), if κ0\kappa \ne 0 then B=β˙×β¨κs˙3=β˙×β¨β˙×β¨ \begin{align*} B =& {{ \dot{\beta} \times \ddot{\beta} } \over { \kappa \dot{s}^{3} }} \\ =& {{ \dot{\beta} \times \ddot{\beta} } \over { \left| \dot{\beta} \times \ddot{\beta} \right| }} \end{align*}

(c)

It’s obvious.

(e)

Differentiating β¨=s¨T+s˙T˙\ddot{\beta} = \ddot{s} T + \dot{s} \dot{T} once more with tt gives β=sT+s¨T˙+(κs˙2)N+κs˙2N˙=sT+s˙s¨T+(κs˙2)N+κs˙3N=sT+κs˙s¨N+(κs˙2)Nκ2s˙3T+κτs˙3B=(sκ2s˙3)T+(κs˙s¨+(κs˙2))N+κτs˙3B \begin{align*} \overset{\cdot\cdot\cdot}{\beta} =& \overset{\cdot\cdot\cdot}{s} T + \ddot{s} \dot{T}+\left(\kappa \dot{s}^{2}\right)^{\cdot} N+\kappa \dot{s}^{2} \dot{N} \\ =& \overset{\cdot\cdot\cdot}{s} T+\dot{s} \ddot{s} T^{\prime} + \left(\kappa \dot{s}^{2}\right)^{\cdot} N + \kappa \dot{s}^{3} N^{\prime} \\ =& \overset{\cdot\cdot\cdot}{s} T+\kappa \dot{s} \ddot{s} N+\left(\kappa \dot{s}^{2}\right)^{\cdot} N-\kappa^{2} \dot{s}^{3} T+\kappa \tau \dot{s}^{3} \mathbf{B} \\ =& \left(\overset{\cdot\cdot\cdot}{s}-\kappa^{2} \dot{s}^{3}\right) T+\left(\kappa \dot{s} \ddot{s}+\left(\kappa \dot{s}^{2}\right)^{\cdot}\right) N+\kappa \tau \dot{s}^{3} \mathbf{B} \end{align*} As the scalar triple product is [β˙,β¨,β]=<β˙×β¨,β>\left[ \dot{\beta} , \ddot{\beta}, \overset{\cdot\cdot\cdot}{\beta} \right] = \left< \dot{\beta} \times \ddot{\beta} , \overset{\cdot\cdot\cdot}{\beta} \right> and with TBT \perp B, NBN \perp B, one finds [β˙,β¨,β]=<β˙×β¨,β>=<κs˙3B,β>=τ(κs˙3)2=τβ˙×β¨2 \begin{align*} \left[ \dot{\beta} , \ddot{\beta}, \overset{\cdot\cdot\cdot}{\beta} \right] =& \left< \dot{\beta} \times \ddot{\beta} , \overset{\cdot\cdot\cdot}{\beta} \right> \\ =& \left< \kappa \dot{s}^{3} B , \overset{\cdot\cdot\cdot}{\beta} \right> \\ =& \tau \left( \kappa \dot{s}^{3} \right)^{2} \\ =& \tau \left| \dot{\beta} \times \ddot{\beta} \right|^{2} \end{align*} In conclusion τ=[β˙,β¨,β]β˙×β¨2 \tau = {{ \left[ \dot{\beta} , \ddot{\beta}, \overset{\cdot\cdot\cdot}{\beta} \right] } \over { \left| \dot{\beta} \times \ddot{\beta} \right|^{2} }}


  1. Millman. (1977). Elements of Differential Geometry: p46~47. ↩︎