Arclength, Curvature, and Torsion

For a smooth curve \(C\) with parameterization \(\bm{r}\) the arclength of \(C\) between \(t=a\) and \(t=b\) is calculated by the integral \({\int_a^b |\bm{r}'(t)| \,\mathrm{d}t\,.}\) For a curve \(C\) with parameterization \(\bm{r},\) its arclength parameterization is \(\bm{r} \circ s^{-1},\) where \(s\) is its arclength function defined as \(s(\ell) = \int_0^{\ell} |\bm{r}'(t)| \,\mathrm{d}t.\) The arclength parameterization has the property that the point corresponding to a specific value of \(t\) lies at a length of \(t\) along the curve from the “anchor point” where \(t = 0.\) The curvature \(\kappa\) at a point along a smooth curve is a measure of how eccentrically it’s bending/turning at a point — it’s a measure how much the curve fails to lie on a line, its tangent line. Orienting fact: the curvature of a circle of radius \(R\) is \(\frac{1}{R}.\) The osculating circle to a curve at a point is its circular approximation at that point, just like the tangent line to a curve at a point is its linear approximation. The torsion at a point along a smooth curve is a measure of how eccentrically it’s twisting in addition to turning — it’s a measure how much the curve fails to lie in a plane, its osculating plane at that point. Orienting fact: the torsion of a helix of radius \(R\) is \(\frac{1}{2R}.\) For a parameterization \(\bm{r},\) the curvature and torsion at position \(\bm{r}(t)\) are calculated by the formulas:

\(\displaystyle \kappa \!=\! \bigg|\frac{\mathrm{d}{\mathbf{T}}}{\mathrm{d}s}\bigg| \;\;\implies\;\; \kappa(t) % \!=\! \bigg|\frac{\mathrm{d}{\mathbf{T}}/\mathrm{d}t}{\mathrm{d}s/\mathrm{d}t}\bigg| \!=\! \bigg|\frac{\mathbf{T}'(t)}{\bm{r}'(t)}\bigg| \!=\! \frac{\big|\bm{r}'(t) \times \bm{r}''(t)\big|}{\big|\bm{r}'(t)\big|^3} \)
\(\displaystyle \tau \!=\! -\frac{\mathrm{d}\mathbf{B}}{\mathrm{d}s} \cdot \mathbf{N} \;\;\implies\;\; \tau(t) \!=\! - \frac{\mathbf{B}'(t) \cdot \mathbf{N}(t)}{\big|\bm{r}'(t)\big|} \!=\! \frac{\big(\bm{r}'(t) \times \bm{r}''(t)\big) \cdot \bm{r}'''(t)}{\big|\bm{r}'(t) \times\bm{r}''(t) \big|^2} \)

The Frenet-Serret formulas describe the kinematic properties of an object moving along a path/curve in space, irrespective of any absolute coordinate system. I.e. the motion of an object moving smoothly in space is completely determined by its curvature and torsion at a moment in time.

\(\displaystyle \frac{\mathrm{d}\mathbf{T}}{\mathrm{d}s} = \kappa \mathbf{N} \quad \frac{\mathrm{d}\mathbf{N}}{\mathrm{d}s} = -\kappa \mathbf{T} + \tau \mathbf{B} \quad \frac{\mathrm{d}\mathbf{B}}{\mathrm{d}s} = -\tau \mathbf{N} \qquad\implies\qquad \begin{pmatrix} \mathbf{T}' \\ \mathbf{N}' \\ \mathbf{B}' \end{pmatrix} = \begin{pmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{pmatrix} \begin{pmatrix} \mathbf{T} \\ \mathbf{N} \\ \mathbf{B} \end{pmatrix} \)